|
|
A341447
|
|
Heinz numbers of integer partitions whose only even part is the smallest.
|
|
3
|
|
|
3, 7, 13, 15, 19, 29, 33, 37, 43, 51, 53, 61, 69, 71, 75, 77, 79, 89, 93, 101, 107, 113, 119, 123, 131, 139, 141, 151, 161, 163, 165, 173, 177, 181, 193, 199, 201, 217, 219, 221, 223, 229, 239, 249, 251, 255, 263, 271, 281, 287, 291, 293, 299, 309, 311, 317
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are numbers whose only even prime index (counting multiplicity) is the smallest.
|
|
LINKS
|
|
|
EXAMPLE
|
The sequence of partitions together with their Heinz numbers begins:
3: (2) 77: (5,4) 165: (5,3,2)
7: (4) 79: (22) 173: (40)
13: (6) 89: (24) 177: (17,2)
15: (3,2) 93: (11,2) 181: (42)
19: (8) 101: (26) 193: (44)
29: (10) 107: (28) 199: (46)
33: (5,2) 113: (30) 201: (19,2)
37: (12) 119: (7,4) 217: (11,4)
43: (14) 123: (13,2) 219: (21,2)
51: (7,2) 131: (32) 221: (7,6)
53: (16) 139: (34) 223: (48)
61: (18) 141: (15,2) 229: (50)
69: (9,2) 151: (36) 239: (52)
71: (20) 161: (9,4) 249: (23,2)
75: (3,3,2) 163: (38) 251: (54)
|
|
MATHEMATICA
|
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], EvenQ[First[primeMS[#]]]&&And@@OddQ[Rest[primeMS[#]]]&]
|
|
CROSSREFS
|
These partitions are counted by A087897, shifted left once.
Terms of A340933 can be factored into elements of this sequence.
A026805 counts partitions whose least part is even, ranked by A340933.
A061395 selects greatest prime index.
A066207 lists numbers with all even prime indices.
A112798 lists the prime indices of each positive integer.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|