A322144 a(n) = Sum_{i=1..phi(n)-1} (r(i+1)-r(i))^2 where r(1) = 1 < ... < n-1 = r(phi(n)) are the phi(n) integers relatively prime to n.

0, 0, 1, 4, 3, 16, 5, 12, 11, 24, 9, 36, 11, 32, 29, 28, 15, 56, 17, 52, 39, 48, 21, 76, 31, 56, 41, 68, 27, 128, 29, 60, 59, 72, 57, 116, 35, 80, 69, 108, 39, 168, 41, 100, 95, 96, 45, 156, 59, 136, 89, 116, 51, 176, 85, 140, 99, 120, 57, 260, 59, 128, 125, 124, 99

Offset: 1

Author

Michel Marcus, Nov 28 2018

Keywords

• nonn

Examples

			a(1) and a(2) are 0, since we have an empty sum.
For a(3), the integers < 3, coprime to 3, are 1 and 2, so a(3) = (2-1)^2 = 1.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• Paul Erdos, Some Unconventional Problems in Number Theory, Mathematics Magazine, Vol. 52, No. 2, Mar., 1979, pp. 67-70. See Problem 12. p. 70.

Crossrefs

Cf. A000010 (phi), A038566 (rows of r).

Cf. A040976 (prime(n)-2), A132952 (isolated totatives).

Programs

Mathematica

a[n_] := Total[Differences[Select[Range[n], GCD[n,#]==1 &]]^2]; Array[a, 50] (* Amiram Eldar, Nov 28 2018 *)

PARI

a(n) = {v = select(x->gcd(x,n)==1, vector(n, k, k)); sum(i=1, #v-1, (v[i+1] - v[i])^2);}

PARI

a(n) = {my(res = 0, io = 1, in = 2); while(in < n, while(gcd(in, n) > 1, in++); res += (in - io)^2; io = in; in++); res}
first(n) = {my(res = vector(n)); for(i = 1, n, c = factorback(factor(i)[, 1]); if(c == i, res[i] = a(i), res[i] = res[c] * (i / c) + 4 * (i / c - 1))); res } \\ David A. Corneth, Nov 28 2018

Formula

a(p) = p-2, for p prime.

a(k^2 * m) = k * a(k * m) + 4 * (k - 1). - David A. Corneth, Nov 28 2018