A345926 Number of distinct possible alternating sums of permutations of the multiset of prime indices of n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 1, 2, 1, 4, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3

Offset

1,6

Comments

First differs from A096825 at a(90) = 3, A096825(90) = 4.

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 alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. Of course, the alternating sum of prime indices is also the reverse-alternating sum of reversed prime indices.

Also the number of possible values of A056239(d) where d is a divisor of n with half as many prime factors (rounded up) as n.

Links

Table of n, a(n) for n=1..100.

Example

Grouping the 12 permutations of {1,2,2,3} by alternating sum k gives:
  k = -2: (1223) (1322) (2213) (2312)
  k =  0: (1232) (2123) (2321) (3212)
  k =  2: (2132) (2231) (3122) (3221)
so a(90) = 3.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]], {i, Length[y]}];
Table[Length[Union[ats/@Permutations[primeMS[n]]]], {n, 100}]

Prog

(Python)
from sympy import factorint, primepi
from sympy.utilities.iterables import multiset_combinations
def A345926(n):
    fs = dict((primepi(a), b) for (a, b) in factorint(n).items())
    return len(set(sum(d) for d in multiset_combinations(fs, (sum(fs.values())+1)//2))) # Chai Wah Wu, Aug 23 2021

Crossrefs

The version for prime factors instead of indices is A343943.

A000005 counts divisors.

A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

A001414 adds up prime factors, row sums of A027746.

A056239 adds up prime indices, row sums of A112798.

A097805 counts compositions by alternating (or reverse-alternating) sum.

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

A316524 gives the alternating sum of prime indices (reverse: A344616).

A345197 counts compositions by length and alternating sum.

A344610 counts partitions by sum and positive reverse-alternating sum.

Cf. A008549, A032443, A083399, A096825, A239830, A344607, A344609, A344651, A345957, A345960, A345961.

Sequence in context: A064372 A343943 A370817 * A096825 A318369 A007875

Adjacent sequences: A345923 A345924 A345925 * A345927 A345928 A345929

Keyword

nonn

Author

Gus Wiseman, Jul 14 2021

Status

approved