OFFSET
1,2
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
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).
The enumeration of these partitions by sum is given by A129654.
LINKS
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
11: {5}
13: {6}
15: {2,3}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
29: {10}
31: {11}
32: {1,1,1,1,1}
37: {12}
41: {13}
43: {14}
MATHEMATICA
primeptn[n_]:=If[n==1, {}, Reverse[Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]];
aug[y_]:=Table[If[i<Length[y], y[[i]]-y[[i+1]]+1, y[[i]]], {i, Length[y]}];
Select[Range[100], And@@Table[SameQ@@Differences[aug[primeptn[#]], k], {k, 0, PrimeOmega[#]}]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 03 2019
STATUS
approved