A001065 -id:A001065

A212373

Numbers n such that A001065(x)*x = n has at least two solutions, where x is an abundant number.

+20
3

141440, 2286080, 6680960, 1119849500, 2081125376, 3991520000, 4141021500, 9644638208, 23664804800, 32662630400, 37516855536, 67994319888, 577461690368, 618169892864, 627198842816, 4132702579824, 4949713492400, 5025386326400, 5579119296000, 9013476012156

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

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..20.

EXAMPLE

Example: For n=141440, A001065(340)*340 = 141440, A001065(320)*320 = 141440, A001065(340) > 340, A001065(320) > 320, therefore 141440 is included in this sequence.

CROSSREFS

Cf. A212327, A005101, A212489, A212490.

KEYWORD

nonn

AUTHOR

Naohiro Nomoto, May 18 2012

EXTENSIONS

a(9)-a(20) from Donovan Johnson, May 21 2012

STATUS

approved

A212489

Numbers n such that A001065(x)*x = n has at least two solutions, where x is a deficient number.

+20
3

1245335, 318047135, 358491735, 533718135, 709131500, 1122571695, 1814416175, 3565970135, 4486354631, 14336906175, 14827262351, 22805269551, 36360557831, 43971297884, 72370166375, 99254203895, 102204949847, 145262865020, 156161459559, 173741271935, 231267964895

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

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..21.

EXAMPLE

Example: For n=1245335, A001065(1955)*1955 = 1245335, A001065(2093)*2093 = 1245335, A001065(1955) < 1955, A001065(2093) < 2093, therefore 1245335 is included in this sequence.

CROSSREFS

Cf. A005100, A212490, A212373, A212327.

KEYWORD

nonn

AUTHOR

Naohiro Nomoto, May 18 2012

EXTENSIONS

a(6)-a(21) from Donovan Johnson, May 21 2012

STATUS

approved

A291765

Compound filter (sum of proper divisors & prime signature): a(n) = P(A001065(n), A046523(n)), where P(n,k) is sequence A000027 used as a pairing function.

+20
3

0, 2, 2, 18, 2, 61, 2, 98, 25, 86, 2, 367, 2, 115, 100, 450, 2, 517, 2, 550, 131, 185, 2, 1747, 42, 226, 203, 769, 2, 2527, 2, 1922, 205, 320, 166, 4060, 2, 373, 248, 2678, 2, 3457, 2, 1315, 979, 491, 2, 7579, 63, 1474, 346, 1642, 2, 3982, 248, 3805, 401, 698, 2, 13969, 2, 775, 1367, 7938, 295, 5749, 2, 2404, 523, 5327, 2, 18844, 2, 1030, 1819, 2839, 295

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

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16385

FORMULA

a(n) = (1/2)*(2 + ((A001065(n)+A046523(n))^2) - A001065(n) - 3*A046523(n)).

PROG

(PARI)

A001065(n) = (sigma(n)-n);

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ This function from Charles R Greathouse IV, Aug 17 2011

A291765(n) = (1/2)*(2 + ((A001065(n)+A046523(n))^2) - A001065(n) - 3*A046523(n));

CROSSREFS

Cf. A000027, A000203, A001065, A046523, A286360, A286592.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Sep 10 2017

STATUS

approved

A318503

Xor-Moebius transform of A001065, the sum of proper divisors.

+20
3

0, 1, 1, 2, 1, 6, 1, 4, 5, 8, 1, 20, 1, 10, 9, 8, 1, 22, 1, 28, 11, 14, 1, 48, 7, 16, 9, 20, 1, 44, 1, 16, 15, 20, 13, 52, 1, 22, 17, 32, 1, 48, 1, 36, 45, 26, 1, 96, 9, 36, 21, 60, 1, 94, 17, 88, 23, 32, 1, 76, 1, 34, 39, 32, 19, 72, 1, 44, 27, 68, 1, 120, 1, 40, 63, 84, 19, 92, 1, 80, 37, 44, 1, 184, 23, 46, 33, 112, 1, 132

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

OFFSET

1,4

COMMENTS

Unique sequence satisfying SumXOR_{d divides n} a(d) = sigma(n)-n for all n > 0, where SumXOR is the analog of summation under the binary XOR operation. See A295901 for a list of some of the properties of Xor-Moebius transform.

LINKS

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

FORMULA

a(n) = A318501(n) XOR A318502(n).

PROG

(PARI) A318503(n) = { my(v=0); fordiv(n, d, if(issquarefree(n/d), v=bitxor(v, sigma(d)-d))); (v); } \\ after code in A295901.

CROSSREFS

Cf. A001065, A051953, A256739, A295901, A296203, A318501, A318502.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Aug 28 2018

STATUS

approved

A324535

An analog of sigma(n)-n (A001065) for nonstandard factorization based on the sieve of Eratosthenes (A083221).

+20
3

0, 1, 1, 3, 1, 6, 1, 7, 4, 8, 1, 16, 1, 10, 9, 15, 1, 21, 1, 22, 13, 14, 1, 36, 6, 16, 11, 28, 1, 42, 1, 31, 33, 20, 13, 55, 1, 22, 15, 50, 1, 66, 1, 40, 40, 26, 1, 76, 8, 43, 49, 46, 1, 54, 31, 64, 41, 32, 1, 108, 1, 34, 17, 63, 17, 144, 1, 58, 105, 74, 1, 123, 1, 40, 21, 64, 19, 78, 1, 106, 57, 44, 1, 172, 73, 46, 87, 92, 1, 201, 57, 76, 121

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

OFFSET

1,4

LINKS

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

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Index entries for sequences generated by sieves

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A001065(A250246(n)) = A324545(n) - A250246(n).

a(n) = A250246(n) - A324546(n).

PROG

(PARI)

up_to = 65537;

ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om, invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om, invec[i], (1+pt))); outvec; };

A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639

A055396(n) = if(1==n, 0, primepi(A020639(n)));

v078898 = ordinal_transform(vector(up_to, n, A020639(n)));

A078898(n) = v078898[n];

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

A250246(n) = if(1==n, n, my(k = 2*A250246(A078898(n)), r = A055396(n)); if(1==r, k, while(r>1, k = A003961(k); r--); (k)));

A001065(n) = (sigma(n)-n);

A324535(n) = A001065(A250246(n));

CROSSREFS

Cf. A001065, A250246, A324545, A324546.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 08 2019

STATUS

approved

A325377

a(n) = A001065(A228058(n)), where A001065(n) gives the sum of proper divisors of n.

+20
3

33, 65, 81, 97, 129, 109, 161, 177, 321, 133, 225, 257, 193, 161, 305, 205, 369, 193, 253, 401, 417, 253, 449, 465, 277, 641, 561, 349, 609, 801, 641, 289, 397, 397, 705, 289, 737, 785, 801, 481, 353, 469, 385, 337, 929, 945, 977, 2241, 565, 1041, 1281, 1089, 613, 1121, 637, 1137, 481, 1185, 673, 685, 1265, 709, 1281, 421, 2717, 545, 1601

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

OFFSET

1,1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..25000

FORMULA

a(n) = A001065(A228058(n)).

a(n) > A325320(n) for all n.

PROG

(PARI)

up_to = 25000;

isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));

A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(k<up_to, n++; if(isA228058(n), k++; v[k] = n)); (v); };

v228058 = A228058list(up_to);

A228058(n) = v228058[n];

A001065(n) = (sigma(n)-n);

A325377(n) = A001065(A228058(n));

CROSSREFS

Cf. A001065, A228058, A325320, A325380.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 22 2019

STATUS

approved

A325380

Numbers k in A228058 such that also A001065(k) is in A228058.

+20
3

801, 1377, 1773, 2525, 3725, 4689, 4753, 6309, 6425, 7209, 7677, 8577, 8957, 9477, 11133, 11225, 11493, 11925, 12393, 12429, 12789, 13077, 15381, 15777, 18873, 19269, 19845, 20025, 20629, 21213, 24201, 26073, 26721, 28037, 28989, 29277, 29961, 30037, 30213, 31925, 32553, 33273, 34425, 34677, 36369, 36441, 38725, 39249, 40329

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

OFFSET

1,1

COMMENTS

If any odd perfect number exists, then it must occur in this sequence.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences where any odd perfect numbers must occur

PROG

(PARI)

A001065(n) = (sigma(n)-n);

isA228058(n) = if(!(n%2)||(omega(n)<2), 0, my(f=factor(n), y=0); for(i=1, #f~, if(1==(f[i, 2]%4), if((1==y)||(1!=(f[i, 1]%4)), return(0), y=1), if(f[i, 2]%2, return(0)))); (y));

k=0; n=0; while(k<100, n++; if(isA228058(n)&&isA228058(A001065(n)), k++; print1(n, ", ")));

CROSSREFS

Cf. A001065, A228058, A325377.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 22 2019

STATUS

approved

A326063

Composite numbers n such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

+20
3

4, 6, 9, 25, 28, 49, 117, 121, 169, 289, 361, 496, 529, 775, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 8128, 9409, 10201, 10309, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729, 51529, 52441

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

OFFSET

1,1

COMMENTS

Composite numbers n such that A318505(n) [sum of divisors of n excluding n itself and the second largest of them, A032742(n)] divides A060681(n) [the largest difference between consecutive divisors of n, = n - A032742(n)].

Numbers k such that A326062(k) = A318505(k).

Question: Is it possible that this sequence could contain a term with more than one non-unitary prime factor? If not, then there are no odd perfect numbers. (See e.g., A326137).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..1001

Index entries for sequences where any odd perfect numbers must occur

Index entries for sequences related to sigma(n)

EXAMPLE

For n = 9 = 3*3, its divisors are [1, 3, 9], thus A318505(9) = 1 and A060681(9) = 9-3 = 6, and 1 divides 6, so 9 is included, like all squares of primes.

For n = 117 = 3^2 * 13,its divisors are [1, 3, 9, 13, 39, 117], thus A318505(117) = 1+3+9+13 = 26 and A060681(117) = (117-39) = 78, which is a multiple of 26, thus 117 is included in the sequence.

PROG

(PARI)

A032742(n) = if(1==n, n, n/vecmin(factor(n)[, 1]));

isA326063(n) = (gcd((sigma(n)-A032742(n))-n, n-A032742(n)) == (sigma(n)-A032742(n))-n);

(PARI)

A060681(n) = (n-A032742(n));

A318505(n) = if(1==n, 0, (sigma(n)-A032742(n))-n);

isA326063(n) = { my(t=A318505(n)); (t && !(A060681(n)%t)); };

CROSSREFS

Cf. A000203, A001065, A032742, A060681, A318505, A325981, A326062, A326137.

Subsequences: A000396, A001248, A326064 (odd terms that are not squares of primes).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jun 06 2019

STATUS

approved

A326188

a(n) = A001065(n) - A003557(n), where A001065(n) = the sum of proper divisors of n, and A003557(n) = n divided by its largest squarefree divisor.

+20
3

-1, 0, 0, 1, 0, 5, 0, 3, 1, 7, 0, 14, 0, 9, 8, 7, 0, 18, 0, 20, 10, 13, 0, 32, 1, 15, 4, 26, 0, 41, 0, 15, 14, 19, 12, 49, 0, 21, 16, 46, 0, 53, 0, 38, 30, 25, 0, 68, 1, 38, 20, 44, 0, 57, 16, 60, 22, 31, 0, 106, 0, 33, 38, 31, 18, 77, 0, 56, 26, 73, 0, 111, 0, 39, 44, 62, 18, 89, 0, 98, 13, 43, 0, 138, 22, 45, 32, 88, 0, 141, 20

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

OFFSET

1,6

LINKS

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

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

FORMULA

a(n) = A326187(n) - n = A000203(n) - A003557(n) - n.

a(n) = A001065(n) - A003557(n).

PROG

(PARI)

A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = f[i, 2]-1); factorback(f); }; \\ From A003557

A326188(n) = ((sigma(n)-A003557(n))-n);

CROSSREFS

Cf. A000203, A001065, A003557, A010848, A326187, A326189.

Cf. also A326185.

KEYWORD

sign

AUTHOR

Antti Karttunen, Jul 11 2019

STATUS

approved

A336552

Numbers k such that A003557(k)-1 either divides A326143(k) [= A001065(k) - A007947(k)], or both are zero. Numbers k such that gcd(A336551(k), A326143(k)) is equal to A336551(k).

+20
3

4, 6, 12, 20, 24, 28, 44, 45, 48, 52, 60, 63, 68, 76, 84, 90, 92, 96, 99, 116, 117, 120, 124, 126, 132, 140, 144, 147, 148, 150, 153, 156, 164, 168, 171, 172, 188, 192, 198, 204, 207, 212, 220, 228, 234, 236, 244, 260, 261, 264, 268, 272, 276, 279, 284, 292, 294, 306, 308, 312, 315, 316, 325, 332, 333, 340, 342, 348, 350

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

OFFSET

1,1

COMMENTS

Numbers k such that either A336551(k) and A326143(k) are both zero (in which case k is squarefree), or A336551(k) divides A326143(k) (in which case k is not squarefree).

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

PROG

(PARI)

A007947(n) = factorback(factorint(n)[, 1]);

A326143(n) = (sigma(n)-A007947(n)-n);

A336551(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 2] = f[i, 2]-1); (factorback(f)-1); };

isA336552(n) = { my(u=A336551(n)); (u==gcd(u, A326143(n))); };

CROSSREFS

Cf. A007947, A326143, A336550, A336551, A336553 (odd terms).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jul 28 2020

STATUS

approved