A294936 -id:A294936

A294934

Characteristic function for deficient numbers (A005100): a(n) = 1 if A001065(n) < n, 0 otherwise.

+10
17

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions.

Index entries for sequences related to sums of divisors.

FORMULA

a(n) = 1 if A033879(n) > 0, 0 otherwise.

a(n) = 1 - A294936(n).

a(n) = 1 - sign(floor(sigma(n)/(2*n))), where sigma is the sum of the divisors of n (A000203). - Wesley Ivan Hurt, Oct 01 2020

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A318172. - Amiram Eldar, Jul 25 2022

MATHEMATICA

Table[1 - Sign[Floor[DivisorSigma[1, n]/(2 n)]], {n, 100}] (* Wesley Ivan Hurt, Oct 02 2020 *)

CROSSREFS

Cf. A001065, A033879, A294935, A294936, A294937, A318172.

Cf. A005100 (positions of 1's), A023196 (of 0's).

Cf. A000203 (sigma).

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Nov 12 2017

STATUS

approved

A294937

Characteristic function for abundant numbers (A005101): a(n) = 1 if A001065(n) > n, 0 otherwise.

+10
17

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for characteristic functions.

Index entries for sequences related to sums of divisors.

FORMULA

a(n) = 1 if A033880(n) > 0, 0 otherwise.

a(n) = 1 - A294935(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A302991. - Amiram Eldar, Jul 25 2022

MATHEMATICA

a[n_] := If[DivisorSigma[1, n] > 2*n, 1, 0]; Array[a, 100] (* Amiram Eldar, Jul 25 2022 *)

PROG

(PARI) a(n) = sigma(n) > 2*n; \\ Michel Marcus, Jul 25 2022

CROSSREFS

Cf. A001065, A033880, A294934, A294935, A294936, A302991.

Cf. A005101 (positions of ones), A263837 (of zeros).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 12 2017

STATUS

approved

A336835

Number of iterations of x -> A003961(x) needed before the result is deficient (sigma(x) < 2x), when starting from x=n.

+10
15

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,120

COMMENTS

It holds that a(n) <= A336836(n) for all n, because sigma(n) <= A003961(n) for all n (see A286385 for a proof).

The first 3 occurs at n = 19399380, the first 4 at n = 195534950863140268380. See A336389.

If x and y are relatively prime (i.e., gcd(x,y) = 1), then a(x*y) >= max(a(x),a(y)). Compare to a similar comment in A336915.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

FORMULA

If A294934(n) = 1, a(n) = 0, otherwise a(n) = 1 + a(A003961(n)).

From Antti Karttunen, Aug 21-Sep 01 2020: (Start)

For all n >= 1,

a(A046523(n)) >= a(n).

a(A071364(n)) >= a(n).

a(A108951(n)) = A337474(n).

a(A025487(n)) = A337475(n).

(End)

EXAMPLE

For n = 120, sigma(120) = 360 >= 2*120, thus 120 is not deficient, and we get the next number by applying the prime shift, A003961(120) = 945, and sigma(945) = 1920 >= 945*2, so neither 945 is deficient, so we prime shift once again, and A003961(945) = 9625, which is deficient, as sigma(9625) = 14976 < 2*9625. Thus after two iteration steps we encounter a deficient number, and therefore a(120) = 2.

MATHEMATICA

Array[-1 + Length@ NestWhileList[If[# == 1, 1, Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]] &, #, DivisorSigma[1, #] >= 2 # &] &, 120] (* Michael De Vlieger, Aug 27 2020 *)

PROG

(PARI)

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961

A336835(n) = { my(i=0); while(sigma(n) >= (n+n), i++; n = A003961(n)); (i); };

CROSSREFS

Cf. A003961, A005100, A005940, A023196, A046523, A047802, A071364, A108951, A286385, A294934, A336834, A337474, A337475.

Cf. A336389 (position of the first occurrence of a term >= n).

Differs from A294936 for the first time at n=120.

Cf. also A246271, A252459, A336836 and A336915 for similar iterations.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 07 2020

STATUS

approved

A294935

Characteristic function for nonabundant numbers (A263837): a(n) = 1 if A001065(n) <= n, 0 otherwise.

+10
10

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions.

Index entries for sequences related to sums of divisors.

FORMULA

a(n) = 1 if A033879(n) >= 0, 0 otherwise.

a(n) = 1 - A294937(n).

MATHEMATICA

a[n_] := Boole[DivisorSigma[1, n] <= 2*n]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

CROSSREFS

Cf. A001065, A033879, A294934, A294936, A294937.

Cf. A263837 (positions of ones), A005101 (of zeros).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 12 2017

STATUS

approved

A294927

Number of proper divisors of n that are nondeficient (A023196).

+10
9

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 1, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,24

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

FORMULA

a(n) = Sum_{d|n, d<n} A294936(d).

a(n) + A294926(n) = A032741(n).

MATHEMATICA

a[n_] := DivisorSum[n, 1 &, # < n && DivisorSigma[1, #] >= 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

PROG

(PARI) A294927(n) = sumdiv(n, d, (d<n)*(sigma(d)>=(2*d)));

CROSSREFS

Cf. A023196, A032741, A071395, A294887, A294926, A294928, A294929, A294936.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 14 2017

STATUS

approved

A341620

Number of nondeficient divisors of n.

+10
8

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 3, 0, 2, 0, 0, 0, 5, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 6, 0, 0, 0, 0, 0, 2, 0, 3, 0, 0, 0, 5, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 6, 0, 0, 0, 3, 0, 2, 0, 0, 0, 0, 0, 8

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

COMMENTS

Number of nondeficient numbers (A023196) dividing n.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for sequences related to sigma(n)

FORMULA

a(n) = Sum_{d|n} A294936(d).

a(n) = A294927(n) + A294936(n).

a(n) = A080224(n) + A080225(n) = A000005(n) - A080226(n).

a(n) >= A337690(n) for all n.

a(n) = 1 iff A341619(n) = 1.

MATHEMATICA

a[n_] := DivisorSum[n, 1 &, DivisorSigma[1, #] >= 2*# &]; Array[a, 120] (* Amiram Eldar, Feb 22 2021 *)

PROG

(PARI)

A294936(n) = (sigma(n, -1)>=2); \\ From A294936.

A341620(n) = sumdiv(n, d, A294936(d));

(PARI) A341620(n) = sumdiv(n, d, (sigma(d)>=(2*d)));

CROSSREFS

Cf. A000005, A006039 (positions of ones), A023196, A080224, A080225, A080226, A294927, A294936, A337690, A341619.

Differs from a derived sequence A341624 for the first time at n=120, where a(120)=8, while A341624(120)=1.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 21 2021

STATUS

approved

A294887

Sum of nondeficient proper divisors of n.

+10
7

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 18, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 36, 0, 0, 0, 20, 0, 6, 0, 0, 0, 0, 0, 42, 0, 0, 0, 0, 0, 24, 0, 28, 0, 0, 0, 68, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 96, 0, 0, 0, 0, 0, 6, 0, 60, 0, 0, 0, 88, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 90, 0, 0, 0, 20, 0, 6, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

COMMENTS

Sum of divisors n smaller than n that are nondeficient numbers (in A023196).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Index entries for sequences related to sums of divisors.

FORMULA

a(n) = Sum_{d|n, d<n} A294936(d)*d. - Typo in A-number corrected by Antti Karttunen, Apr 04 2022

a(n) + A294886(n) = A001065(n).

EXAMPLE

Proper divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, and 45. Of these 6, 18 and 30 are in A023196, thus a(90) = 6+18+30 = 54.

MATHEMATICA

a[n_] := DivisorSum[n, # &, # < n && DivisorSigma[1, #] >= 2*# &]; Array[a, 100] (* Amiram Eldar, Mar 14 2024 *)

PROG

(PARI) A294887(n) = sumdiv(n, d, (d<n)*(sigma(d)>=(2*d))*d);

CROSSREFS

Cf. A001065, A023196, A294927, A294936, A294886, A294888, A294889.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 14 2017

STATUS

approved

A368991

Doudna-gram for nondeficient numbers; Partial sums of A368990.

+10
3

0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 25, 25, 25, 25, 26, 26, 27, 27, 28

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,12

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..65537

Index entries for sequences related to sigma(n)

FORMULA

a(0) = 0, and for n > 0, a(n) = a(n-1) + A368990(n).

PROG

(PARI)

up_to = 65537;

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); (t); };

A294936(n) = (sigma(n)>=(2*n));

A368990(n) = A294936(A005940(1+n));

A368991list(up_to) = { my(v=vector(up_to), s=A368990(0)); for(i=1, up_to, s += A368990(i); v[i] = s); (v); };

v368991 = A368991list(up_to);

A368991(n) = if(!n, A368990(0), v368991[n]);

CROSSREFS

Partial sums of A368990.

Cf. A005940, A023196, A294936, A324055.

Cf. also A368910, A368993.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 11 2024

STATUS

approved

A060862

a(n) = 0 if n is deficient, 1 if n is abundant, 2 if n is perfect.

+10
2

0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

EXAMPLE

a(6) = 2 because 6 = 1+2+3 and is perfect.

a(10) = 0 because 10 > 1+2+5 and is deficient.

a(12) = 1 because 12 < 1+2+3+4+6 and is abundant.

MATHEMATICA

Table[Sign[DivisorSigma[1, n] - 2*n], {n, 1, 100}] /. {-1 -> 0, 0 -> 2, 1 -> 1} (* Amiram Eldar, Mar 15 2024 *)

PROG

(PARI) A060862(n) = { my(def=((2*n)-sigma(n))); if(def>0, 0, if(def<0, 1, 2)); }; \\ Antti Karttunen, Sep 27 2018

CROSSREFS

Cf. A000396, A005100, A005101, A082551, A294934, A294935, A294936, A294937.

KEYWORD

easy,nonn

AUTHOR

Jason Earls, May 04 2001

EXTENSIONS

More terms from Erich Friedman, Jun 01 2001

STATUS

approved

A082551

Denote sigma(n)-n by s(n); a(n)=1 if s(n)>n, a(n)=0 if s(n)=n, a(n)=-1 if s(n)<n.

+10
2

-1, -1, -1, -1, -1, 0, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, 1, -1, 1, -1, -1, -1, 1, -1, -1, -1, 0, -1, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, -1, -1, 1, -1, 1, -1, -1, -1, 1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65537

FORMULA

a(n) = sign(A033880(n)).

MAPLE

A082551 := proc(n)

signum( numtheory[sigma](n)-2*n) ;

end proc: # R. J. Mathar, Sep 28 2011

MATHEMATICA

Table[Sign[DivisorSigma[1, n] - 2*n], {n, 1, 100}] (* Amiram Eldar, Mar 15 2024 *)

PROG

(PARI) A082551(n) = sign(sigma(n)-(2*n)); \\ Antti Karttunen, Sep 28 2018

CROSSREFS

Cf. A000203, A001065, A033879, A033880, A060862, A294934, A294935, A294936, A294937.

KEYWORD

easy,sign

AUTHOR

Hanoch M. bin (hanochb(AT)shiron.com), May 12 2003

STATUS

approved