cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A156942 Odd abundant numbers whose abundance is odd.

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Feb 18 2009

Keywords

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

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

A347889 Numbers k such that sigma(k) > 2*k and A003415(sigma(k)) < k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors function.

Original entry on oeis.org

18, 36, 100, 144, 324, 400, 576, 784, 900, 1296, 1458, 1600, 1936, 2304, 2500, 2704, 2916, 3136, 3600, 4624, 5184, 5202, 5776, 6400, 7744, 8464, 9216, 9604, 10000, 10404, 10816, 11664, 12100, 13122, 13456, 14400, 15376, 17424, 18496, 19044, 23104, 25600, 26244, 28900, 30258, 30276, 30976, 32400, 33856, 36864, 38416
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Numbers k such that A033880(k) is positive but A342926(k) is negative.

Links

Crossrefs

Intersection of A005101 and A343216. Subsequence A347890 gives the odd terms.
Cf. A000203, A003415, A033880, A342926.

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 40000], DivisorSigma[1, #] > 2*# && ad[DivisorSigma[1, #]] < # &] (* Amiram Eldar, Sep 19 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347889(n) = ((A003415(sigma(n))(2*n)));

A347891 Odd numbers k such that sigma(k^2) > 2*k^2 and A003415(sigma(k^2)) < k^2.

Original entry on oeis.org

495, 735, 945, 1485, 1755, 2205, 2475, 2625, 2805, 3315, 3675, 3795, 4455, 4785, 4845, 5145, 5265, 5445, 6615, 6765, 7425, 7605, 7755, 8085, 8415, 8505, 8745, 8775, 9735, 11025, 12375, 12495, 13365, 13965, 14025, 15435, 15795, 16065, 16335, 16905, 17595, 18375, 19845, 20295, 21315, 22185, 22275, 22785, 22815, 23265
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Odd numbers whose square is abundant and present in A343216.
The first term that is not a multiple of 5 is a(146) = 82467.

Links

Crossrefs

Square roots of A347890. Subsequence of A174830.
Cf. A000196, A000203, A003415, A005231, A343216.

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 24000, 2], DivisorSigma[1, #^2] > 2*#^2 && ad[DivisorSigma[1, #^2]] < #^2 &] (* Amiram Eldar, Sep 19 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA347891(n) = if(!(n%2),0,my(u=n*n); (A003415(sigma(u))(2*u)));

Formula

a(n) = A000196(A347890(n)).
Showing 1-3 of 3 results.

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