OFFSET
1,1
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. Of course, the reverse-alternating sum of prime indices is also the alternating sum of reversed prime indices.
EXAMPLE
The initial terms and their prime indices:
3: {2}
10: {1,3}
12: {1,1,2}
21: {2,4}
27: {2,2,2}
30: {1,2,3}
40: {1,1,1,3}
48: {1,1,1,1,2}
55: {3,5}
70: {1,3,4}
75: {2,3,3}
84: {1,1,2,4}
90: {1,2,2,3}
91: {4,6}
108: {1,1,2,2,2}
120: {1,1,1,2,3}
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[100], sats[primeMS[#]]==2&]
CROSSREFS
Below we use k to indicate reverse-alternating sum.
The k > 0 version is A000037.
These multisets are counted by A000097.
These are the positions of 2's in A344616.
The k = -1 version is A345912.
The k = 1 version is A345958.
A000070 counts partitions with alternating sum 1.
A027187 counts partitions with reverse-alternating sum <= 0.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A316524 gives the alternating sum of prime indices.
A344606 counts alternating permutations of prime indices.
A344610 counts partitions by sum and positive reverse-alternating sum.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 12 2021
STATUS
approved