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A307824 Heinz numbers of integer partitions whose augmented differences are all equal.

%I A307824 #18 May 03 2019 21:25:14
%S A307824 1,2,3,4,5,7,8,11,13,15,16,17,19,23,29,31,32,37,41,43,47,53,55,59,61,
%T A307824 64,67,71,73,79,83,89,97,101,103,105,107,109,113,119,127,128,131,137,
%U A307824 139,149,151,157,163,167,173,179,181,191,193,197,199,211,223,227
%N A307824 Heinz numbers of integer partitions whose augmented differences are all equal.
%C A307824 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A307824 The augmented differences aug(y) of an integer partition y of length k are given by aug(y)_i = y_i - y_{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).
%C A307824 The enumeration of these partitions by sum is given by A129654.
%H A307824 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A307824 The sequence of terms together with their prime indices begins:
%e A307824     1: {}
%e A307824     2: {1}
%e A307824     3: {2}
%e A307824     4: {1,1}
%e A307824     5: {3}
%e A307824     7: {4}
%e A307824     8: {1,1,1}
%e A307824    11: {5}
%e A307824    13: {6}
%e A307824    15: {2,3}
%e A307824    16: {1,1,1,1}
%e A307824    17: {7}
%e A307824    19: {8}
%e A307824    23: {9}
%e A307824    29: {10}
%e A307824    31: {11}
%e A307824    32: {1,1,1,1,1}
%e A307824    37: {12}
%e A307824    41: {13}
%e A307824    43: {14}
%t A307824 primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A307824 aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];
%t A307824 Select[Range[100],And@@Table[SameQ@@Differences[aug[primeptn[#]],k],{k,0,PrimeOmega[#]}]&]
%Y A307824 Cf. A049988, A056239, A093641, A112798, A129654, A325327, A325328, A325351, A325359, A325366, A325389, A325394, A325395, A325396.
%K A307824 nonn
%O A307824 1,2
%A A307824 _Gus Wiseman_, May 03 2019