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%I #12 May 31 2019 05:33:26
%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 32,33,34,35,37,38,39,41,43,46,47,49,51,53,55,57,58,59,61,62,64,65,67,
%U 69,71,73,74,77,79,81,82,83,85,86,87,89,91,93,94,95,97
%N Heinz numbers of finite arithmetic progressions (integer partitions with equal differences).
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C The enumeration of these partitions by sum is given by A049988.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression.</a>
%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%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 36: {1,1,2,2}
%e 40: {1,1,1,3}
%e 42: {1,2,4}
%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 60: {1,1,2,3}
%e 63: {2,2,4}
%e 66: {1,2,5}
%e 68: {1,1,7}
%e 70: {1,3,4}
%e For example, 60 is the Heinz number of (3,2,1,1), which has differences (-1,-1,0), which are not equal, so 60 does not belong to the sequence.
%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t Select[Range[100],SameQ@@Differences[primeptn[#]]&]
%Y Cf. A000961, A007862, A049988, A056239, A112798, A130091, A240026, A289509, A307824, A325327, A325352, A325368, A325405, A325407.
%K nonn
%O 1,2
%A _Gus Wiseman_, Apr 23 2019
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