login
A076610
Numbers having only prime factors of form prime(prime); a(1)=1.
117
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
1,2
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)
LINKS
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
EXAMPLE
99 = 11*3*3 = A000040(A000040(3))*A000040(A000040(1))^2, therefore 99 is a term.
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 *)
PROG
(PARI) isok(k) = my(f = factor(k)[, 1]); sum(i=1, #f, isprime(primepi(f[i]))) == #f; \\ Michel Marcus, Sep 16 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Oct 21 2002
STATUS
approved