A325395 - OEIS

%I #4 May 03 2019 08:37:05

%S 1,2,3,5,7,9,11,13,17,19,23,25,29,31,35,37,41,43,47,49,53,59,61,67,71,

%T 73,77,79,83,89,91,97,101,103,107,109,113,121,127,131,137,139,143,149,

%U 151,157,163,167,169,173,179,181,187,191,193,197,199,209,211,221

%N Heinz numbers of integer partitions whose augmented differences are strictly increasing.

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%C 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 The enumeration of these partitions by sum is given by A325357.

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e The sequence of terms together with their prime indices begins:

%e     1: {}

%e     2: {1}

%e     3: {2}

%e     5: {3}

%e     7: {4}

%e     9: {2,2}

%e    11: {5}

%e    13: {6}

%e    17: {7}

%e    19: {8}

%e    23: {9}

%e    25: {3,3}

%e    29: {10}

%e    31: {11}

%e    35: {3,4}

%e    37: {12}

%e    41: {13}

%e    43: {14}

%e    47: {15}

%e    49: {4,4}

%t primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t aug[y_]:=Table[If[i<Length[y],y[[i]]-y[[i+1]]+1,y[[i]]],{i,Length[y]}];

%t Select[Range[100],Less@@aug[primeptn[#]]&]

%Y Cf. A056239, A093641, A112798, A240027, A325351, A325357, A325366, A325389, A325394, A325396, A325398, A325456, A325460.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 02 2019