A387158 -id:A387158 - cp's OEIS frontend

A386424 Numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

1, 2, 5, 12, 13, 26, 29, 37, 41, 44, 56, 61, 73, 74, 76, 90, 101, 109, 113, 122, 137, 146, 153, 157, 172, 173, 181, 193, 218, 229, 236, 257, 268, 277, 281, 312, 313, 314, 317, 353, 362, 373, 386, 389, 397, 401, 409, 421, 433, 457, 458, 461, 509, 522, 524, 528, 541, 554, 560, 569, 601, 613, 617, 626, 641, 652, 653

Offset: 1

Antti Karttunen, Aug 17 2025

Conjecture 1: the initial 1 is the only square in this sequence, and a(2) = 2 is the only term that is twice a square.

Conjecture 2: A323653 is a subsequence (which would follow from conjecture 1 (c) given there).

Cf. A000203, A003557, A057521, A387156.
Subsequences: A323653 (conjectured), A351549, A386425 (odd composites), A386426 (nondeficient terms).
Cf. also A006872, A351446, A387158.

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[653],a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386424(n) = (A057521(sigma(n))==A057521(n));

{k | A057521(A000203(k)) = A057521(k)}, or equally, {k | A387156(k) = A003557(k)}.

A387157 a(n) = A173557(sigma(n)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

Original entry on oeis.org

1, 2, 1, 6, 2, 2, 1, 8, 12, 2, 2, 6, 6, 2, 2, 30, 2, 24, 4, 12, 1, 2, 2, 8, 30, 12, 4, 6, 8, 2, 1, 12, 2, 2, 2, 72, 18, 8, 6, 8, 12, 2, 10, 12, 24, 2, 2, 30, 36, 60, 2, 6, 2, 8, 2, 8, 4, 8, 8, 12, 30, 2, 12, 126, 12, 2, 16, 12, 2, 2, 2, 96, 36, 36, 30, 24, 2, 12, 4, 60, 10, 12, 12, 6, 2, 20, 8, 8, 8, 24, 6, 12, 1, 2, 8

Offset: 1

Antti Karttunen, Aug 19 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000203, A003958, A080398, A173557, A387158 (positions where equal to A173557(n)).

Cf. also A351442.

A387157[n_] := If[n == 1, 1, Times @@ (FactorInteger[DivisorSigma[1, n]][[All, 1]] - 1)];
Array[A387157, 100] (* Paolo Xausa, Aug 20 2025 *)

A387157(n) = factorback(apply(p -> p-1,factor(sigma(n))[,1]));

a(n) = A003958(A080398(n)).

A387159 Odd numbers k such that A173557(k) = A173557(sigma(k)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

Original entry on oeis.org

1, 63, 135, 351, 875, 891, 999, 1647, 1859, 1971, 4239, 5211, 7479, 8451, 10719, 11367, 12339, 14607, 16317, 16551, 17847, 18171, 19791, 20439, 22103, 23679, 26919, 27951, 29511, 31131, 31407, 31487, 32427, 32751, 33399, 35667, 37287, 39231, 43767, 44739, 47331, 50571, 52191, 53811, 54459, 57319, 57699, 63207, 66771

Offset: 1

Antti Karttunen, Aug 19 2025

nonn

Odd numbers k for which A173557(k) == A387157(k).

Odd terms of A387158.
Cf. A000203, A173557, A387157.
Cf. also A351443, A353679, A386425.

A387159Q[k_] := OddQ[k] && #[k] == #[DivisorSigma[1, k]] & [Times @@ (FactorInteger[#][[All, 1]] - 1) &];
Select[Range[100000], A387159Q] (* Paolo Xausa, Aug 20 2025 *)

A173557(n) = factorback(apply(p -> p-1,factor(n)[,1]));
is_A387159(n) = (n%2 && (A173557(sigma(n))==A173557(n)));

cp's OEIS Frontend

A386424 Numbers k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

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A387157 a(n) = A173557(sigma(n)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

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A387159 Odd numbers k such that A173557(k) = A173557(sigma(k)), where A173557(n) is multiplicative with a(p^e) = p-1 and sigma is the sum of divisors function.

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