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A325327 Heinz numbers of multiples of triangular partitions, or finite arithmetic progressions with offset 0.

%I A325327 #14 May 25 2019 05:44:48
%S A325327 1,2,3,5,6,7,11,13,17,19,21,23,29,30,31,37,41,43,47,53,59,61,65,67,71,
%T A325327 73,79,83,89,97,101,103,107,109,113,127,131,133,137,139,149,151,157,
%U A325327 163,167,173,179,181,191,193,197,199,210,211,223,227,229,233,239
%N A325327 Heinz numbers of multiples of triangular partitions, or finite arithmetic progressions with offset 0.
%C A325327 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A325327 Also numbers of the form Product_{k = 1..b} prime(k * c) for some b >= 0 and c > 0.
%C A325327 The enumeration of these partitions by sum is given by A007862.
%H A325327 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts</a>.
%e A325327 The sequence of terms together with their prime indices begins:
%e A325327     1: {}
%e A325327     2: {1}
%e A325327     3: {2}
%e A325327     5: {3}
%e A325327     6: {1,2}
%e A325327     7: {4}
%e A325327    11: {5}
%e A325327    13: {6}
%e A325327    17: {7}
%e A325327    19: {8}
%e A325327    21: {2,4}
%e A325327    23: {9}
%e A325327    29: {10}
%e A325327    30: {1,2,3}
%e A325327    31: {11}
%e A325327    37: {12}
%e A325327    41: {13}
%e A325327    43: {14}
%e A325327    47: {15}
%e A325327    53: {16}
%t A325327 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A325327 Select[Range[100],SameQ@@Differences[Append[primeptn[#],0]]&]
%Y A325327 Cf. A000961, A007294, A007862, A049988, A056239, A112798, A130091, A289509, A307824, A325328, A325367, A325390, A325407.
%K A325327 nonn
%O A325327 1,2
%A A325327 _Gus Wiseman_, Apr 23 2019