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A325400
Heinz numbers of reversed integer partitions whose k-th differences are weakly increasing for all k >= 0.
10
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74
OFFSET
1,2
COMMENTS
First differs from A109427 in lacking 54.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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).
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.
The enumeration of these partitions by sum is given by A325354.
EXAMPLE
Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
18: {1,2,2}
36: {1,1,2,2}
50: {1,3,3}
54: {1,2,2,2}
60: {1,1,2,3}
70: {1,3,4}
72: {1,1,1,2,2}
75: {2,3,3}
90: {1,2,2,3}
98: {1,4,4}
100: {1,1,3,3}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
126: {1,2,2,4}
140: {1,1,3,4}
144: {1,1,1,1,2,2}
147: {2,4,4}
150: {1,2,3,3}
154: {1,4,5}
162: {1,2,2,2,2}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], And@@Table[Greater@@Differences[primeptn[#], k], {k, 0, PrimeOmega[#]}]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 02 2019
STATUS
approved