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 1-4 of 4 results.

A342925 a(n) = A003415(sigma(n)), where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

0, 1, 4, 1, 5, 16, 12, 8, 1, 21, 16, 32, 9, 44, 44, 1, 21, 16, 24, 41, 80, 60, 44, 92, 1, 41, 68, 92, 31, 156, 80, 51, 112, 81, 112, 20, 21, 92, 92, 123, 41, 272, 48, 124, 71, 156, 112, 128, 22, 34, 156, 77, 81, 244, 156, 244, 176, 123, 92, 332, 33, 272, 164, 1, 124, 384, 72, 165, 272, 384, 156, 119, 39, 101, 128, 188
Offset: 1

Views

Author

Antti Karttunen, Apr 07 2021

Keywords

Links

Crossrefs

Cf. A023194 (positions of ones, which is a subsequence of prime powers, A000961).
Cf. A342922, A342923, A342924, A342926, A351568, A351569, A351570, A351571.
Cf. A342021 (fixed points), A343216 [positions k where a(k) < k], A343217 [a(k) >= k], A343218 [a(k) > k].
Cf. A347870 (parity of terms), A347872, A347873, A347877 (positions of odd terms), A347878 (of even terms), A343218, A343220, A344024.

Programs

  • Mathematica
    Array[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &@ DivisorSigma[1, #] &, 76] (* Michael De Vlieger, Apr 08 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));

Formula

a(A023194(n)) = 1.
If gcd(m,n) = 1, a(m*n) = sigma(m)*A003415(sigma(n)) + sigma(n)*A003415(sigma(m)) = sigma(m)*a(n) + sigma(n)*a(m).
a(n) = (A351568(n)*A351571(n)) + (A351569(n)*A351570(n)). - Antti Karttunen, Feb 23 2022

A342926 a(n) = A003415(sigma(n)) - n, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

-1, -1, 1, -3, 0, 10, 5, 0, -8, 11, 5, 20, -4, 30, 29, -15, 4, -2, 5, 21, 59, 38, 21, 68, -24, 15, 41, 64, 2, 126, 49, 19, 79, 47, 77, -16, -16, 54, 53, 83, 0, 230, 5, 80, 26, 110, 65, 80, -27, -16, 105, 25, 28, 190, 101, 188, 119, 65, 33, 272, -28, 210, 101, -63, 59, 318, 5, 97, 203, 314, 85, 47, -34, 27, 53, 112, 195
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2021

Keywords

Links

Crossrefs

Cf. A342925, A342924, A343223 [= gcd(A003415(n), a(n))].
Cf. A342021 (positions of 0's), A343216 (of negative terms), A343217 (of nonnegative terms), A343218 (of positive terms).

Programs

  • Mathematica
    Array[If[#2 < 2, 0, #2 Total[#2/#1 & @@@ FactorInteger[#2]]] - #1 & @@ {#, DivisorSigma[1, #]} &, 77] (* Michael De Vlieger, Apr 08 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);

Formula

a(n) = A342925(n) - n = A003415(A000203(n)) - n.

A343216 Numbers k such that A003415(sigma(k)) < k, where A003415(x) gives the arithmetic derivative of x.

Original entry on oeis.org

1, 2, 4, 9, 13, 16, 18, 25, 36, 37, 49, 50, 61, 64, 73, 81, 97, 100, 101, 109, 113, 121, 137, 144, 157, 169, 173, 181, 193, 225, 229, 241, 242, 256, 257, 277, 281, 289, 313, 317, 324, 325, 333, 337, 353, 361, 373, 397, 400, 401, 409, 421, 433, 441, 457, 484, 512, 529, 541, 549, 576, 577, 578, 601, 613, 617, 625, 641
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2021

Keywords

Links

Crossrefs

Cf. A000203, A003415, A343217 (complement), A342925.
Positions of negative terms in A342926.

Programs

  • Mathematica
    Select[Range[641], If[#2 < 2, 0, #2 Total[#2/#1 & @@@ FactorInteger[#2]]] < #1 & @@ {#, DivisorSigma[1, #]} &] (* Michael De Vlieger, Apr 08 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
isA343216(n) = (A003415(sigma(n))

A342021 Numbers k such that A003415(sigma(k)) = k, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

Original entry on oeis.org

5, 8, 41, 47057
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2021

Keywords

Comments

a(5) > 2^33, if it exists.
Terms of this sequence and A230165 group into pairs (m, sigma(m)), where m is a term of this sequence and sigma(m) is a term of A230165. - Max Alekseyev, Feb 13 2025

Crossrefs

Fixed points of A342925.
Positions of 0's in A342926.
Subsequence of A343217.
Cf. A000203, A003415, A229342, A230164, A230165, A347889.

Formula

a(n) = A003415(A230165(n)). - Max Alekseyev, Feb 13 2025
Showing 1-4 of 4 results.

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