cp's OEIS Frontend

A325369 Numbers with no two prime exponents appearing the same number of times in the prime signature.

%I A325369 #4 May 02 2019 16:05:03
%S A325369 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,30,31,
%T A325369 32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,60,61,62,
%U A325369 64,65,66,67,69,70,71,73,74,77,78,79,81,82,83,84,85,86
%N A325369 Numbers with no two prime exponents appearing the same number of times in the prime signature.
%C A325369 The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
%C A325369 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 whose multiplicities appear with distinct multiplicities. The enumeration of these partitions by sum is given by A325329.
%e A325369 Most small numbers are in the sequence. However the sequence of non-terms together with their prime indices begins:
%e A325369   12: {1,1,2}
%e A325369   18: {1,2,2}
%e A325369   20: {1,1,3}
%e A325369   24: {1,1,1,2}
%e A325369   28: {1,1,4}
%e A325369   40: {1,1,1,3}
%e A325369   44: {1,1,5}
%e A325369   45: {2,2,3}
%e A325369   48: {1,1,1,1,2}
%e A325369   50: {1,3,3}
%e A325369   52: {1,1,6}
%e A325369   54: {1,2,2,2}
%e A325369   56: {1,1,1,4}
%e A325369   63: {2,2,4}
%e A325369   68: {1,1,7}
%e A325369   72: {1,1,1,2,2}
%e A325369   75: {2,3,3}
%e A325369   76: {1,1,8}
%e A325369   80: {1,1,1,1,3}
%e A325369   88: {1,1,1,5}
%e A325369 For example, the prime indices of 1260 are {1,1,2,2,3,4}, whose multiplicities give the prime signature {1,1,2,2}, and since 1 and 2 appear the same number of times, 1260 is not in the sequence.
%t A325369 Select[Range[100],UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]
%Y A325369 Cf. A056239, A098859, A112798, A118914, A130091, A317090, A319161, A325326, A325329, A325331, A325337, A325370, A325371.
%K A325369 nonn
%O A325369 1,2
%A A325369 _Gus Wiseman_, May 02 2019