%I #43 Feb 26 2024 01:54:48
%S 1,2,8,9,12,18,32,72,80,96,108,128,243,288,448,486,512,625,720,768,
%T 864,972,1152,1200,1250,1620,1944,2000,2025,2048,2560,2592,3888,4032,
%U 4050,4608,5000,5625,6144,6561,6912,7500,7776,8192,8748,9408,10800,11250
%N Numbers k whose number of divisors d(k) is divisible by every prime factor of k.
%C 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
%C 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
%H Amiram Eldar, <a href="/A120737/b120737.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..100 from Paolo P. Lava)
%e d(32) = 6. 2 is the only prime dividing 32. 2 divides 6, so 32 is in the sequence.
%p 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 ;
%p # _R. J. Mathar_, Aug 17 2006
%t divQ[n_] := AllTrue[FactorInteger[n][[;; , 1]], Divisible[DivisorSigma[0, n], #] &]; Select[Range[10^4], divQ] (* _Amiram Eldar_, Nov 08 2020 *)
%o (PARI) isok(k) = Mod(numdiv(k), k)^eulerphi(k) == 0; \\ _Michel Marcus_, May 11 2019
%Y Cf. A000005, A070226, A007947, A120736.
%K nonn
%O 1,2
%A _Leroy Quet_, Jun 29 2006
%E More terms from _R. J. Mathar_, Aug 17 2006
%E Name simplified by _Jon E. Schoenfield_, Mar 03 2019