A329347 -id:A329347 - cp's OEIS frontend

A323599 Dirichlet convolution of the identity function with omega.

0, 1, 1, 3, 1, 7, 1, 7, 4, 9, 1, 19, 1, 11, 10, 15, 1, 25, 1, 25, 12, 15, 1, 43, 6, 17, 13, 31, 1, 54, 1, 31, 16, 21, 14, 67, 1, 23, 18, 57, 1, 68, 1, 43, 37, 27, 1, 91, 8, 49, 22, 49, 1, 79, 18, 71, 24, 33, 1, 142, 1, 35, 45, 63, 20, 96, 1, 61, 28, 90, 1, 151, 1, 41, 55

Offset: 1

Torlach Rush, Jan 18 2019

nonn

a(n) = omega(n) = 1 iff n is prime.
a(n) = A323600(n) = 1 iff n is prime.
a(n) = A323600(n) - 1 = 1 iff n is the square of a prime.
a(n) = A323600(n) - 2 = 2 iff n is a squarefree semiprime.
a(n) = A323600(n) - (p + 2) if n is the cube of a prime p.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A001221, A180253, A276085, A319684, A323600, A329347, A329380.

Inverse Möbius transform of A069359.

with(numtheory):
a:= n-> add(d*nops(factorset(n/d)), d=divisors(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 28 2019

Table[DivisorSum[n, # PrimeNu[n/#] &], {n, 75}] (* Michael De Vlieger, Jan 27 2019 *)

a(n) = sumdiv(n, d, d*omega(n/d)); \\ Michel Marcus, Jan 22 2019

a(n) = Sum_{d|n} d * A001221(n/d).
a(n) = Sum_{p|n} sigma(n/p) where p is prime and sigma(n) = A000203(n). - Ridouane Oudra, Apr 28 2019
a(n) = Sum_{d|n} A069359(d), a(n) = A276085(A329380(n)). - Antti Karttunen, Nov 12 2019
From Torlach Rush, Mar 23 2024: (Start)
For p in primes: (Start)
a(p^(k+1)) = a(p^k) + p^k, k >= 0.
a(p^2) = p + 1.
(End)
a(2^k) = 2^k - 1, k >= 0.
(End)

A329371 Dirichlet convolution of the identity function with A246277.

Original entry on oeis.org

0, 1, 1, 4, 1, 8, 1, 12, 5, 12, 1, 28, 1, 16, 11, 32, 1, 37, 1, 44, 15, 24, 1, 80, 7, 28, 19, 60, 1, 82, 1, 80, 21, 36, 15, 128, 1, 40, 27, 128, 1, 114, 1, 92, 49, 48, 1, 208, 9, 89, 33, 108, 1, 146, 21, 176, 39, 60, 1, 284, 1, 64, 69, 192, 25, 174, 1, 140, 45, 170, 1, 364, 1, 76, 70, 156, 21, 210, 1, 336, 65, 84, 1, 396, 33, 88, 55, 272, 1, 368, 25, 188, 63

Offset: 1

Antti Karttunen, Nov 12 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences computed from indices in prime factorization

Cf. A246277.

Cf. also A305796, A323599, A329347, A329372, A329373.

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2);
A329371(n) = sumdiv(n,d,(n/d)*A246277(d));

a(n) = Sum_{d|n} d * A246277(n/d).

A329346 a(n) = A322356(A324886(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 7, 1, 1, 1, 1, 1, 1, 5, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 7, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 7, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 19, 1, 1, 1, 1, 13, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 7, 1, 1, 13, 1, 1, 1, 1, 1, 1, 19, 1, 1, 1, 1, 7, 1, 13, 1, 13, 1, 1, 1, 1, 13

Offset: 1

Antti Karttunen, Nov 11 2019

nonn

			For n = 128 = 2^7, A108951(128) = A034386(2)^7 = 128. As 128 = 4 * 30 + 1*6 + 1* 2, A276086(128) = 36015 = 7^4 * 5^1 * 3^1, and there are two such primes that both p and p-2 divide n, and p-2 is also prime, namely, 7 and 5, thus a(128) = 7*5 = 35. This is also the first occurrence of composite number in this sequence.

Cf. A002110, A034386, A108951, A276086, A324886, A329040, A322356, A329343, , A329344, A329347, A329348.

A034386(n) = prod(i=1, primepi(n), prime(i));
A108951(n) = { my(f=factor(n)); prod(i=1, #f~, A034386(f[i, 1])^f[i, 2]) };  \\ From A108951
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A324886(n) = A276086(A108951(n));
A322356(n) = { my(f = factor(n), m=1); for(i=1, #f~, if(isprime(f[i,1]+2)&&!(n%(f[i,1]+2)), m *= (f[i,1]+2))); (m); };
A329346(n) = A322356(A324886(n));

a(n) = A322356(A324886(n)).

A369743 a(n) = Sum_{p|n, p prime} p * Omega(n/p).

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 4, 3, 7, 0, 10, 0, 9, 8, 6, 0, 10, 0, 14, 10, 13, 0, 15, 5, 15, 6, 18, 0, 20, 0, 8, 14, 19, 12, 15, 0, 21, 16, 21, 0, 24, 0, 26, 16, 25, 0, 20, 7, 14, 20, 30, 0, 15, 16, 27, 22, 31, 0, 30, 0, 33, 20, 10, 18, 32, 0, 38, 26, 28, 0, 20, 0, 39, 16

Offset: 1

Wesley Ivan Hurt, Jan 30 2024

Cf. A001222 (Omega), A329347, A369741.

Table[DivisorSum[n, #*PrimeOmega[n/#] &, PrimeQ[#] &], {n, 100}]

a(p^k) = p*(k-1), for p prime and k >= 1. - Wesley Ivan Hurt, Jun 26 2024

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A323599 Dirichlet convolution of the identity function with omega.

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A329371 Dirichlet convolution of the identity function with A246277.

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A329346 a(n) = A322356(A324886(n)).

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A369743 a(n) = Sum_{p|n, p prime} p * Omega(n/p).

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