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%I #30 Sep 16 2022 12:27:39
%S 1,3,5,9,11,15,17,25,27,31,33,41,45,51,55,59,67,75,81,83,85,93,99,109,
%T 121,123,125,127,135,153,155,157,165,177,179,187,191,201,205,211,225,
%U 241,243,249,255,275,277,279,283,289,295,297,327,331,335,341,353,363
%N Numbers having only prime factors of form prime(prime); a(1)=1.
%C 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
%C Multiplicative closure of A006450.
%C Sequence A064988 sorted into ascending order. - _Antti Karttunen_, Aug 08 2017
%C From _David A. Corneth_, Sep 28 2020: (Start)
%C Product_{p in A006450} p/(p-1) where primepi(p) <= 10^k for k = 3..10 respectively is
%C 2.7609365004752546...
%C 2.8489587563778631...
%C 2.9038201166664191...
%C 2.9413699333962213...
%C 2.9687172228411300...
%C 2.9895324403761206...
%C 3.0059192857697702...
%C 3.0191633206253085... (End)
%H Amiram Eldar, <a href="/A076610/b076610.txt">Table of n, a(n) for n = 1..10000</a>
%F 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
%e 99 = 11*3*3 = A000040(A000040(3))*A000040(A000040(1))^2, therefore 99 is a term.
%p 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
%t {1}~Join~Select[Range@ 400, AllTrue[PrimePi@ First@ Transpose@ FactorInteger@ #, PrimeQ] &] (* _Michael De Vlieger_, May 09 2015, Version 10 *)
%o (PARI) isok(k) = my(f = factor(k)[,1]); sum(i=1, #f, isprime(primepi(f[i]))) == #f; \\ _Michel Marcus_, Sep 16 2022
%Y Cf. A000040, A006450, A064988.
%K nonn
%O 1,2
%A _Reinhard Zumkeller_, Oct 21 2002
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