This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Select[Range[1200000],Divisible[DivisorSigma[12,#],EulerPhi[#]]&] (* Harvey P. Dale, Dec 04 2015 *)
Select[Range[250000],Divisible[DivisorSigma[14,#],EulerPhi[#]]&] (* Harvey P. Dale, Feb 02 2019 *)
with(numtheory): A015764:=n->`if`(sigma[6](n) mod phi(n) = 0,n,NULL): seq(A015764(n), n=1..10^5); # Wesley Ivan Hurt, Mar 10 2015
Select[Range[10^5], Divisible[DivisorSigma[6, #], EulerPhi[#]] &] (* Amiram Eldar, Jan 20 2019 *)
Select[Range[40000],Divisible[DivisorSigma[8,#],EulerPhi[#]]&] (* Harvey P. Dale, Jan 31 2016 *)
These terms appear as 5th entries in A020492, A015759-A015774. k = {1, 2, 3, 6} are solutions to Min{k : Mod[sigma[n, k], phi[k]]=0}. First nontrivial solutions are larger: for odd n, k = 12 is solution; for even powers larger numbers arise like 22, 249, 9897, 19689, etc. Certain power-sums of divisors proved to be hard to find.
f[k_, x_] := DivisorSigma[k, x]/EulerPhi[x]; Table[k=7; While[!IntegerQ[f[n, k]], k++]; k, {n, 1, 15}] (* corrected by Jason Yuen, Jun 27 2025 *)
ok(n,k)=my(f=factor(n), r=sigma(f,k)/eulerphi(f)); r>=7 && denominator(r)==1 a(n)=my(k=7); while(!ok(k, n), k++); k \\ Charles R Greathouse IV, Nov 27 2013
from sympy import divisors, totient as phi
def a(n):
k, pk = 7, phi(7)
while sum(pow(d, n, pk) for d in divisors(k, generator=True))%pk != 0:
k += 1
pk = phi(k)
return k
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Dec 22 2021
n=6: prime(6)=13, cases of sigma(13,x)/phi(x) is an integer are listed in A015771: 1, 2, 3, 6, 12, etc.; the first term which is non-balanced, i.e., not in A020492, is a(6) = 749 = A020492(31); a(29) = 700854 and a(45) = 510759 are remarkably large.
Table[fl=1; Do[s1=DivisorSigma[1, n]/EulerPhi[n]; sk=DivisorSigma[Prime[k], n]/EulerPhi[n]; If[ !IntegerQ[s1]&&IntegerQ[sk]&&Equal[fl, 1], Print[{n, Prime[k]}]; fl=0], {n, 1, 1000000}], {k, 1, 100}]
lista(nmax) = {my(ps = primes(nmax), pmax = ps[#ps], v = vector(pmax), c = 0, k = 2, f, e, p); while(c < nmax, f = factor(k); e = eulerphi(f); if(sigma(f) % e > 0, for(i = 1, nmax, p = ps[i]; if(!(sigma(f, p) % e) && v[p] == 0, c++; v[p] = k))); k++); for(i = 1, pmax, if(v[i] > 0, print1(v[i], ", ")));} \\ Amiram Eldar, Aug 29 2024
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