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

Author

Vladimir Shevelev, Aug 20 2010

Keywords

• nonn
• easy

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

Examples

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

Links

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

Crossrefs

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

Programs

Mathematica

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

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

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.

Extensions

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