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

Author

Leroy Quet, Jun 29 2006

Keywords

• nonn

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

Examples

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

Links

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

Crossrefs

Cf. A000005, A070226, A007947, A120736.

Programs

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 *)

PARI

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

Extensions

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

Name simplified by Jon E. Schoenfield, Mar 03 2019