A357629 Half-alternating sum of the prime indices of n.

0, 1, 2, 2, 3, 3, 4, 1, 4, 4, 5, 0, 6, 5, 5, 0, 7, 1, 8, -1, 6, 6, 9, -1, 6, 7, 2, -2, 10, 0, 11, 1, 7, 8, 7, -2, 12, 9, 8, -2, 13, -1, 14, -3, 1, 10, 15, 2, 8, 1, 9, -4, 16, -1, 8, -3, 10, 11, 17, -3, 18, 12, 0, 2, 9, -2, 19, -5, 11, 0, 20, 1, 21, 13, 2, -6

Offset

1,3

Comments

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

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.

Links

Table of n, a(n) for n=1..76.

Example

The prime indices of 525 are {2,3,3,4} so a(525) = 2 + 3 - 3 - 4 = -2.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Table[halfats[primeMS[n]], {n, 30}]

Crossrefs

The original alternating sum is A316524, reverse A344616.

The version for standard compositions is A357621, reverse A357622.

The skew-alternating form is A357630, reverse A357634.

Positions of zeros are A357631, reverse A357635.

The reverse version is A357633.

These partitions are counted by A357637, skew A357638.

A056239 adds up prime indices, row sums of A112798.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357641 counts comps w/ half-alt sum 0, partitions A357639, even A357642.

Cf. A003963, A053251, A055932, A357189, A357485-A357488, A357623-A357626, A357632, A357636, A357640.

Sequence in context: A071505 A071508 A326031 * A357633 A322997 A367315

Adjacent sequences: A357626 A357627 A357628 * A357630 A357631 A357632

Keyword

sign

Author

Gus Wiseman, Oct 08 2022

Status

approved