A345002 -id:A345002 - cp's OEIS frontend

A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

-1, 0, -1, 3, -3, 5, -5, 11, 1, 5, -9, 20, -11, 5, 2, 31, -15, 24, -17, 26, 0, 5, -21, 56, -9, 5, 13, 32, -27, 43, -29, 79, -4, 5, -10, 79, -35, 5, -6, 78, -39, 53, -41, 44, 27, 5, -45, 140, -27, 38, -10, 50, -51, 93, -22, 100, -12, 5, -57, 140, -59, 5, 29, 191, -28, 73, -65, 62, -16, 63, -69, 207, -71, 5, 29, 68

Offset: 1

Antti Karttunen, Jun 05 2021

Coincides with A003415 only on perfect numbers (A000396).

Cf. A000203, A000396, A001065, A003415, A033879, A168036, A344999, A345002, A345003, A345004, A345005, A345049.
Cf. also A051709, A344753 (analogous sequences).
Cf. A013661, A136141.

A003415[n_] := If[n < 2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
a[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
Array[a, 80] (* Jean-François Alcover, Jun 12 2021 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A345001(n) = (sigma(n)+A003415(n)-(2*n));

a(n) = A003415(n) - A033879(n) = A000203(n) + A003415(n) - 2*n.
a(n) = A001065(n) + A168036(n).
a(n) = A344999(n) / A048250(n) = A345049(n) / A173557(n).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 + A136141 - 2 = 0.418090735898... . - Amiram Eldar, Dec 08 2023

A344995 Numbers k such that A051709(k)/A173557(k) is a positive natural number and a divisor of k.

Original entry on oeis.org

4, 6, 12, 28, 42, 312, 456, 496, 588, 828, 1080, 1216, 1242, 1377, 1560, 4560, 5964, 7320, 7480, 7584, 8128, 11400, 13728, 14784, 23760, 33462, 59400, 59520, 124020, 147840, 188600, 277648, 321000, 543552, 1288224, 1510272, 1596048, 1964544, 2038140, 3323736, 3611520, 3780672, 3909816, 6137440, 9034032, 10783890

Offset: 1

Antti Karttunen, Jun 05 2021

nonn

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A051709, A173557.
Subsequence of A344994.
Cf. also A344755, A345002.

A051709(n) = ((sigma(n) + eulerphi(n)) - (2*n));
A173557(n) = factorback(apply(p -> p-1, factor(n)[, 1]));
isA344995(n) = { my(u=A051709(n),t=A173557(n),r=u/t); ((u>0)&&(1==denominator(r)&&!(n%r))); };

cp's OEIS Frontend

A345001 a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula