

A076610


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


45



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
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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. ith 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/(p1) 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

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


FORMULA

Sum_{n>=1} 1/a(n) = Product_{p in A006450} p/(p1) 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 *)


CROSSREFS

Cf. A000040, A006450, A064988.
Sequence in context: A120696 A071156 A274212 * A069205 A319987 A319985
Adjacent sequences: A076607 A076608 A076609 * A076611 A076612 A076613


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Oct 21 2002


STATUS

approved



