A076610 Numbers having only prime factors of form prime(prime); a(1)=1.

1, 3, 5, 9, 11, 15, 17, 25, 27, 31, 33, 41, 45, 51, 55, 59, 67, 75, 81, 83, 85, 93, 99, 109, 121, 123, 125, 127, 135, 153, 155, 157, 165, 177, 179, 187, 191, 201, 205, 211, 225, 241, 243, 249, 255, 275, 277, 279, 283, 289, 295, 297, 327, 331, 335, 341, 353, 363

Offset: 1

Author

Reinhard Zumkeller, Oct 21 2002

Keywords

• nonn

Comments

Numbers n such that the partition B(n) has only prime parts. For n>=2, B(n) is defined as the partition obtained by taking the prime decomposition of n and replacing each prime factor p by its index i (i.e. i-th prime = p); also B(1) = the empty partition. For example, B(350) = B(2*5^2*7) = [1,3,3,4]. B is a bijection between the positive integers and the set of all partitions. In the Maple program the command B(n) yields B(n). - Emeric Deutsch, May 09 2015

Multiplicative closure of A006450.

Sequence A064988 sorted into ascending order. - Antti Karttunen, Aug 08 2017

From David A. Corneth, Sep 28 2020: (Start)

Product_{p in A006450} p/(p-1) where primepi(p) <= 10^k for k = 3..10 respectively is

2.7609365004752546...

2.8489587563778631...

2.9038201166664191...

2.9413699333962213...

2.9687172228411300...

2.9895324403761206...

3.0059192857697702...

3.0191633206253085... (End)

Examples

			99 = 11*3*3 = A000040(A000040(3))*A000040(A000040(1))^2, therefore 99 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A006450, A064988.

Programs

Maple

with(numtheory): B := proc (n) local pf: pf := op(2, ifactors(n)): [seq(seq(pi(op(1, op(i, pf))), j = 1 .. op(2, op(i, pf))), i = 1 .. nops(pf))] end proc: S := {}: for r to 400 do s := 0: for t to nops(B(r)) do if isprime(B(r)[t]) = false then s := s+1 else end if end do: if s = 0 then S := `union`(S, {r}) else end if end do: S; # Emeric Deutsch, May 09 2015

Mathematica

{1}~Join~Select[Range@ 400, AllTrue[PrimePi@ First@ Transpose@ FactorInteger@ #, PrimeQ] &] (* Michael De Vlieger, May 09 2015, Version 10 *)

PARI

isok(k) = my(f = factor(k)[,1]); sum(i=1, #f, isprime(primepi(f[i]))) == #f; \\ Michel Marcus, Sep 16 2022

Formula

Sum_{n>=1} 1/a(n) = Product_{p in A006450} p/(p-1) converges since the sum of the reciprocals of A006450 converges. - Amiram Eldar, Sep 27 2020