cp's OEIS Frontend

A325128 Numbers in whose prime factorization the exponent of prime(k) is less than k for all prime indices k.

%I A325128 #10 Feb 02 2021 04:35:45
%S A325128 1,3,5,7,11,13,15,17,19,21,23,25,29,31,33,35,37,39,41,43,47,49,51,53,
%T A325128 55,57,59,61,65,67,69,71,73,75,77,79,83,85,87,89,91,93,95,97,101,103,
%U A325128 105,107,109,111,113,115,119,121,123,127,129,131,133,137,139,141
%N A325128 Numbers in whose prime factorization the exponent of prime(k) is less than k for all prime indices k.
%C A325128 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 A325128 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions where each part k appears fewer than k times. Such partitions are counted by A087153.
%C A325128 The asymptotic density of this sequence is Product_{k>=1} (1 - 1/prime(k)^k) = 0.44070243286030291209... - _Amiram Eldar_, Feb 02 2021
%H A325128 Amiram Eldar, <a href="/A325128/b325128.txt">Table of n, a(n) for n = 1..10000</a>
%e A325128 The sequence of terms together with their prime indices begins:
%e A325128    1: {}
%e A325128    3: {2}
%e A325128    5: {3}
%e A325128    7: {4}
%e A325128   11: {5}
%e A325128   13: {6}
%e A325128   15: {2,3}
%e A325128   17: {7}
%e A325128   19: {8}
%e A325128   21: {2,4}
%e A325128   23: {9}
%e A325128   25: {3,3}
%e A325128   29: {10}
%e A325128   31: {11}
%e A325128   33: {2,5}
%e A325128   35: {3,4}
%e A325128   37: {12}
%e A325128   39: {2,6}
%e A325128   41: {13}
%e A325128   43: {14}
%e A325128   47: {15}
%e A325128   49: {4,4}
%t A325128 Select[Range[100],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k<PrimePi[p]]&]
%Y A325128 Cf. A056239, A062457, A087153, A109298, A112798, A115584, A118914, A276078.
%Y A325128 Cf. A324524, A324525, A324571, A325127, A325130.
%K A325128 nonn
%O A325128 1,2
%A A325128 _Gus Wiseman_, Apr 01 2019