A334353 Least positive integer m relatively prime to n such that sigma(m*n) is a fourth power, where sigma(k) is the sum of the divisors of k.

1, 255, 170, 3783, 102, 85, 31, 39063, 34711, 51, 85, 1261, 1164, 53, 34, 417067, 30, 716125, 499, 55563, 127, 345, 34, 13021, 417067, 55563, 3493, 117273, 10776, 17, 7, 34359, 230, 15, 321, 10549987, 2469230, 13021, 388, 8483, 28128, 187, 5323, 30865, 314758, 17, 230, 1345225, 1481538, 9473379, 10, 291, 14, 82445, 17, 60615, 1999, 7495, 5960, 18521

Offset

1,2

Comments

Conjecture: For any positive integers k and m, there is a positive integer n relatively prime to m such that sigma(m*n) is a k-th power.

This implies that a(n) exists for every n = 1,2,3,....

See also A344350 for a similar conjecture involving Euler's totient function (A000010).

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..127

Example

a(2) = 255 with gcd(2, 255) = 1 and sigma(2*255) = sigma(2)*sigma(255) = 3*432 = 1296 = 6^4.
a(64) = 1851519543 with gcd(64, 1851519543) = 1 and sigma(64*1851519543) = sigma(64)*sigma(1851519543) = 127*2654704368 = 337147454736 = 762^4.

Mathematica

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

Prog

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

Crossrefs

Cf. A000010, A000203, A000583, A006532, A019422, A020477, A334350.

Sequence in context: A145715 A145587 A283188 * A028526 A132828 A296899

Adjacent sequences: A334350 A334351 A334352 * A334354 A334355 A334356

Keyword

nonn

Author

Zhi-Wei Sun, Apr 24 2020

Status

approved