OFFSET
1,1
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
See A257989 or the program for a definition of crank of a partition.
EXAMPLE
The sequence of terms together with their prime indices begins:
6: {1,2} 106: {1,16} 218: {1,29}
10: {1,3} 118: {1,17} 220: {1,1,3,5}
14: {1,4} 122: {1,18} 226: {1,30}
22: {1,5} 134: {1,19} 254: {1,31}
26: {1,6} 140: {1,1,3,4} 260: {1,1,3,6}
34: {1,7} 142: {1,20} 262: {1,32}
38: {1,8} 146: {1,21} 274: {1,33}
46: {1,9} 158: {1,22} 278: {1,34}
58: {1,10} 166: {1,23} 298: {1,35}
62: {1,11} 178: {1,24} 300: {1,1,2,3,3}
74: {1,12} 194: {1,25} 302: {1,36}
82: {1,13} 196: {1,1,4,4} 308: {1,1,4,5}
86: {1,14} 202: {1,26} 314: {1,37}
94: {1,15} 206: {1,27} 326: {1,38}
100: {1,1,3,3} 214: {1,28} 334: {1,39}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ck[y_]:=With[{w=Count[y, 1]}, If[w==0, Max@@y, Count[y, _?(#>w&)]-w]];
Select[Range[100], ck[primeMS[#]]==0&]
CROSSREFS
Indices of zeros in A257989.
A000005 counts constant partitions.
A001522 counts partitions of positive crank.
A003242 counts anti-run compositions.
A064391 counts partitions by crank.
A064428 counts partitions of nonnegative crank.
A224958 counts compositions with alternating parts unequal.
A257989 gives the crank of the partition with Heinz number n.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Apr 05 2021
STATUS
approved