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
Keywords
Examples
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.
Links
- Zhi-Wei Sun, Table of n, a(n) for n = 1..127
Programs
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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 -
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
Comments