A116550 -id:A116550

A293184

Numbers k such that bphi(k) = bphi(k+1), where bphi(k) is the bi-unitary analog of Euler's totient function (A116550).

+20
7

1, 14, 20, 57, 187, 188, 916, 1603, 93928, 142891, 432976, 549815, 692259, 773887, 872191, 4297168, 9478088, 127162432, 127991488, 129015616, 132527167

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

OFFSET

1,2

COMMENTS

187 is the first solution to bphi(k) = bphi(k+1) = bphi(k+2).

a(22) > 1.6*10^9, if it exists. - Amiram Eldar, Jul 16 2022

LINKS

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

EXAMPLE

14 is in the sequence since bphi(14) = bphi(15) = 9.

MATHEMATICA

bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a={}; b1=0; Do[b2 = bphi[k]; If[b1 == b2, a = AppendTo[a, k - 1]]; b1 = b2, {k, 1, 10^3}]; a (* after Jean-François Alcover at A116550 *)

PROG

(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }

gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));

biuphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));

isok(n) = biuphi(n) == biuphi(n+1);

lista(nn) = {x = biuphi(1); for (n=2, nn, y = biuphi(n); if (x==y, print1(n-1, ", ")); x = y; ); } \\ Michel Marcus, Nov 09 2017

CROSSREFS

Cf. A001274, A116550, A287055, A294030.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Oct 01 2017

EXTENSIONS

a(10) from Michel Marcus, Nov 11 2017

a(11) from Michel Marcus, Nov 12 2017

a(12)-a(21) from Amiram Eldar, Jul 16 2022

STATUS

approved

A005424

Smallest number that requires n iterations of the bi-unitary totient function (A116550) to reach 1.
(Formerly M0530)

+20
5

2, 3, 4, 5, 8, 9, 13, 16, 17, 24, 25, 35, 44, 63, 64, 91, 97, 128, 193, 221, 259, 324, 353, 391, 477, 702, 929, 1188, 1269, 1589, 1613, 2017, 2309, 2623, 3397, 4064, 4781, 5468, 6515, 6887, 9213, 12286, 12887, 14009, 16564, 16897, 17803, 30428, 36256

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

OFFSET

1,1

COMMENTS

Let p(n) = number of unitary divisors k of n, k<n, that are relatively prime to n. Let p_1(n) = p(n), p_r(n) = p(p_{r-1}(n)). Sequence gives minimal r such that p_r(n)=1.

REFERENCES

M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..97 (terms 1..78 from Donovan Johnson)

MAPLE

L := [seq(0, i=0..100)] ;

for n from 1 do

itr := A225320(n) ;

if itr < nops(L) then

if op(itr, L) = 0 then

L := subsop(itr=n, L) ;

print(L) ;

end if;

end do: # R. J. Mathar, May 02 2013

MATHEMATICA

A116550[1] = 1; A116550[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; A225320[n_] := A225320[n] = If[n == 1, 0, 1+A225320[A116550[n]]]; L = Array[0&, 100]; For[n = 1, n <= 40000, n++, itr = A225320[n]; If[itr < Length[L], If[L[[itr]] == 0, L = ReplacePart[L, itr -> n]; Print[Select[L, Positive] // Last]]]]; Select[L, Positive] (* Jean-François Alcover, Jan 13 2014, after R. J. Mathar *)

CROSSREFS

Cf. A116550, A225175, A225176.

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

A225175

Largest number which requires n iterations of the bi-unitary totient function (A116550) to reach 1.

+20
3

1, 2, 3, 6, 10, 11, 12, 18, 30, 42, 78, 106, 210, 366, 550, 603, 750, 1290, 2562, 4398, 4305, 7470, 9090, 14322, 24558, 35382, 55482, 78020, 141190, 207519, 301642, 429870, 552693, 684846, 1060710, 1391390, 2385246, 3454044

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

OFFSET

0,2

COMMENTS

a(26) >= 55482. a(27) >= 78020. - R. J. Mathar, May 05 2013

REFERENCES

M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

LINKS

Table of n, a(n) for n=0..37.

CROSSREFS

Cf. A116550, A005424, A225176, A225174, A047994, A003271, A225172, A225173.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, May 01 2013

EXTENSIONS

a(26)-a(37) from Donovan Johnson, Dec 07 2013

STATUS

approved

A225176

Number of numbers which require n iterations of the bi-unitary totient function (A116550) to reach 1.

+20
3

1, 1, 1, 2, 3, 2, 2, 4, 7, 6, 13, 12, 16, 24, 31, 51, 66, 87, 126, 139, 187, 260, 331, 412, 551, 693

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

OFFSET

1,4

REFERENCES

M. Lal, H. Wareham and R. Mifflin, Iterates of the bi-unitary totient function, Utilitas Math., 10 (1976), 347-350.

LINKS

Table of n, a(n) for n=1..26.

CROSSREFS

Cf. A116550, A005424, A225175, A225174, A047994, A003271, A225172, A225173.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, May 01 2013

STATUS

approved

A306070

Partial sums of A116550: Sum_{k=1..n} bphi(k) where bphi(k) is the bi-unitary analog of Euler's totient function.

+20
3

1, 2, 4, 7, 11, 14, 20, 27, 35, 41, 51, 59, 71, 80, 89, 104, 120, 132, 150, 164, 178, 193, 215, 232, 256, 274, 300, 321, 349, 364, 394, 425, 448, 472, 497, 526, 562, 589, 617, 648, 688, 709, 751, 786, 820, 853, 899, 935, 983, 1019, 1056, 1098, 1150, 1189

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

OFFSET

1,2

COMMENTS

The bi-unitary version of A002088 and A177754.

LINKS

Table of n, a(n) for n=1..54.

László Tóth, On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function, Journal of Integer Sequences, Vol. 12 (2009), Article 09.5.2.

FORMULA

a(n) = A*n^2/2 + O(n*log(n)^2), where A = A306071.

MATHEMATICA

a[1] = 1; a[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; Accumulate[Table[a[n], {n, 1, 100}]] (* after Jean-François Alcover at A116550 *)

PROG

(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }

gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));

bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));

a(n) = sum(k=1, n, bphi(k)); \\ Michel Marcus, Jun 20 2018

CROSSREFS

Cf. A002088, A177754, A116550, A306071.

KEYWORD

nonn

AUTHOR

Amiram Eldar, Jun 19 2018

STATUS

approved

A225320

The number of iterations of the bi-unitary totient A116550 needed to reach 1 starting with n.

+20
2

0, 1, 2, 3, 4, 3, 4, 5, 6, 4, 5, 6, 7, 7, 7, 8, 9, 7, 8, 8, 8, 8, 9, 10, 11, 8, 9, 9, 10, 8, 9, 10, 10, 11, 12, 11, 12, 10, 10, 10, 11, 9, 10, 13, 12, 11, 12, 12, 13, 12, 13, 10, 11, 11, 10, 12, 10, 10, 11, 12, 13, 13, 14, 15, 13, 13, 14, 13, 14, 11, 12, 14, 15, 12, 13, 15, 14, 10, 11, 14

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

OFFSET

1,3

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000

FORMULA

The smallest x such that A116550^x(n) = 1, where the operation Op^x denotes x nestings of the operator Op.

EXAMPLE

a(6) = 3 because the first step is A116550(6) = 3, the second A116550(3) = 2, the third A116550(2) = 1, where 1 is reached.

MAPLE

A225320 := proc(n)

option remember;

if n = 1 then

0;

else

1+procname(A116550(n)) ;

end if;

end proc:

MATHEMATICA

A116550[1] = 1; A116550[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a[n_] := a[n] = If[n == 1, 0, 1 + a[A116550[n]]]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Dec 16 2013 *)

CROSSREFS

Cf. A005424 (positions of records), A116550.

KEYWORD

nonn

AUTHOR

R. J. Mathar, May 05 2013

STATUS

approved

A294030

Values of bphi(k) = bphi(k+1), where bphi is the bi-unitary analog of Euler's totient function (A116550).

+20
1

1, 9, 14, 42, 161, 161, 798, 1400, 86156, 123656, 419430, 387868, 508797, 772121, 870233, 4162866, 8754569, 126168912, 126991491, 128007618, 131491736

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

OFFSET

1,2

COMMENTS

The bi-unitary totient function of numbers k such that k and k+1 have the same function value (A293184).

LINKS

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

FORMULA

a(n) = A116550(A293184(n)).

EXAMPLE

9 is in the sequence since 9 = bphi(14) = bphi(15).

MATHEMATICA

bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; a={}; b1=0; Do[b2 = bphi[k]; If[b1 == b2, a = AppendTo[a, b1]]; b1 = b2, {k, 1, 10^2}]; a (* after Jean-François Alcover at A116550 *)

CROSSREFS

The bi-unitary version of A003275.

Cf. A116550, A293184.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Oct 22 2017

EXTENSIONS

a(10)-a(11) from Michel Marcus, Nov 14 2017

a(12)-a(21) from Amiram Eldar, Jul 16 2022

STATUS

approved

A356747

Numbers m that divide A306070(m) = Sum_{k=1..m} bphi(k), where bphi is the bi-unitary totient function (A116550).

+20
1

1, 2, 141, 1035, 2388, 3973, 5157, 14160, 37023, 68861, 99889, 116106, 117939, 627400, 1561944, 1626983, 5901444, 10054091, 12260525, 32619981, 49775099

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

OFFSET

1,2

COMMENTS

The corresponding quotients A306070(m)/m are 1, 1, 57, 418, ... (see the link for more values).

a(22) > 6.5*10^8, if it exists.

LINKS

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

Amiram Eldar, Table of n, a(n), A306070(a(n))/a(n) for n = 1..21.

MATHEMATICA

phi[x_, n_] := DivisorSum[n, MoebiusMu[#]*Floor[x/#] &]; bphi[n_] := DivisorSum[n, (-1)^PrimeNu[#]*phi[n/#, #] &, CoprimeQ[#, n/#] &]; seq = {}; s = 0; Do[s = s + bphi[n]; If[Divisible[s, n], AppendTo[seq, n]], {n, 1, 10^6}]; seq

CROSSREFS

Cf. A116550, A306070.

Similar sequences: A048290, A306950.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Aug 25 2022

STATUS

approved

A298759

Numbers k such that bphi(k) = k/2, where bphi is the bi-unitary analog of Euler's totient function (A116550).

+20
0

2, 6, 30, 42, 1722, 1806, 19977474

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

OFFSET

1,1

COMMENTS

With Euler's totient function, phi(k) = k/2 only for powers of 2 (A000079, except for 1). With the unitary totient function (A047994) the corresponding sequence is A030163.

a(8) > 2*10^9, if it exists. - Amiram Eldar, Jul 16 2022

LINKS

Table of n, a(n) for n=1..7.

EXAMPLE

42 is in the sequence since bphi(42) = 21 = 42/2.

MATHEMATICA

bphi[1] = 1; bphi[n_] := With[{pp = Power @@@ FactorInteger[n]}, Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; aQ[n_] := bphi[n] == n/2; Select[Range[10000], aQ]

PROG

(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }

gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));

bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));

isok(n) = bphi(n) == n/2; \\ Michel Marcus, Jan 26 2018

CROSSREFS

Cf. A047994, A030163, A116550.

KEYWORD

nonn,more

AUTHOR

Amiram Eldar, Jan 26 2018

EXTENSIONS

a(7) from Amiram Eldar, Jul 16 2022

STATUS

approved