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A120737
Numbers k whose number of divisors d(k) is divisible by every prime factor of k.
6
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
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