cp's OEIS Frontend

A325130 Numbers in whose prime factorization the exponent of prime(k) is not equal to k for any prime index k.

%I A325130 #19 Jan 09 2021 04:48:19
%S A325130 1,3,4,5,7,8,11,12,13,15,16,17,19,20,21,23,24,25,27,28,29,31,32,33,35,
%T A325130 37,39,40,41,43,44,47,48,49,51,52,53,55,56,57,59,60,61,64,65,67,68,69,
%U A325130 71,73,75,76,77,79,80,81,83,84,85,87,88,89,91,92,93,95,96
%N A325130 Numbers in whose prime factorization the exponent of prime(k) is not equal to k for any prime index k.
%C A325130 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A325130 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of the integer partitions counted by A276429.
%C A325130 The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k + 1/prime(k)^(k+1)) = 0.68974964705635552968... - _Amiram Eldar_, Jan 09 2021
%H A325130 Alois P. Heinz, <a href="/A325130/b325130.txt">Table of n, a(n) for n = 1..10000</a>
%e A325130 The sequence of terms together with their prime indices begins:
%e A325130    1: {}
%e A325130    3: {2}
%e A325130    4: {1,1}
%e A325130    5: {3}
%e A325130    7: {4}
%e A325130    8: {1,1,1}
%e A325130   11: {5}
%e A325130   12: {1,1,2}
%e A325130   13: {6}
%e A325130   15: {2,3}
%e A325130   16: {1,1,1,1}
%e A325130   17: {7}
%e A325130   19: {8}
%e A325130   20: {1,1,3}
%e A325130   21: {2,4}
%e A325130   23: {9}
%e A325130   24: {1,1,1,2}
%e A325130   25: {3,3}
%e A325130   27: {2,2,2}
%e A325130   28: {1,1,4}
%p A325130 q:= n-> andmap(i-> numtheory[pi](i[1])<>i[2], ifactors(n)[2]):
%p A325130 a:= proc(n) option remember; local k; for k from 1+
%p A325130      `if`(n=1, 0, a(n-1)) while not q(k) do od; k
%p A325130     end:
%p A325130 seq(a(n), n=1..80);  # _Alois P. Heinz_, Oct 28 2019
%t A325130 Select[Range[100],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k!=PrimePi[p]]&]
%Y A325130 Cf. A056239, A087153, A112798, A124010, A276078, A276429.
%Y A325130 Cf. A324525, A324571, A325127, A325128, A325130, A325131.
%Y A325130 Complement of A276936.
%K A325130 nonn
%O A325130 1,2
%A A325130 _Gus Wiseman_, Apr 01 2019