login
A316153
Heinz numbers of integer partitions of prime numbers into a prime number of prime parts.
2
15, 33, 45, 93, 153, 177, 275, 327, 369, 405, 425, 537, 603, 605, 775, 831, 891, 1025, 1059, 1125, 1413, 1445, 1475, 1641, 1705, 1719, 1761, 2057, 2075, 2319, 2511, 2577, 2979, 3175, 3179, 3189, 3459, 3485, 3603, 3609, 3825, 3925, 4299, 4475, 4497, 4565, 4581
OFFSET
1,1
COMMENTS
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
Sequence of integer partitions of prime numbers into a prime number of prime parts, preceded by their Heinz numbers, begins:
15: (3,2)
33: (5,2)
45: (3,2,2)
93: (11,2)
153: (7,2,2)
177: (17,2)
275: (5,3,3)
327: (29,2)
369: (13,2,2)
405: (3,2,2,2,2)
425: (7,3,3)
537: (41,2)
603: (19,2,2)
605: (5,5,3)
775: (11,3,3)
831: (59,2)
891: (5,2,2,2,2)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], And[PrimeQ[PrimeOmega[#]], PrimeQ[Total[primeMS[#]]], And@@PrimeQ/@primeMS[#]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 25 2018
STATUS
approved