A325130 - OEIS

%I #19 Jan 09 2021 04:48:19

%S 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 37,39,40,41,43,44,47,48,49,51,52,53,55,56,57,59,60,61,64,65,67,68,69,

%U 71,73,75,76,77,79,80,81,83,84,85,87,88,89,91,92,93,95,96

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

%C 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 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 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 Alois P. Heinz, <a href="/A325130/b325130.txt">Table of n, a(n) for n = 1..10000</a>

%e The sequence of terms together with their prime indices begins:

%e    1: {}

%e    3: {2}

%e    4: {1,1}

%e    5: {3}

%e    7: {4}

%e    8: {1,1,1}

%e   11: {5}

%e   12: {1,1,2}

%e   13: {6}

%e   15: {2,3}

%e   16: {1,1,1,1}

%e   17: {7}

%e   19: {8}

%e   20: {1,1,3}

%e   21: {2,4}

%e   23: {9}

%e   24: {1,1,1,2}

%e   25: {3,3}

%e   27: {2,2,2}

%e   28: {1,1,4}

%p q:= n-> andmap(i-> numtheory[pi](i[1])<>i[2], ifactors(n)[2]):

%p a:= proc(n) option remember; local k; for k from 1+

%p      `if`(n=1, 0, a(n-1)) while not q(k) do od; k

%p     end:

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Oct 28 2019

%t Select[Range[100],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k!=PrimePi[p]]&]

%Y Cf. A056239, A087153, A112798, A124010, A276078, A276429.

%Y Cf. A324525, A324571, A325127, A325128, A325130, A325131.

%Y Complement of A276936.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 01 2019