A342924 Composite numbers k such that A003415(sigma(k)) = k + p*A003415(k), for some prime p, where A003415 is the arithmetic derivative, and sigma is the sum of divisors of n.

6, 28, 120, 496, 672, 963, 1036, 5871, 8128, 10479, 164284, 264768, 523776, 2308203, 6511664, 33550336, 41240261, 75384301, 400902412, 459818240, 581013140, 1253768516, 1476304896, 2114464203, 8589869056

Offset

1,1

Comments

Composite numbers k for which A342926(k) = p*A003415(k), for some prime p.

Corresponding prime p for the first 25 terms is: 2, 2, 3, 2, 3, 3, 3, 11, 2, 11, 2, 3, 3, 5, 2, 2, 101, 397, 2, 3, 5, 7, 3, 5, 2. - Antti Karttunen, Feb 25 2022

Links

Table of n, a(n) for n=1..25.

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

Mathematica

Block[{f}, f[n_] := If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Select[Range[4, 10^6], And[CompositeQ[#], PrimeQ[(f[DivisorSigma[1, #]] - #)/f[#] ]] &]] (* Michael De Vlieger, Apr 08 2021 *)

Prog

(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));
isA342924(n) = if((n<2)||isprime(n), 0, my(q=(A342925(n)-n)/A003415(n)); ((1==denominator(q))&&isprime(q)));

Crossrefs

Odd terms in this sequence form a subsequence of A347884.

Cf. A342925, A342926.

Cf. A000396, A005820, A046060, A065997 (subsequences).

Cf. also A342922, A342923, A007691.

Sequence in context: A055715 A026031 A002694 * A007691 A348031 A260508

Adjacent sequences: A342921 A342922 A342923 * A342925 A342926 A342927

Keyword

nonn,more

Author

Antti Karttunen, Apr 08 2021

Extensions

Terms a(21) - a(25) from Antti Karttunen, Feb 25 2022

Status

approved