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

1,2

Comments

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

Links

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

Formula

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.

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A345270 A132065 A157020 * A264786 A055225 A054791

Adjacent sequences: A180250 A180251 A180252 * A180254 A180255 A180256

Keyword

nonn,easy

Author

Vladimir Shevelev, Aug 20 2010

Extensions

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

Status

approved