A353764 Numbers k for which A353749(sigma(k)) is a multiple of A353749(k), where A353749(k) = phi(k) * A064989(k), and A064989 shifts the prime factorization one step towards lower primes.

1, 2, 4, 6, 8, 10, 18, 20, 24, 28, 30, 32, 40, 60, 72, 84, 90, 108, 120, 128, 200, 216, 224, 234, 252, 360, 384, 496, 600, 640, 672, 864, 936, 1080, 1120, 1152, 1170, 1488, 1800, 1920, 2016, 2176, 3200, 3360, 3456, 4320, 4464, 4680, 5148, 5600, 5760, 6048, 6528, 6552, 8128, 9600, 10080, 10880, 14976, 16800, 17280

Offset: 1

Author

Antti Karttunen, May 10 2022

Keywords

• nonn

Comments

Question: Are there any odd terms after the initial one? See A353789, A353796, A353797.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..229; terms <= 2^32
• Index entries for sequences where odd perfect numbers must occur, if they exist at all
• Index entries for sequences computed from indices in prime factorization
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000010, A000203, A006872, A062401, A064989, A336549, A336550, A353757, A353761, A353766 (characteristic function), A353789, A353796, A353797.

Positions of 1's in A353762. Cf. also A353765.

Subsequence of A353759. Cf. A007691 (a subsequence).

Programs

Mathematica

f[p_, e_] := (p - 1)*p^(e - 1)*If[p == 2, 1, NextPrime[p, -1]^e]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[20000], Divisible[s[DivisorSigma[1, #]], s[#]] &] (* Amiram Eldar, May 10 2022 *)

PARI

A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
A353749(n) = (eulerphi(n)*A064989(n));
isA353764(n) = { my(s=sigma(n)); !(A353749(s)%A353749(n)); };