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.

A347883 a(n) = A342926(n) mod 3.

Original entry on oeis.org

2, 2, 1, 0, 0, 1, 2, 0, 1, 2, 2, 2, 2, 0, 2, 0, 1, 1, 2, 0, 2, 2, 0, 2, 0, 0, 2, 1, 2, 0, 1, 1, 1, 2, 2, 2, 2, 0, 2, 2, 0, 2, 2, 2, 2, 2, 2, 2, 0, 2, 0, 1, 1, 1, 2, 2, 2, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 1, 2, 2, 1, 2, 2, 0, 2, 1, 0, 2, 1, 0, 1, 2, 2, 1, 2, 0, 1, 2, 1, 0, 2, 0, 1, 2, 2, 0, 1, 1, 1, 1, 2, 0, 1, 2, 1
Offset: 1

Views

Author

Antti Karttunen, Sep 18 2021

Keywords

Links

Crossrefs

Cf. A000203, A003415, A342926, A347880 (positions of zeros).
Cf. also A347871.

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); a[n_] := Mod[ad[DivisorSigma[1, n]] - n, 3]; Array[a, 105] (* Amiram Eldar, Sep 18 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(n))-n);
A347883(n) = (A342926(n)%3);

Formula

a(n) = A342926(n) mod 3.

A347884 Odd composites k for which A003415(sigma(k))-k is strictly positive and a multiple of A003415(k). Here A003415 is the arithmetic derivative.

Original entry on oeis.org

963, 969, 5871, 10479, 2308203, 41240261, 52024391, 69989429, 75384301, 319255721, 634457761, 781718149, 1184197307, 1190942957, 1195786661, 2114464203
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2021

Keywords

Comments

Odd nonprimes k for which A343223(k) = A003415(k).
Any odd terms of A065997, including odd perfect numbers, odd triperfect numbers and odd 5-multiperfect numbers, should occur in this sequence, if such numbers exist at all.
The odd terms present in A342924 form a subsequence of this sequence.

Links

Crossrefs

Cf. A000203, A003415, A007691, A065997, A342925, A342926, A343223.
Subsequence of A343218.
Cf. also A000396, A005820, A046060, A342922, A342923, A342924, A347872, A347880.

Programs

  • Mathematica
    ad[1] = 0; ad[n_] := n * Total@(Last[#]/First[#]& /@ FactorInteger[n]); Select[Range[1, 2.5*10^6, 2], CompositeQ[#] && (d = ad[DivisorSigma[1, #]] - #) > 0 && Divisible[d, ad[#]] &] (* 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]));
isA347884(n) = if(!(n%2)||isprime(n),0,my(u=(A003415(sigma(n))-(n))); ((u>0)&&!(u%A003415(n))));

A351562 Odd composites k such that A342926(2*k) is a multiple of 3.

Original entry on oeis.org

15, 33, 45, 51, 69, 87, 99, 105, 123, 135, 141, 147, 153, 159, 165, 175, 177, 195, 207, 213, 231, 249, 255, 261, 267, 285, 297, 303, 315, 321, 325, 339, 345, 357, 369, 375, 393, 405, 411, 423, 429, 435, 441, 447, 459, 465, 475, 477, 483, 495, 501, 507, 519, 531, 537, 555, 561, 573, 585, 591, 609, 615, 621, 627, 639
Offset: 1

Views

Author

Antti Karttunen, Feb 23 2022

Keywords

Links

Crossrefs

Cf. A000203, A003415, A342925, A342926, A347880.
Cf. A347874 (the intersection of this sequence and A347872).

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(n))-n);
isA351562(n) = ((n%2)&&!isprime(n)&&!(A342926(2*n)%3));
Showing 1-3 of 3 results.

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