A353797 -id:A353797 - cp's OEIS frontend

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

Antti Karttunen, May 10 2022

nonn

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

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).

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 *)

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)); };

A353796 Numbers k such that k divides A353790(k), where A353790(n) = phi(A003973(n)) * A064989(A003973(n)).

Original entry on oeis.org

1, 2, 4, 8, 12, 24, 36, 44, 72, 96, 112, 128, 132, 160, 180, 220, 288, 336, 352, 360, 384, 396, 480, 528, 560, 640, 660, 880, 1044, 1056, 1152, 1232, 1344, 1404, 1440, 1680, 1760, 1920, 1980, 2088, 2352, 2376, 2464, 2496, 2640, 3168, 3600, 3696, 3920, 4032, 4400, 4736, 5220, 5280, 5376, 5760, 5824, 6075, 6144, 6160

Offset: 1

Antti Karttunen, May 12 2022

nonn

Of 5263 initial terms (terms < 2^32), only 67 are odd, and of these, only two, 1 and 1525391261 (= 503^2 * 6029) are in A007310. Of 5263 initial terms, 4653 are multiples of 3, 2331 are multiples of 81, and 3780 are multiples of 5.

Cf. A000010, A000203, A003961, A003973, A353790, A353797 (subsequence).

Cf. also A353795.

A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A064989(n) = { my(f=factor(n>>valuation(n, 2))); for(i=1, #f~, f[i, 1] = precprime(f[i, 1]-1)); factorback(f); };
A353790(n) = { my(s=sigma(A003961(n))); (eulerphi(s)*A064989(s)); };
isA353796(n) = !(A353790(n)%n);

cp's OEIS Frontend

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.

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