cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A319677 Denominator of A047994(n)/n where A047994 is the unitary totient function.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 2, 13, 7, 15, 16, 17, 9, 19, 5, 7, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 32, 33, 17, 35, 3, 37, 19, 13, 10, 41, 7, 43, 22, 45, 23, 47, 8, 49, 25, 51, 13, 53, 27, 11, 4, 19, 29, 59, 5, 61, 31, 21, 64, 65, 33, 67, 17, 69, 35, 71
Offset: 1

Views

Author

Michel Marcus, Sep 26 2018

Keywords

Links

Crossrefs

Cf. A047994, A030163, A305678, A319481, A319676 (numerators), A323409, A331177 (ordinal transform).
Cf. A002110, A060753.

Programs

  • Mathematica
    uphi[n_] := Product[{p, e} = pe; p^e - 1, {pe, FactorInteger[n]}];
a[n_] := Denominator[uphi[n]/n];
Array[a, 100] (* Jean-François Alcover, Jan 10 2022 *)
  • PARI
    a(n)=my(f=factor(n)~); denominator(prod(i=1, #f, f[1, i]^f[2, i]-1)/n);

Formula

a(p) = p, for p prime.
a(A002110(n)) = A060753(n).
a(n) = n / A323409(n) = n / gcd(n, A047994(n)). - Antti Karttunen, Jan 11 2020

A319676 Numerator of A047994(n)/n where A047994 is the unitary totient function.

Original entry on oeis.org

1, 1, 2, 3, 4, 1, 6, 7, 8, 2, 10, 1, 12, 3, 8, 15, 16, 4, 18, 3, 4, 5, 22, 7, 24, 6, 26, 9, 28, 4, 30, 31, 20, 8, 24, 2, 36, 9, 8, 7, 40, 2, 42, 15, 32, 11, 46, 5, 48, 12, 32, 9, 52, 13, 8, 3, 12, 14, 58, 2, 60, 15, 16, 63, 48, 10, 66, 12, 44, 12, 70, 7, 72, 18, 16
Offset: 1

Views

Author

Michel Marcus, Sep 26 2018

Keywords

Crossrefs

Cf. A047994, A030163, A065463, A305678, A319481, A319677 (denominators).
Cf. A002110, A038110, A006093.

Programs

  • Mathematica
    uphi[n_] := Product[{p, e} = pe; p^e - 1, {pe, FactorInteger[n]}];
a[n_] := If[n == 1, 1, Numerator[uphi[n]/n]];
Array[a, 100] (* Jean-François Alcover, Jan 10 2022 *)
  • PARI
    a(n)=my(f=factor(n)~); numerator(prod(i=1, #f, f[1, i]^f[2, i]-1)/n);

Formula

a(p) = p-1, for p prime (see A006093).
a(A002110(n)) = A038110(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A319677(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.7044422... (A065463). - Amiram Eldar, Nov 21 2022

A318842 a(n) is the least integer m such that A047994(m) = ((n-1)/n)*m where A047994 is the unitary totient function, or 0 if there is no such m.

Original entry on oeis.org

2, 3, 4, 5, 144, 7, 8, 9, 400, 11, 64281600, 13, 84672, 129600, 16, 17, 518400, 19, 4327213363200, 254016, 6326996189184000000, 23, 300174920860041216000, 25, 2747437056, 27, 3136, 29
Offset: 2

Views

Author

Michel Marcus, Sep 04 2018

Keywords

Comments

If it is not 0, a(30) > 10^30. - Michel Marcus, Sep 08 2018

Links

Crossrefs

Cf. A000961 (primepowers), A047994 (unitary totient).
Cf. A030163, A305678.
Cf. A145680 (analog with unitary sigma).

Programs

  • PARI
    \\ uses the "solve_uphi pari code", see links
a(n) = {my(lim = 1, v); while (1, v = solve_uphi(n-1, n, lim); if (#v, return (v[1])); lim *= 10;);}

Formula

a(pp) = pp iff pp is a prime power (A000961) > 1.

A319313 a(n) is the least integer m such that A047994(m) = ((n-2)/n)*m where A047994 is the unitary totient function, or 0 if there is no such m.

Original entry on oeis.org

6, 2, 20, 3, 1008, 4, 72, 5, 4400, 144, 835660800, 7, 10800, 8, 272, 9, 9849600, 400, 208039104, 11, 145520912351232000000, 64281600, 3608344625286776094720000, 13, 1296, 84672, 90944, 129600
Offset: 3

Views

Author

Michel Marcus, Sep 17 2018

Keywords

Links

  • Yasutoshi Kohmoto, UnitaryPhi, SeqFan list, Aug-Sep 2018.

Crossrefs

Cf. A047994 (unitary totient).
Cf. A030163, A305678, A318842.

Programs

  • PARI
    \\ uses the "solve_uphi pari code", see A318842 links
a(n) = {my(lim = 1, v); while (1, v = solve_uphi(n-2, n, lim); if (#v, return (v[1])); lim *= 10; ); }

Formula

a(2n) = A318842(n).
Showing 1-4 of 4 results.

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