A325405 - OEIS

%I #7 Jun 07 2019 16:33:56

%S 1,2,3,5,7,10,11,13,14,15,17,19,22,23,26,29,31,33,34,35,37,38,39,41,

%T 43,46,47,51,53,55,57,58,59,61,62,67,69,71,73,74,77,79,82,83,85,86,87,

%U 89,91,93,94,95,97,101,103,106,107,109,111,113,115,118,119,122

%N Heinz numbers of integer partitions y such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

%C First differs from A325388 in lacking 130.

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

%C The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).

%C The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.

%C The enumeration of these partitions by sum is given by A325404.

%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    10: {1,3}

%e    11: {5}

%e    13: {6}

%e    14: {1,4}

%e    15: {2,3}

%e    17: {7}

%e    19: {8}

%e    22: {1,5}

%e    23: {9}

%e    26: {1,6}

%e    29: {10}

%e    31: {11}

%e    33: {2,5}

%e    34: {1,7}

%e    35: {3,4}

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

%t Select[Range[100],UnsameQ@@Join@@Table[Differences[primeMS[#],k],{k,0,PrimeOmega[#]}]&]

%Y A subsequence of A005117.

%Y Cf. A056239, A112798, A279945, A325325, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325404, A325406, A325467.

%K nonn

%O 1,2

%A _Gus Wiseman_, May 02 2019