A353160 -id:A353160 - cp's OEIS frontend

A353161 Product_{n>=1} (1 + x^n)^a(n) = 1 + x + Sum_{n>=2} prime(n-1) * x^n.

1, 2, 1, 3, 1, 2, -1, 1, -3, -1, 4, 1, 5, 2, -4, -4, -9, 0, -3, 14, 19, 4, 6, -38, -27, -17, 5, 59, 50, 103, -49, -100, -142, -222, 83, 138, 468, 362, 0, -313, -1215, -599, -526, 961, 2572, 1837, 1673, -2858, -4516, -6182, -3880, 5981, 9282, 18218, 7414, -8554, -24446

Offset: 1

Ilya Gutkovskiy, Apr 28 2022

sign

Inverse weigh transform of {1, primes}.

Cf. A008578, A030011, A305871, A353160.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= proc(n) option remember; `if`(n=1, 1, ithprime(n-1))-b(n, n-1) end:
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 28 2022

p[n_] := If[n == 1, 1, Prime[n - 1]]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j] b[n - i j, i - 1], {j, 0, n/i}]]]; a[n_] := a[n] = p[n] - b[n, n - 1]; Table[a[n], {n, 1, 57}]

A353169 Expansion of Product_{k>=1} (1 + x^k)^prime(k+1).

Original entry on oeis.org

1, 3, 8, 23, 57, 137, 317, 705, 1524, 3224, 6667, 13521, 26980, 52985, 102624, 196248, 370849, 693159, 1282537, 2350584, 4269912, 7692044, 13748080, 24390170, 42966637, 75187515, 130737631, 225957706, 388279308, 663533206, 1127936772, 1907676978, 3210783522, 5378798428

Offset: 0

Ilya Gutkovskiy, Apr 28 2022

nonn

Weigh transform of odd primes.

Cf. A061152, A065091, A353065, A353160, A353170.

nmax = 33; CoefficientList[Series[Product[(1 + x^k)^Prime[k + 1], {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d + 1) d Prime[d + 1], {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 33}]

cp's OEIS Frontend

A353161 Product_{n>=1} (1 + x^n)^a(n) = 1 + x + Sum_{n>=2} prime(n-1) * x^n.

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