A334337 -id:A334337 - cp's OEIS frontend

A334339 Least positive integer m such that sigma(m * n) is a cube, where sigma(k) is the sum of the divisors of k.

1, 51, 34, 291, 22, 17, 1, 1347, 597, 11, 10, 97, 892, 51, 46, 1758, 6, 3540, 343, 1649, 34, 5, 30, 449, 2928, 446, 199, 291, 472, 23, 34, 879, 235, 3, 22, 1770, 8661, 356, 3007, 1593, 884, 17, 241, 298, 1416, 15, 22, 586, 133, 1464, 2, 223, 3, 1180, 2, 1347, 711, 236, 232, 1062, 1200, 17, 597, 96771, 586, 265, 577, 485, 10, 11

Offset: 1

Zhi-Wei Sun, Apr 23 2020

nonn

Conjecture: a(n) exists for any n > 0. In other words, for any positive integer n, there is a positive integer m with sigma(m * n) equal to a cube.
The author's conjecture in A259915 implies that for each n = 1, 2, 3, ... there is a positive integer m with sigma(m * n) equal to a square.
See also A334337 for a similar conjecture.

			a(2) = 51 with sigma(2*51) = 216 = 6^3.
a(4) = 291 with sigma(4*291) = 2744 = 14^3.
a(578) = 34312749 with sigma(578*34312749) = 42144192000 = 3480^3.
a(673) = 49061802 with sigma(673*49061802) = 66135317184 = 4044^3.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. See also arXiv, arXiv:1211.1588 [math.NT], 2012-2017. (Cf. Conjecture 4.5.)

Cf. A000203, A000578, A020477, A259915, A334337.

cubeQ[n_] := cubeQ[n] = IntegerQ[n^(1/3)];
sigma[n_] := sigma[n] = DivisorSigma[1, n];
tab = {}; Do[m = 0; Label[aa]; m = m + 1; If[cubeQ[sigma[m * n]], tab = Append[tab, m], Goto[aa]], {n, 70}]; tab
lpi[n_]:=Module[{k=1},While[!IntegerQ[Surd[DivisorSigma[1,n*k],3]],k++]; k]; Array[lpi,70] (* Harvey P. Dale, Nov 05 2020 *)

a(n) = my(m=1); while (!ispower(sigma(n*m), 3), m++); m; \\ Michel Marcus, Apr 23 2020

Corrected and extended by Harvey P. Dale, Nov 05 2020

A334350 Least positive integer m relatively prime to n such that phi(mn) = phi(m)phi(n) is a fourth power, where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 1, 16, 15, 8, 703, 247, 5, 247, 489, 1255, 5, 109, 247, 4, 3, 1, 247, 73, 3, 109, 1255, 13315, 163, 753, 109, 73, 109, 1373, 163, 27331, 1, 625, 1, 81, 109, 57, 73, 1295, 1, 251, 109, 74663, 625, 949, 13315, 1557377, 1, 74663, 753, 16, 81, 175765, 73, 251, 81, 37, 1373, 243895, 1

Offset: 1

Zhi-Wei Sun, Apr 24 2020

nonn

Conjecture: For any positive integers k and m, there is a positive integer n relatively prime to m such that phi(m*n) = phi(m)*phi(n) is a k-th power.
This conjecture implies that a(n) exists for every n = 1,2,3,....
See also A334353 for a similar conjecture involving the sigma function (A000203).
a(n) = 1 if and only if n is in A078164. - Charles R Greathouse IV, Apr 24 2020

			a(3) = 16 with gcd(3,16) = 1 and phi(3*16) = phi(3)*phi(16) = 2*8 = 2^4.
a(167) = 370517977 with gcd(167, 370517977) = 1 and phi(167*370517977) = phi(167)*phi(370517977) = 166*370517976 = 61505984016 = 498^4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..226
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. See also arXiv, arXiv:1211.1588 [math.NT], 2012-2017.

Cf. A000010, A000203, A000583, A039770, A039771, A259915, A259916, A334337, A280988, A334339, A334353, A078164.

QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
phi[n_]:=phi[n]=EulerPhi[n];
tab={};Do[m=0;Label[aa];m=m+1;If[GCD[m,n]==1&&QQ[phi[m]*phi[n]],tab=Append[tab,m],Goto[aa]],{n,1,60}];tab

a(n) = my(m=1,e=eulerphi(n)); while (!((gcd(n, m) == 1) && ispower(e*eulerphi(m), 4)), m++); m; \\ Michel Marcus, Apr 25 2020

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A334339 Least positive integer m such that sigma(m * n) is a cube, where sigma(k) is the sum of the divisors of k.

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A334350 Least positive integer m relatively prime to n such that phi(mn) = phi(m)phi(n) is a fourth power, where phi is Euler's totient function (A000010).

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cp's OEIS Frontend

A334339 Least positive integer m such that sigma(m * n) is a cube, where sigma(k) is the sum of the divisors of k.

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A334350 Least positive integer m relatively prime to n such that phi(m*n) = phi(m)*phi(n) is a fourth power, where phi is Euler's totient function (A000010).

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Mathematica

PARI

A334350 Least positive integer m relatively prime to n such that phi(mn) = phi(m)phi(n) is a fourth power, where phi is Euler's totient function (A000010).