A386425 Odd composites k such that sigma(k) has the same powerful part as k, where sigma is the sum of divisors function.

153, 801, 1773, 3725, 4689, 4753, 5013, 6957, 8577, 8725, 9549, 9873, 11493, 13437, 14409, 15381, 18621, 19269, 21213, 21537, 23481, 25101, 26073, 26225, 28989, 29161, 29313, 29961, 32229, 33849, 34173, 36117, 38061, 39033, 40653, 42597, 43893, 47457, 47781, 48725, 48753, 51669, 52317, 54261, 56953, 57177, 57501

Offset: 1

Author

Antti Karttunen, Aug 17 2025

Keywords

• nonn
• new

Comments

By definition, the sequence contains all odd perfect numbers, and also includes any hypothetical odd triperfect number that is not a multiple of 3 (see A005820 and A347391), and similarly, any odd term of A046060 that is not a multiple of 5, etc. If there are no squares in this sequence (see conjecture in A386424), then the latter categories of numbers certainly do not exist, and this is then a subsequence of A228058.

The first nondeficient term is a(32315) = 81022725. See A386426.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..25000
• Index entries for sequences where odd perfect numbers must occur, if they exist at all
• Index entries for sequences related to sigma(n)

Crossrefs

Intersection of A071904 and A386424.

Nonsquare terms form a subsequence of A228058.

Cf. A000203, A003557, A057521, A386426 (nondeficient terms).

Cf. also A324647, A349749.

Programs

Mathematica

rad[n_] := Times @@ First /@ FactorInteger[n];a057521[n_] := n/Denominator[n/rad[n]^2];Select[Range[9,57501,2],!PrimeQ[#]&&a057521[DivisorSigma[1,#]]==a057521[#]&] (* James C. McMahon, Aug 18 2025 *)

PARI

A057521(n)=my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1))
isA386425(n) = ((n>1) && (n%2) && !isprime(n) && (A057521(sigma(n))==A057521(n)));

Formula

{k | k is odd composite and A003557(A000203(k)) = A003557(k)}.