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A325327
Heinz numbers of multiples of triangular partitions, or finite arithmetic progressions with offset 0.
14
1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 133, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 210, 211, 223, 227, 229, 233, 239
OFFSET
1,2
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers of the form Product_{k = 1..b} prime(k * c) for some b >= 0 and c > 0.
The enumeration of these partitions by sum is given by A007862.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
5: {3}
6: {1,2}
7: {4}
11: {5}
13: {6}
17: {7}
19: {8}
21: {2,4}
23: {9}
29: {10}
30: {1,2,3}
31: {11}
37: {12}
41: {13}
43: {14}
47: {15}
53: {16}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Select[Range[100], SameQ@@Differences[Append[primeptn[#], 0]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 23 2019
STATUS
approved