cp's OEIS Frontend

A325370 Numbers whose prime signature has multiplicities covering an initial interval of positive integers.

%I A325370 #11 Nov 30 2020 10:57:20
%S A325370 1,2,3,4,5,7,8,9,11,12,13,16,17,18,19,20,23,24,25,27,28,29,31,32,37,
%T A325370 40,41,43,44,45,47,48,49,50,52,53,54,56,59,60,61,63,64,67,68,71,72,73,
%U A325370 75,76,79,80,81,83,84,88,89,90,92,96,97,98,99,101,103,104
%N A325370 Numbers whose prime signature has multiplicities covering an initial interval of positive integers.
%C A325370 First differs from A319161 in lacking 420.
%C A325370 The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
%C A325370 Numbers whose prime signature covers an initial interval are given by A317090.
%C A325370 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 have multiplicities covering an initial interval of positive integers. The enumeration of these partitions by sum is given by A325330.
%e A325370 The sequence of terms together with their prime indices begins:
%e A325370     1: {}
%e A325370     2: {1}
%e A325370     3: {2}
%e A325370     4: {1,1}
%e A325370     5: {3}
%e A325370     7: {4}
%e A325370     8: {1,1,1}
%e A325370     9: {2,2}
%e A325370    11: {5}
%e A325370    12: {1,1,2}
%e A325370    13: {6}
%e A325370    16: {1,1,1,1}
%e A325370    17: {7}
%e A325370    18: {1,2,2}
%e A325370    19: {8}
%e A325370    20: {1,1,3}
%e A325370    23: {9}
%e A325370    24: {1,1,1,2}
%e A325370    25: {3,3}
%e A325370    27: {2,2,2}
%e A325370 For example, the prime indices of 1890 are {1,2,2,2,3,4}, whose multiplicities give the prime signature {1,1,1,3}, and since this does not cover an initial interval (2 is missing), 1890 is not in the sequence.
%t A325370 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A325370 Select[Range[100],normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&]
%Y A325370 Cf. A000009, A055932, A056239, A112798, A118914, A317081, A317089, A317090, A319161, A325326, A325330, A325337, A325369, A325371.
%K A325370 nonn
%O A325370 1,2
%A A325370 _Gus Wiseman_, May 02 2019