A339427
Number of compositions (ordered partitions) of n into an odd number of powers of 2.
Original entry on oeis.org
0, 1, 1, 1, 4, 4, 9, 17, 26, 50, 88, 150, 274, 478, 841, 1497, 2634, 4650, 8234, 14518, 25654, 45340, 80040, 141414, 249822, 441192, 779422, 1376752, 2431772, 4295678, 7587761, 13402881, 23675186, 41819442, 73869802, 130483966, 230485902, 407130212, 719154602
Offset: 0
a(5) = 4 because we have [2, 2, 1], [2, 1, 2], [1, 2, 2] and [1, 1, 1, 1, 1].
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b:= proc(n, t) option remember; `if`(n=0, t,
add(b(n-2^i, 1-t), i=0..ilog2(n)))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..42); # Alois P. Heinz, Dec 03 2020
-
nmax = 38; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}]) - 1/(1 + Sum[x^(2^k), {k, 0, Floor[Log[2, nmax]] + 1}])), {x, 0, nmax}], x]
A364525
a(n) is the number of distinct ways to partition the set {1,2,...,n} into nonempty subsets such that the sum of the pi(x)*(pi(x) + 1)/2 values of each subset's size x equals n, where pi() is the prime counting function given by A000720.
Original entry on oeis.org
0, 0, 1, 1, 2, 5, 9, 18, 36, 73, 145, 290, 580, 1159, 2319, 4637, 9273, 18544, 37083, 74157, 148330, 296658, 593311, 1186613, 2373208, 4746380, 9492687, 18985447, 37970821, 75941497, 151882704, 303764828, 607528497, 1215054675, 2430104713, 4860217541
Offset: 1
Cf.
A365062 (sum of pi(x) + 1 for n>0).
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p[n_] := p[n] = PrimePi[n];
pv[n_] := pv[n] = p[n]*(p[n] + 1)/2;
v[n_, k_] := v[n, k] = Module[{c = 0, i = 1}, If[k == 1, Return[If[pv[n] == n, 1, 0]]]; While[i < n - k + 2, If[pv[i] <= n, c += v[n - i, k - 1]]; i++]; c];
a[n_] := a[n] = Module[{c = 0, k = 1}, While[k <= n, c += v[n, k]; k++]; c]; Table[a[n], {n, 1, 36}]