A091285
Numbers k such that sigma_3(k) is divisible by the square of phi(k).
Original entry on oeis.org
1, 2, 3, 6, 14, 42, 3810, 318990, 13243560, 1108809240, 1719507048, 25330080090, 271984504290
Offset: 1
k = 14: phi(k)^2 = 36, sigma_3(k) = 3096 = 36*86.
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Empirical test for very high powers of divisors is: t = {1, 2, 3, 6, 14, 42, 3810, 13243560} Table[{6*j+3, Union[Table[IntegerQ[DivisorSigma[6*j + 3, Part[t, k]]/EulerPhi[Part[t, k]]^2], {k, 1, 8}]]}, {j, 1, 300}]; output={exponent, True}.
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for(n = 1, 10^9, if(sigma(n, 3) % (eulerphi(n)^2) == 0, print1(n, ", "))) \\ Ryan Propper, Jan 18 2008
A078538
Smallest k > 6 such that sigma_n(k)/phi(k) is an integer.
Original entry on oeis.org
12, 22, 12, 249, 12, 22, 12, 19689, 12, 22, 12, 249, 12, 22, 12
Offset: 1
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.
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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 *)
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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
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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
A078540
Least non-balanced x (i.e., not in A020492) such that sigma(p(n),x)/phi(x) is an integer, where p(n) = n-th prime.
Original entry on oeis.org
22, 38, 46, 295, 235, 749, 3687, 6128, 1415, 4254, 10451, 8351, 334, 4511, 3398, 1286, 148870, 11015, 35519, 10239, 14072, 76088, 5991, 718, 11654, 30761, 7431, 20993, 700654, 22169, 5095, 4198, 27415, 26744, 14318, 48368, 180878, 16991, 173123, 4166, 5033, 7246
Offset: 1
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.
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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}]
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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
a(18) corrected and more terms added by
Amiram Eldar, Aug 29 2024
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