cp's OEIS Frontend

A325371 Numbers whose prime signature has multiplicities of its parts all distinct and covering an initial interval of positive integers.

%I A325371 #11 Nov 30 2020 10:57:14
%S A325371 1,2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43,47,49,53,
%T A325371 59,60,61,64,67,71,73,79,81,83,84,89,90,97,101,103,107,109,113,120,
%U A325371 121,125,126,127,128,131,132,137,139,140,149,150,151,156,157,163
%N A325371 Numbers whose prime signature has multiplicities of its parts all distinct and covering an initial interval of positive integers.
%C A325371 The first term that is not 1 or a prime power is 60.
%C A325371 The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
%C A325371 Numbers whose prime signature has distinct parts that cover an initial interval are given by A325337.
%C A325371 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 that cover an initial interval of positive integers. The enumeration of these partitions by sum is given by A325331.
%e A325371 The sequence of terms together with their prime indices begins:
%e A325371     1: {}
%e A325371     2: {1}
%e A325371     3: {2}
%e A325371     4: {1,1}
%e A325371     5: {3}
%e A325371     7: {4}
%e A325371     8: {1,1,1}
%e A325371     9: {2,2}
%e A325371    11: {5}
%e A325371    13: {6}
%e A325371    16: {1,1,1,1}
%e A325371    17: {7}
%e A325371    19: {8}
%e A325371    23: {9}
%e A325371    25: {3,3}
%e A325371    27: {2,2,2}
%e A325371    29: {10}
%e A325371    31: {11}
%e A325371    32: {1,1,1,1,1}
%e A325371    37: {12}
%t A325371 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A325371 Select[Range[100],normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&&UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]
%Y A325371 Cf. A055932, A056239, A098859, A112798, A118914, A130091, A317090, A325329, A325330, A325331, A325337, A325369, A325370.
%K A325371 nonn
%O A325371 1,2
%A A325371 _Gus Wiseman_, May 02 2019