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 1-6 of 6 results.

A072011 Numbers k such that A072010(k) = k.

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 35, 64, 70, 128, 140, 256, 280, 323, 512, 560, 646, 899, 1024, 1120, 1225, 1292, 1763, 1798, 2048, 2240, 2450, 2584, 3526, 3596, 4096, 4480, 4900, 5168, 7052, 7192, 8192, 8960, 9800, 10336, 10403, 11305
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 05 2002

Keywords

Comments

If A072010(n) = n then also A072010(n*3^i) = n and A072010(n*2^j) = n*2^j.
For m=(4*k+1)*(4*k+3), product of twin prime pairs: A072010(m) = m, as well as for values m in the free monoid generated by the range of A071697.

Examples

			395675 is a term as f(395675) = f(323*1225) = f((17*19)*(5*7)^2) = f(17*19)*(f(5*7))^2 = f(17)*f(19)*(f(5)*f(7))^2 = 19*17*(7*5)^2 = 323*1225 = 395675 for f = A072010.

Links

Crossrefs

Cf. A002144, A002145, A071695, A071696, A071697, A072010.

A072012 a(n) = A072010(A072010(n)).

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 1, 4, 7, 14, 5, 16, 17, 2, 19, 20, 7, 2, 5, 8, 25, 14, 1, 28, 29, 10, 31, 32, 1, 34, 35, 4, 15, 38, 7, 40, 41, 14, 43, 4, 5, 10, 7, 16, 49, 50, 17, 28, 63, 2, 5, 56, 19, 58, 17, 20, 5, 62, 7, 64, 35, 2, 105, 68, 5, 70, 21, 8, 49, 30
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 05 2002

Keywords

Links

Crossrefs

Cf. A072010, A072013.

Programs

  • Mathematica
    b[n_] := If[n == 1, 1, Product[{p, e} = pe; Which[
     Mod[p, 4] == 1, p + 2,
     Mod[p, 4] == 3, p - 2,
     True, 2]^e, {pe, FactorInteger[n]}]];
a[n_] := b[b[n]];
Array[a, 100] (* Jean-François Alcover, Nov 21 2021 *)

A072014 Minima when the mapping of A072010 is applied to n repeatedly.

Original entry on oeis.org

1, 2, 1, 4, 5, 2, 5, 8, 1, 10, 1, 4, 5, 10, 5, 16, 17, 2, 17, 20, 5, 2, 5, 8, 25, 10, 1, 20, 29, 10, 29, 32, 1, 34, 35, 4, 5, 34, 5, 40, 41, 10, 41, 4, 5, 10, 5, 16, 25, 50, 17, 20, 5, 2, 5, 40, 17, 58, 17, 20, 5, 58, 5, 64, 35, 2, 35, 68, 5, 70, 5, 8, 25, 10, 25, 68, 5
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 05 2002

Keywords

Examples

			The trajectory of 71 under the A072010-mapping is:
71 ->69 ->21 ->5 ->7 ->5 ..., therefore a(71) = 5.

Crossrefs

Cf. A072015.

A072015 Maxima when the mapping of A072010 is applied to n repeatedly.

Original entry on oeis.org

1, 2, 1, 4, 7, 2, 7, 8, 1, 14, 9, 4, 15, 14, 7, 16, 19, 2, 19, 28, 7, 18, 21, 8, 49, 30, 1, 28, 31, 14, 31, 32, 9, 38, 35, 4, 39, 38, 15, 56, 43, 14, 43, 36, 7, 42, 45, 16, 49, 98, 19, 60, 63, 2, 63, 56, 19, 62, 57, 28, 63, 62, 7, 64, 105, 18, 105, 76, 21, 70
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 05 2002

Keywords

Examples

			The trajectory of 89 under the A072010-mapping is:
89 ->91 ->75 ->49 ->25 ->49 ..., therefore a(89) = 91.

Crossrefs

Cf. A072014.

A072013 Numbers k such that A072012(k) = k.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 14, 16, 17, 19, 20, 25, 28, 29, 31, 32, 34, 35, 38, 40, 41, 43, 49, 50, 56, 58, 62, 64, 68, 70, 76, 80, 82, 85, 86, 95, 98, 100, 101, 103, 112, 116, 119, 124, 125, 128, 133, 136, 137, 139, 140, 145, 149, 151, 152, 155, 160, 164
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 05 2002

Keywords

Links

Crossrefs

Cf. A072010, A072011, A072012.

A116912 In prime factorization of n replace all primes of form k*6+1 with k*6+5 and primes of form k*6+5 with k*6+1.

Original entry on oeis.org

1, 2, 3, 4, 1, 6, 11, 8, 9, 2, 7, 12, 17, 22, 3, 16, 13, 18, 23, 4, 33, 14, 19, 24, 1, 34, 27, 44, 25, 6, 35, 32, 21, 26, 11, 36, 41, 46, 51, 8, 37, 66, 47, 28, 9, 38, 43, 48, 121, 2, 39, 68, 49, 54, 7, 88, 69, 50, 55, 12, 65, 70, 99, 64
Offset: 1

Views

Author

Jonathan Vos Post, Mar 18 2006

Keywords

Comments

Primes of form 6n + 1 are also primes of the form 3n+1 and -3 is a quadratic residue mod a prime p iff p is in this sequence. Primes of the form 6n + 5 are the same as A003627 Primes of form 3n-1, except that the latter sequence starts with 2. Every twin prime after (3,5) is of the form (6n+5, 6n+1) hence the current sequence exchanges lesser twin primes with greater twin primes. See also: A072010 In prime factorization of n replace all primes of form k*4+1 by k*4+3 and primes of form k*4+3 by k*4+1. See also: A002476 Primes of form 6n + 1.

Examples

			a(5) = 1 because 5 is a prime of the form 6n + 5 (with n = 0), so is replaced with 6n + 1 (with n = 0), namely 1.
a(7) = 11 because 7 is a prime of the form 6n + 1 (with n = 1), so is replaced with 6n + 5 (with n = 1), namely 11.
a(11) = 7 because 11 is a prime of the form 6n + 5 (with n = 1), so is replaced with 6n + 1 (with n = 1), namely 7.
a(13) = 17 because 13 is a prime of the form 6n + 1 (with n = 2), so is replaced with 6n + 5 (with n = 2), namely 17.
a(14) = 22 because 14 = 2 * 7; but 7 is a prime of the form 6n + 1 (with n = 1), so is replaced with 6n + 5 (with n = 1), namely 11; giving 2 * 11 = 22.

Links

Crossrefs

Cf. A002476, A003627, A072010.

Programs

  • Mathematica
    f[p_, e_] := If[p < 5, p^e, (p + 6 - 2 * Mod[p, 6])^e]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 64] (* Amiram Eldar, Jan 10 2020 *)

Formula

Multiplicative. a(2^n) = 2^n, a(3^n) = 3^n, a(5^n) = 1, a(7^n) = 11^n, a(11^n) = 7^n, a(13^n) = 17^n, a(17^n) = 13^n, a(19^n) = 23^n, a(23^n) = 19^n, a(29^n) = 5^(2n), a(31^n) = (5^n)*(7^n), a(37^n) = 41^n, a(41^n) = 37^n, a(43^n) = 47^n, a(47^n) = 43^n, a(53^n) = 7^(2n), a(59^n) = (5^n)*(11^n), a(61^n) = (5^n)*(13^n), ...

Extensions

Data corrected by Amiram Eldar, Jan 10 2020
Showing 1-6 of 6 results.

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