OFFSET
1,2
COMMENTS
Sequence is identical to A048751 except for terms 1 and 2 that are included here. - Michel Marcus, Jun 06 2014
Numbers k such that tau(k) = A000005(k) divides the product of the divisors of k (A007955). - Jaroslav Krizek, Sep 05 2017
Based on empirical data up to 10^7 terms in the sequence, it appears that the natural density of this sequence is approximately 0.351. a(10) = 22, a(10^2) = 276, a(10^3) = 2810, a(10^4) = 28410, a(10^5) = 284742, a(10^6) = 2848546, and a(10^7) = 28485598. This gives natural densities of 0.454545, 0.362319, 0.355872, 0.351989, 0.351195, 0.351056, and 0.351055, respectively. This is unlike the similar sequence A033950, which has a density of 0 (Kennedy and Cooper, 1990). - Abingdon Apel, Jun 07 2026
LINKS
Paolo P. Lava, Table of n, a(n) for n = 1..10000
Robert E. Kennedy and Curtis N. Cooper, Tau numbers, natural density and Hardy and Wright's Theorem 437, International Journal of Mathematics and Mathematical Sciences, Vol. 13, No. 2 (1990), pp. 383-386.
EXAMPLE
d(26) = 4. 2 is the only prime dividing 4. 2 divides 26, so 26 is in the sequence.
MAPLE
isA120736 := proc(n) local d, p; d := numtheory[tau](n) ; p := 2 ; while p <= n do if ( d mod p ) = 0 then if ( n mod p ) <> 0 then RETURN(false) ; fi ; fi ; p := nextprime(p) ; od ; RETURN(true) ; end: for n from 1 to 200 do if isA120736(n) then printf("%d, ", n) ; fi ; od ; # R. J. Mathar, Aug 17 2006
MATHEMATICA
Select[Range@ 168, Divisible[Times @@ Divisors@ #, DivisorSigma[0, #]] &] (* Michael De Vlieger, Sep 05 2017 *)
PROG
(Magma) [n: n in [1..1000] | Denominator(&*[d: d in Divisors(n)] / #[d: d in Divisors(n)]) eq 1]; // Jaroslav Krizek, Sep 05 2017
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Leroy Quet, Jun 29 2006
EXTENSIONS
More terms from R. J. Mathar, Aug 17 2006
STATUS
approved