A075443 a(n)=(1/2)Sum_h |h-h'| with h and h' in [1,n], gcd(h,n)=1, hh'=1 (mod n).

0, 0, 0, 0, 0, 1, 0, 4, 0, 6, 4, 10, 0, 25, 4, 12, 16, 33, 12, 46, 8, 42, 32, 58, 0, 101, 44, 60, 56, 97, 12, 130, 64, 126, 72, 98, 72, 247, 80, 108, 80, 243, 48, 310, 64, 162, 196, 312, 96, 354, 172, 228, 168, 417, 120, 302, 176, 378, 284, 444, 120, 729, 188, 294, 352

Offset

0,8

Comments

For a given n, a(n) is half the sum for h ranging over the set of least positive residues coprime with n of |h-h'|, where h' is the (unique) number in the same set such that hh'=1 (mod n).

If h and h' are chosen randomly from [1,n] then the expected value of |h-h'|/2 is n/6. So it is plausible that a(n) ~ n*phi(n)/6 and numerical evidence seems to support that.

Links

Ivan Neretin, Table of n, a(n) for n = 0..10000

M. Dondi, Plot of A075443(n)/phi(n) (Euler's totient function) against the line y=x/6 in the range [0,100].

M. Dondi, Plot of A075443(n)/phi(n) (Euler's totient function) against the line y=x/6 in the range [0,1000].

M. Dondi, Plot of A075443(n)/phi(n) (Euler's totient function) against the line y=x/6 in the range [0,10000].

M. Dondi, Plot of A075443(n)/phi(n) (Euler's totient function) against the line y=x/6 in the range [0,10000] showing only one point out of every 5.

Mathematica

a[n_] := Sum[If[GCD[h, n]==1, Abs[h-PowerMod[h, -1, n]], 0], {h, 1, n}]/2

Crossrefs

Cf. A075444-A075452.

Sequence in context: A374969 A278210 A291540 * A021250 A073758 A133995

Adjacent sequences: A075440 A075441 A075442 * A075444 A075445 A075446

Keyword

nonn

Author

Michele Dondi (bik.mido(AT)tiscalinet.it), Sep 18 2002

Extensions

Edited by Dean Hickerson, Sep 20 2002

Status

approved