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 1-5 of 5 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.

A230164 Numbers k such that k' = sigma(k), where k' is the arithmetic derivative of k.

Original entry on oeis.org

17296, 24016, 334144656, 358585488, 2955423888, 311063879024
Offset: 1

Views

Author

Paolo P. Lava, Oct 14 2013

Keywords

Comments

a(7) > 10^12. - Giovanni Resta, Mar 11 2014

Examples

			If k = 17296 then k' = sigma(k) = 35712. If k = 24016 then k' = sigma(k) = 49600.

Crossrefs

Cf. A003415, A229342, A230165, A342021.

Programs

  • Maple
    with(numtheory); P:= proc(q) local a,n,p;
for n from 1 to q do a:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
if sigma(n)=a then print(n); fi; od; end: P(10^9);

Extensions

a(3)-a(5) from Giovanni Resta, Oct 14 2013
a(6) from Giovanni Resta, Mar 11 2014

A343217 Numbers k such that A003415(sigma(k)) >= k, where A003415(x) gives the arithmetic derivative of x.

Original entry on oeis.org

3, 5, 6, 7, 8, 10, 11, 12, 14, 15, 17, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 98, 99
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2021

Keywords

Links

Crossrefs

Cf. A000203, A003415, A342925, A343216 (complement).
Disjoint union of A342021 and A343218.
Positions of nonnegative terms in A342926.

Programs

  • Mathematica
    Select[Range[100], 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]));
isA343217(n) = (A003415(sigma(n))>=n);

A230165 Numbers k such that k = sigma(k'), where k' is the arithmetic derivative of k.

Original entry on oeis.org

6, 15, 42, 47058
Offset: 1

Views

Author

Paolo P. Lava, Oct 14 2013

Keywords

Comments

a(5) > 10^10. - Giovanni Resta, Oct 14 2013
Terms of this sequence and A342021 group into pairs (m, m'), where m is a term of this sequence and m' is a term of A342021. - Max Alekseyev, Feb 13 2025

Examples

			Arithmetic derivative of 15 is 8 and sigma(8) = 15.

Crossrefs

Cf. A003415, A229342, A230164, A342021.

Programs

  • Maple
    with(numtheory); P:=proc(q) local n; for n from 1 to q do
if n=sigma(n*add(op(2,p)/op(1,p),p=ifactors(n)[2])) then print(n);
fi; od; end: P(10^9);

Formula

a(n) = sigma(A342021(n)). - Max Alekseyev, Feb 13 2025
Showing 1-5 of 5 results.

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