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 1-10 of 10 results.

A319481 Integers k such that uphi(k) divides k, where uphi is the unitary phi function (A047994).

Original entry on oeis.org

1, 2, 6, 12, 168, 240, 14880, 65280, 4294901760, 7608944640, 1125874137169920, 18446744069414584320
Offset: 1

Views

Author

Michel Marcus, Sep 20 2018

Keywords

Links

Crossrefs

Cf. A047994.
A030163 is a subsequence.

Programs

  • PARI
    uphi(n) = my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1);
isok(n) = frac(n/uphi(n)) == 0;

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

A305678 Solutions x of uphi(x)/x = 2/3, where uphi is the unitary phi function (A047994).

Original entry on oeis.org

3, 36, 504, 720, 1404, 19656, 28080, 44640, 181440, 195840, 1740960, 7637760, 46085760, 65862720, 105960960, 12884705280, 16729130880, 22826833920, 38463828480, 502503505920, 890246522880, 3377622411509760, 131727274048880640, 55340232208243752960, 2158269056121506365440
Offset: 1

Views

Author

Michel Marcus, Jun 08 2018

Keywords

Comments

Apparently, these is a single x, 6, such that uphi(x)/x = 1/3.

Examples

			uphi(3) = 2 so uphi(3)/3 = 2/3;
uphi(36) = 24 so uphi(36)/36 = 2/3.

Links

Crossrefs

Cf. A047994 (uphi), A030163 (uphi(x)/x = 1/2).

Programs

  • PARI
    solve_uphi(2, 3, 10^40) \\ see A030163

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.

A071008 Numbers n such that uphi(uphi(n)) = n/2.

Original entry on oeis.org

2, 4, 16, 256, 364, 1456, 3276, 13104, 21600, 23296, 65536, 209664, 249984, 367200, 1285632, 3110400, 5963776, 6596304, 9749376, 23046144, 27216000, 33022080, 52876800, 53673984, 76639680, 94370400, 105540864, 119992320, 245765520, 285405120, 426037248
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Comments

If n = Product p_i^r_i then uphi(n) = Product (p_i^r_i-1); for example uphi(12) = (4-1)*(3-1) = 6.
If 2^n+1 is a Fermat prime then 2^(2*n) is a solution of the equation.
3110400 and 4294967296 are also in the sequence.

Links

Crossrefs

Cf. A030163, A047994.

Programs

Formula

{n: 2*A047994(A047994(n)) = n}.

Extensions

More terms from R. J. Mathar, Alois P. Heinz and M. F. Hasler, Nov 20 2010

A298759 Numbers k such that bphi(k) = k/2, where bphi is the bi-unitary analog of Euler's totient function (A116550).

Original entry on oeis.org

2, 6, 30, 42, 1722, 1806, 19977474
Offset: 1

Views

Author

Amiram Eldar, Jan 26 2018

Keywords

Comments

With Euler's totient function, phi(k) = k/2 only for powers of 2 (A000079, except for 1). With the unitary totient function (A047994) the corresponding sequence is A030163.
a(8) > 2*10^9, if it exists. - Amiram Eldar, Jul 16 2022

Examples

			42 is in the sequence since bphi(42) = 21 = 42/2.

Crossrefs

Cf. A047994, A030163, A116550.

Programs

  • Mathematica
    bphi[1] = 1; bphi[n_] :=  With[{pp = Power @@@ FactorInteger[n]},   Count[Range[n], m_ /; Intersection[pp, Power @@@ FactorInteger[m]] == {}]]; aQ[n_] := bphi[n] == n/2; Select[Range[10000], aQ]
  • PARI
    udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
bphi(n) = if (n==1, 1, sum(k=1, n-1, gcud(n, k) == 1));
isok(n) = bphi(n) == n/2; \\ Michel Marcus, Jan 26 2018

Extensions

a(7) from Amiram Eldar, Jul 16 2022

A144587 Consider pairs m,n such that UnitaryPhi(m) = UnitaryPhi(n) = (1/8)*(5*m^(1/2)-3*n^(1/2))^2, m

Original entry on oeis.org

2152640, 11948993133919100295708672000
Offset: 1

Views

Author

Yasutoshi Kohmoto, Jan 11 2009

Keywords

Comments

If m=n then we get A030163.

Examples

			Factorizations:
2^6*5*7*31^2
2^22*5^3*7^2*11^3*19*23*31^2*683*3^4*13^2

A144588 Consider pairs m,n such that UnitaryPhi(m)=UnitaryPhi(n)=1/8*(5*m^(1/2)-3*n^(1/2))^2, m

Original entry on oeis.org

2439360, 13205954936303752529313792000
Offset: 1

Views

Author

Yasutoshi Kohmoto, Jan 11 2009

Keywords

Comments

If m=n then we get A030163.

Examples

			2^6*5*7*3^2*11^2 and 2^22*5^3*7^2*11^3*19*23*31^2*683*3^2*41^2.

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 1-10 of 10 results.

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