Search: a132952 -id:a132952
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A132953
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a(n) is the sum of the isolated totatives of n.
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+10
2
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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
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OFFSET
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1,4
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COMMENTS
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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).
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(PARI) 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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A322144
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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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+10
2
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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
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OFFSET
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1,4
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LINKS
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FORMULA
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a(p) = p-2, for p prime.
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EXAMPLE
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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.
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MATHEMATICA
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a[n_] := Total[Differences[Select[Range[n], GCD[n, #]==1 &]]^2]; Array[a, 50] (* Amiram Eldar, Nov 28 2018 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A322165
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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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+10
0
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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
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OFFSET
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1,2
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COMMENTS
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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?
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REFERENCES
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Richard K. Guy, Unsolved Problems in Number Theory, 3rd ed., Springer, 2004, Chapter B40, Gaps between totatives, p. 146.
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LINKS
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EXAMPLE
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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..., ...
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MATHEMATICA
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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
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PROG
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(PARI) 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
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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