A120737 Numbers k whose number of divisors d(k) is divisible by every prime factor of k.

1, 2, 8, 9, 12, 18, 32, 72, 80, 96, 108, 128, 243, 288, 448, 486, 512, 625, 720, 768, 864, 972, 1152, 1200, 1250, 1620, 1944, 2000, 2025, 2048, 2560, 2592, 3888, 4032, 4050, 4608, 5000, 5625, 6144, 6561, 6912, 7500, 7776, 8192, 8748, 9408, 10800, 11250

Offset

1,2

Comments

Numbers k such that A000005(k)/A007947(k) is an integer. A070226 is a subsequence of this sequence. Conjecture: If A000005(k) divides A007947(k) for some k, then A007947(k)/A000005(k)=1. - Ctibor O. Zizka, Feb 05 2009

This sequence contains exactly those positive integers k where 1 is the only positive divisor of k that is coprime to d(k). - Leroy Quet, May 23 2009

Links

Amiram Eldar, Table of n, a(n) for n = 1..1000 (terms 1..100 from Paolo P. Lava)

Example

d(32) = 6. 2 is the only prime dividing 32. 2 divides 6, so 32 is in the sequence.

Maple

isA120737 := proc(n) local d, p; d := numtheory[tau](n) ; p := 2 ; while p <= n do if ( n mod p ) = 0 then if ( d mod p ) <> 0 then RETURN(false) ; fi ; fi ; p := nextprime(p) ; od ; RETURN(true) ; end: for n from 1 to 12000 do if isA120737(n) then printf("%d, ", n) ; fi ; od ;
# R. J. Mathar, Aug 17 2006

Mathematica

divQ[n_] := AllTrue[FactorInteger[n][[;; , 1]], Divisible[DivisorSigma[0, n], #] &]; Select[Range[10^4], divQ] (* Amiram Eldar, Nov 08 2020 *)

Prog

(PARI) isok(k) = Mod(numdiv(k), k)^eulerphi(k) == 0; \\ Michel Marcus, May 11 2019

Crossrefs

Cf. A000005, A070226, A007947, A120736.

Sequence in context: A279373 A057529 A336715 * A081381 A235524 A239133

Adjacent sequences: A120734 A120735 A120736 * A120738 A120739 A120740

Keyword

nonn

Author

Leroy Quet, Jun 29 2006

Extensions

More terms from R. J. Mathar, Aug 17 2006

Name simplified by Jon E. Schoenfield, Mar 03 2019

Status

approved