proposed

approved

editing

proposed

Cf. A000070, A025047, A088218, A120452, A341446, A344614, A344617, A344653, A344654, A345957, A345958, A345959.

A344617 gives the sign of a(n) - A346698(n).

A000120 (ones) and A080791 (zeros) count binary digits, 1 and 0, with difference A145037.

A344617 gives the sign of a(n) - A346698(n) (alternating sum of prime indices).

Cf. `A000070, ~A000097 ptns_altsum_2, A025047, ~A035363, A088218, `A097805, `A120452, A341446, ~A343941 strptns_ev_sats4, `A344604 alt_comps, `A344605 alt_normseq, A344614, A344653, `A344654 ptn_no_alt_perm, `A344741, A345957, A345958, A345959, ~A345960 prix_ats_2, ~A345961 prix_sats_2, ~A345962 prix_ats_neg2.

Cf. A000070, A025047, A088218, A120452, A341446, A344614, A344617, A344653, A344654, A345957, A345958, A345959.

allocated for Gus WisemanSum of the odd-indexed parts (odd bisection) of the multiset of prime indices of n.

0, 1, 2, 1, 3, 1, 4, 2, 2, 1, 5, 3, 6, 1, 2, 2, 7, 3, 8, 4, 2, 1, 9, 2, 3, 1, 4, 5, 10, 4, 11, 3, 2, 1, 3, 3, 12, 1, 2, 2, 13, 5, 14, 6, 5, 1, 15, 4, 4, 4, 2, 7, 16, 3, 3, 2, 2, 1, 17, 3, 18, 1, 6, 3, 3, 6, 19, 8, 2, 5, 20, 4, 21, 1, 5, 9, 4, 7, 22, 5, 4, 1

1,3

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.

a(n) = A056239(n) - A346698(n).

a(n) = A316524(n) + A346698(n).

a(n odd omega) = A346699(n).

a(n even omega) = A346700(n).

A344616(n) = A346699(n) - A346700(n).

The prime indices of 1100 are {1,1,3,3,5}, so a(1100) = 1 + 3 + 5 = 9.

The prime indices of 2100 are {1,1,2,3,3,4}, so a(2100) = 1 + 2 + 3 = 6.

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Total[First/@Partition[Append[primeMS[n], 0], 2]], {n, 100}]

The version for standard compositions is A209281(n+1) (even: A346633).

Subtracting the even version gives A316524 (reverse: A344616).

The even version is A346698.

The reverse version is A346699.

The even reverse version is A346700.

A000120 (ones) and A080791 (zeros) count binary digits, with difference A145037.

A000302 counts compositions with odd alternating sum, ranked by A053738.

A001414 adds up prime factors, row sums of A027746.

A029837 adds up parts of standard compositions (alternating: A124754).

A056239 adds up prime indices, row sums of A112798.

A103919 counts partitions by sum and alternating sum (reverse: A344612).

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344606 counts alternating permutations of prime indices.

A344617 gives the sign of a(n)-A346698(n) (alternating sum of prime indices).

Cf. `A000070, ~A000097 ptns_altsum_2, A025047, ~A035363, A088218, `A097805, `A120452, A341446, ~A343941 strptns_ev_sats4, `A344604 alt_comps, `A344605 alt_normseq, A344614, A344653, `A344654 ptn_no_alt_perm, `A344741, A345957, A345958, A345959, ~A345960 prix_ats_2, ~A345961 prix_sats_2, ~A345962 prix_ats_neg2.

allocated

nonn

Gus Wiseman, Aug 01 2021

approved

editing

allocating

allocated

allocated for Gus Wiseman

allocating

approved