A325369 - OEIS

%I #4 May 02 2019 16:05:03

%S 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 32,33,34,35,36,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,60,61,62,

%U 64,65,66,67,69,70,71,73,74,77,78,79,81,82,83,84,85,86

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

%C The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.

%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 integer partitions whose multiplicities appear with distinct multiplicities. The enumeration of these partitions by sum is given by A325329.

%e Most small numbers are in the sequence. However the sequence of non-terms together with their prime indices begins:

%e   12: {1,1,2}

%e   18: {1,2,2}

%e   20: {1,1,3}

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

%e   28: {1,1,4}

%e   40: {1,1,1,3}

%e   44: {1,1,5}

%e   45: {2,2,3}

%e   48: {1,1,1,1,2}

%e   50: {1,3,3}

%e   52: {1,1,6}

%e   54: {1,2,2,2}

%e   56: {1,1,1,4}

%e   63: {2,2,4}

%e   68: {1,1,7}

%e   72: {1,1,1,2,2}

%e   75: {2,3,3}

%e   76: {1,1,8}

%e   80: {1,1,1,1,3}

%e   88: {1,1,1,5}

%e 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 Select[Range[100],UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]

%Y Cf. A056239, A098859, A112798, A118914, A130091, A317090, A319161, A325326, A325329, A325331, A325337, A325370, A325371.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 02 2019