A329354 -id:A329354 - cp's OEIS frontend

A180253 Call two divisors of n adjacent if the larger is a prime times the smaller. a(n) is the sum of elements of all pairs of adjacent divisors of n.

0, 3, 4, 9, 6, 24, 8, 21, 16, 36, 12, 64, 14, 48, 48, 45, 18, 87, 20, 96, 64, 72, 24, 144, 36, 84, 52, 128, 30, 216, 32, 93, 96, 108, 96, 229, 38, 120, 112, 216, 42, 288, 44, 192, 174, 144, 48, 304, 64, 201, 144, 224, 54, 276, 144, 288, 160, 180, 60, 552, 62, 192, 232, 189

Offset: 1

Vladimir Shevelev, Aug 20 2010

The pairs of adjacent divisors of n are counted in A062799(n).

For each divisor d of n we can check in how many pairs it occurs. For each prime divisor p of n, see the exponent of p in the factorization of d. If it's positive (p|d) then it occurs once more. If d*p doesn't divide n, add one to the frequency as well. - David A. Corneth, Dec 17 2018

			a(4) = (1 + 2) + (2 + 4) = 9.
a(120) = a(3*5*2^3) = 4*6*(3*8 + 4*4 + 4*2 + 3) = 1224.

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

Cf. A001221, A062799, A069359, A179926, A180026, A323599, A328260, A329354, A329375.

divisorSumPrime[n_] := DivisorSum[n, 1+1/# &, PrimeQ[#] &]; a[n_] := DivisorSum[n, #*divisorSumPrime[#]& ]; Array[a, 70] (* Amiram Eldar, Dec 17 2018 *)

a(n) = sumdiv(n, d, d*sumdiv(d, p, isprime(p)*(1+1/p))); \\ Michel Marcus, Dec 17 2018

a(n) = my(f = factor(n), res = 0); fordiv(n, d, for(i = 1, #f~, v = valuation(d, f[i, 1]); res+=(d * ((v > 0) + (v < f[i, 2]))))); res \\ David A. Corneth, Dec 17 2018

a(n) = Sum_{d|n} d*Sum_{p|d} (1 + 1/p) where p is restricted to primes.
a(n) = Sum_{d|n} A069359(d) + Sum_{d|n} d*A001221(d).
a(n) = A323599(n) + A329354(n) = A323599(n) + A328260(n) + A329375(n). - Antti Karttunen, Nov 15 2019
a(p^k) = (p^k - 1)*(p + 1)/(p - 1).
a(p_1*p_2*...*p_m) = m*(p_1 + 1)*(p_2 + 1)*...*(p_m + 1).
a(p*q^k) = (p + 1)*(2*q^k + 3*q^(k - 1) + 3*q^(k - 2) + ... + 3*q + 2).
a(p*q*r^k) = (p + 1)*(q + 1)*(3*r^k + 4*r^(k - 1) + 4*r^(k - 2) + ... + 4*r + 3) and similar for a larger number of distinct prime factors of n.

Definition rephrased, entries checked, one example added. - R. J. Mathar, Oct 25 2010

A347104 Dirichlet g.f.: primezeta(s-1) * zeta(s-1) / zeta(s).

Original entry on oeis.org

0, 2, 3, 2, 5, 7, 7, 4, 6, 13, 11, 10, 13, 19, 22, 8, 17, 18, 19, 18, 32, 31, 23, 20, 20, 37, 18, 26, 29, 38, 31, 16, 52, 49, 58, 24, 37, 55, 62, 36, 41, 56, 43, 42, 54, 67, 47, 40, 42, 60, 82, 50, 53, 54, 94, 52, 92, 85, 59, 60, 61, 91, 78, 32, 112, 92, 67, 66, 112, 106, 71, 48, 73, 109, 100

Offset: 1

Ilya Gutkovskiy, Aug 18 2021

nonn

a(n) is the sum of the prime terms in row n of A050873.

Moebius transform of A328260.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000010, A001221, A008472, A010051, A018804, A050873, A061397, A078439, A117494, A122411, A255655, A328260, A329354.

Table[DivisorSum[n, MoebiusMu[n/#] # PrimeNu[#] &], {n, 1, 75}]
Table[DivisorSum[n, # EulerPhi[n/#] &, PrimeQ[#] &], {n, 1, 75}]
Table[Sum[Boole[PrimeQ[GCD[n, k]]] GCD[n, k], {k, 1, n}], {n, 1, 75}]

a(n) = sumdiv(n, d, moebius(n/d)*d*omega(d)); \\ Michel Marcus, Aug 18 2021

a(n) = Sum_{d|n} mu(n/d) * d * omega(d).
a(n) = Sum_{p|n, p prime} p * phi(n/p).
a(n) = Sum_{k=1..n} A010051(gcd(n,k)) * gcd(n,k).

A329375 a(n) = Sum_{d|n, d

Original entry on oeis.org

0, 0, 0, 2, 0, 5, 0, 6, 3, 7, 0, 21, 0, 9, 8, 14, 0, 26, 0, 31, 10, 13, 0, 53, 5, 15, 12, 41, 0, 72, 0, 30, 14, 19, 12, 90, 0, 21, 16, 79, 0, 94, 0, 61, 47, 25, 0, 117, 7, 52, 20, 71, 0, 89, 16, 105, 22, 31, 0, 230, 0, 33, 61, 62, 18, 138, 0, 91, 26, 132, 0, 218, 0, 39, 63, 101, 18, 160, 0, 175, 39, 43, 0, 304, 22, 45, 32, 157, 0, 297, 20

Offset: 1

Antti Karttunen, Nov 15 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001221, A328260, A329354.

PARI
```
A329375(n) = sumdiv(n,d,(d
				
```

a(n) = Sum_{d|n, dA001221(d).

a(n) = A329354(n) - A328260(n).

A369903 a(n) = Sum_{p|n, p prime} n^sigma(n/p).

Original entry on oeis.org

0, 2, 3, 64, 5, 1512, 7, 2097152, 6561, 1001000, 11, 8916136280064, 13, 1475791800, 11441250, 1152921504606846976, 17, 21979796247097344, 19, 262144000000001280000000, 37823053842, 12855002631059864, 23, 442501521100279866178075737690262732800, 244140625

Offset: 1

Wesley Ivan Hurt, Feb 05 2024

Cf. A000203 (sigma), A323599, A329354, A369740.

Table[DivisorSum[n, n^DivisorSigma[1, n/#] &, PrimeQ[#] &], {n, 30}]

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

A369904 a(n) = n * Sum_{p|n, p prime} sigma(n/p) / p.

Original entry on oeis.org

0, 1, 1, 6, 1, 18, 1, 28, 12, 36, 1, 100, 1, 62, 42, 120, 1, 189, 1, 208, 68, 138, 1, 456, 30, 188, 117, 364, 1, 612, 1, 496, 144, 312, 86, 1038, 1, 386, 194, 960, 1, 1080, 1, 820, 477, 558, 1, 1936, 56, 955, 318, 1120, 1, 1782, 162, 1688, 392, 876, 1, 3336, 1, 998, 789

Offset: 1

Wesley Ivan Hurt, Feb 05 2024

Cf. A000203 (sigma), A323599, A329354, A369740, A369903.

Table[n*DivisorSum[n, DivisorSigma[1, n/#]/# &, PrimeQ[#] &], {n, 100}]

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

cp's OEIS Frontend

A180253 Call two divisors of n adjacent if the larger is a prime times the smaller. a(n) is the sum of elements of all pairs of adjacent divisors of n.

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A347104 Dirichlet g.f.: primezeta(s-1) * zeta(s-1) / zeta(s).

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A329375 a(n) = Sum_{d|n, d

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A369903 a(n) = Sum_{p|n, p prime} n^sigma(n/p).

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A369904 a(n) = n * Sum_{p|n, p prime} sigma(n/p) / p.

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