A156942 Odd abundant numbers whose abundance is odd.

11025, 99225, 245025, 275625, 342225, 540225, 893025, 1334025, 1863225, 2205225, 2480625, 3080025, 3186225, 3980025, 4601025, 4862025, 5832225, 6125625, 6890625, 7868025, 8037225, 8555625, 9272025, 9828225, 10595025, 10989225

Offset: 1

Author

Robert G. Wilson v, Feb 18 2009

Keywords

• nonn

Comments

Number of terms <10^n: 0, 0, 0, 0, 2, 7, 24, 83, 250, 792, 2484, 7988, 25383, 80082, ..., . Not all are a multiple of 25, i.e.; 81162081 = 9009^2 = (9*7*11*13)^2. See A156943.

Any term must be an odd square. Square roots are in A174830.

Indeed, the sum of divisors of any number isn't odd unless it's a square or twice a square (A028982), and to get the abundance, twice the number is subtracted, so the parity remains the same. - M. F. Hasler, Jan 26 2020

Question: Is this a subsequence of A379503? (Is A379504(a(n)) > 0 for all n? See A379951). The first 15000 terms are all included there. - Amiram Eldar and Antti Karttunen, Jan 06 2025

Question 2: Is A379505(a(n)) > 1 for all n, especially if there are no quasiperfect numbers (numbers k such that sigma(k) = 2k+1)? - Antti Karttunen, Jan 06 2025

From Amiram Eldar, Jan 16 2025: (Start)

The least term that is not divisible by 5 is a(75) = 81162081.

The least term that is not divisible by 3 is a(296889) = 1382511906801025.

The least term that is coprime to 15 is 15285071557677427358507559514565648611799881. (End)

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)

Crossrefs

Cf. A005101, A005231, A028982, A033880, A156903, A174830, A379503, A379504, A379505, A379951 [= A379504(a(n))].

Subsequences: A156943, A325311 (thus also A379490), A347890, A379949 (terms that are primitive abundant).

Programs

Mathematica

fQ[n_] := Block[{ds = DivisorSigma[1, n] - 2 n}, ds > 0 && OddQ@ ds]; Select[ Range[1, 12006223, 2], fQ @# &]

PARI

is(n)=my(s=sigma(n)); n%2 && s>2*n && (s-2*n)%2 \\ Charles R Greathouse IV, Feb 21 2017

Formula

a(n) = A174830(n)^2. - M. F. Hasler, Jan 26 2020

Extensions

Edited by Robert G. Wilson v at the suggestion of T. D. Noe, Mar 30 2010