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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A083111 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

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Last modified June 6 18:59 EDT 2020. Contains 334832 sequences. (Running on oeis4.)