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A067392 Sum of numbers <= n which have common prime factors with n. 8
0, 2, 3, 6, 5, 15, 7, 20, 18, 35, 11, 54, 13, 63, 60, 72, 17, 117, 19, 130, 105, 143, 23, 204, 75, 195, 135, 238, 29, 345, 31, 272, 231, 323, 210, 450, 37, 399, 312, 500, 41, 651, 43, 550, 495, 575, 47, 792, 196, 775, 510, 754, 53, 999, 440, 924, 627, 899, 59 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Sum of k <= n such that gcd(n,k) > 1.

LINKS

Ivan Neretin, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = n(n+1)/2 - n*phi(n)/2 = A000217(n)-A023896(n), for n>=2.

Not multiplicative.

a(p) = p where p is a prime; a(2^k) = 2^(k-1)*{2^(k-1) + 1).

G.f.: -Sum_{k>=2} mu(k)*k*x^k/(1 - x^k)^3. - Ilya Gutkovskiy, May 28 2019

EXAMPLE

For n=24, a(24) = 2+3+4+6+8+9+10+12+14+15+16+18+20+21+22+24 = 204.

MAPLE

with(numtheory): (0, seq(n*(n+1)/2-n*phi(n)/2, n=2..59)); # Paolo P. Lava, May 08 2018

MATHEMATICA

a[n_] := Plus@@Select[Range[1, n], GCD[ #, n]>1&]

Join[{0}, Table[n (n + 1) / 2 - n EulerPhi@(n) / 2, {n, 2, 60}]] (* Vincenzo Librandi, Jul 19 2019 *)

PROG

(PARI) A067392(n)={a=0; for(i=1, n, if(gcd(i, n)<>1, a=a+i)); a}

(PARI) a(n) = sum(k=1, n, k*(gcd(k, n) != 1)); \\ Michel Marcus, May 08 2018

(MAGMA) [0] cat [n*(n+1)/2-n*EulerPhi(n)/2: n in [2..60]]; // Vincenzo Librandi, Jul 19 2019

CROSSREFS

Cf. A000203, A000217, A023896, A024816.

Sequence in context: A136183 A100211 A071257 * A066449 A276942 A255483

Adjacent sequences:  A067389 A067390 A067391 * A067393 A067394 A067395

KEYWORD

nonn

AUTHOR

Labos Elemer, Jan 22 2002

STATUS

approved

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Last modified June 5 18:48 EDT 2020. Contains 334854 sequences. (Running on oeis4.)