A305614 -id:A305614 - cp's OEIS frontend

A351619 a(n) = Sum_{p|n, p prime} (-1)^p.

0, 1, -1, 1, -1, 0, -1, 1, -1, 0, -1, 0, -1, 0, -2, 1, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, -1, -1, -1, 1, -2, 0, -2, 0, -1, 0, -2, 0, -1, -1, -1, 0, -2, 0, -1, 0, -1, 0, -2, 0, -1, 0, -2, 0, -2, 0, -1, -1, -1, 0, -2, 1, -2, -1, -1, 0, -2, -1, -1, 0, -1, 0, -2, 0, -2, -1, -1, 0, -1, 0, -1, -1, -2, 0, -2, 0, -1, -1, -2, 0, -2, 0, -2, 0, -1, 0, -2, 0, -1

Offset: 1

Seiichi Manyama, Mar 02 2022

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A001221, A048272, A305614.

A351619[n_] := 2*Boole[EvenQ[n]] - PrimeNu[n]; Array[A351619, 100] (* Paolo Xausa, Jan 28 2025 *)

a(n) = my(f=factor(n)); sum(k=1, #f~, (-1)^f[k, 1]);

my(N=99, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, isprime(k)*(-x)^k/(1-x^k))))

from sympy import primefactors
def A351619(n): return (0 if n%2 else 2) - len(primefactors(n)) # Chai Wah Wu, Mar 02 2022

G.f.: Sum_{k>=1} (-x)^prime(k)/(1 - x^prime(k)).

a(n) = -A001221(n) if n is odd and a(n) = 2 - A001221(n) if n is even. - Chai Wah Wu, Mar 02 2022

A347103 G.f.: Sum_{k>=1} k * x^prime(k) / (1 + x^prime(k)).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Aug 18 2021

sign

a(n) is the sum of indices of prime divisors p|n such that n/p is odd, minus the sum of indices of prime divisors p|n such that n/p is even.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Cf. A000720, A066328, A300852, A305614, A332422.

nmax = 72; CoefficientList[Series[Sum[k x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[-DivisorSum[n, (-1)^(n/#) PrimePi[#] &, PrimeQ[#] &], {n, 1, 72}]

a(n) = my(f=factor(n)[,1]); sum(k=1, #f, if ((n/f[k]) % 2, primepi(f[k]), -primepi(f[k]))); \\ Michel Marcus, Aug 19 2021

a(n) = -Sum_{p|n, p prime} (-1)^(n/p) * pi(p), where pi = A000720.

A352003 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k))^(1/prime(k)).

Original entry on oeis.org

1, 0, 1, 2, -3, 44, -35, 1014, -1127, 46808, 153081, 3240170, -30922859, 443621892, 331421077, 121899383774, 691635821745, 19657393214384, 424491327098353, 2132527815161298, -2864544697983059, 3885322666246386140, 22621061924336157261, 882556261002776755142

Offset: 0

Seiichi Manyama, Feb 28 2022

sign

Cf. A168243, A305614, A318913, A352005.

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, (1+isprime(k)*x^k)^(1/k))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, isprime(d)*(-1)^(k/d+1))*x^k/k))))

E.g.f.: exp( Sum_{k>=1} A305614(k)*x^k/k ) where A305614(k) = Sum_{p|k, p prime} (-1)^(k/p+1).

A382511 Expansion of Sum_{p prime} x^p / (1 - x^p)^3.

Original entry on oeis.org

0, 1, 1, 3, 1, 9, 1, 10, 6, 18, 1, 31, 1, 31, 21, 36, 1, 66, 1, 65, 34, 69, 1, 114, 15, 94, 45, 115, 1, 196, 1, 136, 72, 156, 43, 249, 1, 193, 97, 246, 1, 357, 1, 263, 165, 279, 1, 436, 28, 380, 159, 361, 1, 549, 81, 442, 196, 438, 1, 753, 1, 499, 276, 528, 106

Offset: 1

Ilya Gutkovskiy, Mar 30 2025

nonn

Cf. A000217, A001221, A007437, A069359, A305614, A322078.

nmax = 65; CoefficientList[Series[Sum[x^Prime[k]/(1 - x^Prime[k])^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = Sum_{p|n, p prime} A000217(n/p).

a(n) = (A069359(n) + A322078(n)) / 2.

cp's OEIS Frontend

A351619 a(n) = Sum_{p|n, p prime} (-1)^p.

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A347103 G.f.: Sum_{k>=1} k * x^prime(k) / (1 + x^prime(k)).

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Mathematica

PARI

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A352003 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k))^(1/prime(k)).

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Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A382511 Expansion of Sum_{p prime} x^p / (1 - x^p)^3.

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Mathematica

Formula