A132952 -id:A132952 - cp's OEIS frontend

A132953 a(n) is the sum of the isolated totatives of n.

0, 1, 0, 4, 0, 6, 0, 16, 0, 20, 0, 24, 0, 42, 15, 64, 0, 54, 0, 80, 21, 110, 0, 96, 0, 156, 0, 168, 0, 120, 0, 256, 33, 272, 35, 216, 0, 342, 39, 320, 0, 252, 0, 440, 135, 506, 0, 384, 0, 500, 51, 624, 0, 486, 55, 672, 57, 812, 0, 480, 0, 930, 189, 1024, 65, 660, 0, 1088, 69

Offset: 1

Leroy Quet, Sep 05 2007

nonn

An isolated totative, k, of n is a positive integer which is coprime to n, is <= n and is such that neither (k-1) nor (k+1) are coprime to n.

a(2n) = phi(2n)*n, where phi(n) = A000010(n).

			The positive integers which are <= 15 and are coprime to 15 are 1,2,4,7,8,11,13,14. Of these, 1 and 2 are adjacent, 7 and 8 are adjacent and 13 and 14 are adjacent. So the isolated totatives of 15 are 4 and 11. Therefore a(15) = 4 + 11 = 15.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A132952.

fQ[k_, n_] := GCD[k, n] == 1 && GCD[k - 1, n] > 1 && GCD[k + 1, n] > 1; f[n_] := Plus @@ Select[ Rest[ Range@n - 1], fQ[ #, n] &]; Array[f, 69] (* Robert G. Wilson v *)

A132953(n) = { my(s=0,pg=0,g=1,ng); for(k=1,n-1,if((1!=(ng=gcd(n,k+1)))&&(1==g)&&(1!=pg),s += k); pg = g; g = ng); (s); }; \\ Antti Karttunen, Nov 01 2018

a(n) = (n/2) * A132952(n). - Robert G. Wilson v, Sep 13 2007

Edited and extended by Robert G. Wilson v, Sep 13 2007

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.

Original entry on oeis.org

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

Michel Marcus, Nov 28 2018

nonn

			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.

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.

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

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

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

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

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

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

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

A322165 Numbers k that give record values for s(k)*phi(k)/k^2, where s(k) is the sum of squares of the differences between consecutive totatives of k (A322144).

Original entry on oeis.org

1, 3, 4, 6, 10, 12, 15, 18, 20, 21, 30, 42, 60, 70, 105, 210, 385, 770, 1155, 2310, 4620, 5005, 10010, 15015, 30030, 60060, 90090, 120120, 150150, 180180, 210210, 240240, 255255, 510510

Offset: 1

Amiram Eldar, Nov 29 2018

Erdős conjectured that this ratio is bounded and offered $500 for a proof. The conjecture was proved by Montgomery and Vaughan, who won the prize.

Is this sequence infinite? If yes, what is lim_{n->oo} s(a(n))*phi(a(n))/a(n)^2?

			The values of the ratio at the first terms of the sequence are 0, 0.222..., 0.5, 0.888..., 0.96, 1, 1.031..., ...

Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, Chapter B40, Gaps between totatives, p. 146.

Paul Erdős, The difference of consecutive primes, Duke Mathematical Journal, Vol. 6, No. 2 (1940), pp. 438-441, alternative link.
Paul Erdős, On the integers relatively prime to n and on a number-theoretic function considered by Jacobsthal, Mathematica Scandinavica, Vol. 10 (1962), pp. 163-170, alternative link.
H. L. Montgomery & R. C. Vaughan, On the distribution of reduced residues, Annals of Mathematics, Vol. 123, No. 2 (1986), pp. 311-333.

Cf. A000010, A038566, A132952, A322144.

ratio[n_] := Module[{v=Differences[Select[Range[n], GCD[n, #] == 1 &]]^2}, Total[v] * (Length[v]+1) / n^2]; seq={}; rm=-1; Do[r=ratio[n]; If[r>rm, rm=r; AppendTo[seq, n]], {n, 1, 1000}]; seq

s(n) = {v = select(x->gcd(x, n)==1, vector(n, k, k)); sum(i=1, #v-1, (v[i+1] - v[i])^2); } \\ A322144
lista(nn) = {my(m = -1); for (n=1, nn, newm = s(n)*eulerphi(n)/n^2; if (newm > m, print1(n, ", "); m = newm););} \\ Michel Marcus, Nov 29 2018

cp's OEIS Frontend

A132953 a(n) is the sum of the isolated totatives of n.

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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.

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A322165 Numbers k that give record values for s(k)*phi(k)/k^2, where s(k) is the sum of squares of the differences between consecutive totatives of k (A322144).

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