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A349160
Numbers whose sum of prime indices is twice their reverse-alternating sum.
7
1, 10, 12, 39, 63, 66, 88, 112, 115, 190, 228, 255, 259, 306, 325, 408, 434, 468, 517, 544, 609, 620, 783, 793, 805, 832, 870, 931, 946, 1150, 1160, 1204, 1241, 1242, 1353, 1380, 1392, 1534, 1539, 1656, 1691, 1722, 1845, 1900, 2035, 2067, 2208, 2296, 2369
OFFSET
1,2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are also Heinz numbers of partitions whose sum is twice their reverse-alternating sum.
FORMULA
A056239(a(n)) = 2*A344616(a(n)).
A346700(a(n)) = 3*A346699(a(n)).
EXAMPLE
The terms and their prime indices begin:
1: ()
10: (3,1)
12: (2,1,1)
39: (6,2)
63: (4,2,2)
66: (5,2,1)
88: (5,1,1,1)
112: (4,1,1,1,1)
115: (9,3)
190: (8,3,1)
228: (8,2,1,1)
255: (7,3,2)
259: (12,4)
306: (7,2,2,1)
325: (6,3,3)
408: (7,2,1,1,1)
434: (11,4,1)
468: (6,2,2,1,1)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Select[Range[1000], Total[primeMS[#]]==2*sats[primeMS[#]]&]
CROSSREFS
These partitions are counted by A006330 up to 0's.
The negative reverse version is A348617.
An ordered version is A349153, non-reverse A348614.
The non-reverse version is A349159.
A027193 counts partitions with rev-alt sum > 0, ranked by A026424.
A034871, A097805, A345197 count compositions by alternating sum.
A056239 adds up prime indices, row sums of A112798, row lengths A001222.
A103919 counts partitions by alternating sum, reverse A344612.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A346697 adds up odd-indexed prime indices.
A346698 adds up even-indexed prime indices.
Sequence in context: A324745 A367148 A267393 * A248481 A266700 A242508
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 25 2021
STATUS
approved