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 1-5 of 5 results.

A291931 Primitive elements of A290002.

Original entry on oeis.org

1, 10, 18, 54, 70, 78, 110, 162, 174, 198, 222, 230, 234, 246, 294, 414, 426, 438, 450, 470, 486, 534, 594, 666, 702, 770, 846, 858, 882, 910, 1070, 1158, 1218, 1242, 1314, 1350, 1458, 1610, 1722, 1782, 1794, 1866, 1914, 1926, 1938, 1950, 1998, 2058, 2106, 2250, 2442, 2530, 2538, 2574, 2590, 2646, 2886
Offset: 1

Views

Author

Robert Israel, Sep 06 2017

Keywords

Comments

Members k of A290002 such that k/2 is not in A290002.
Includes all members of A025192 except 2 and 6.

Examples

			a(3) = 18 is in the sequence because psi(phi(18)) = phi(psi(18)) = 12 but psi(phi(9)) = 12 <> 4 = phi(psi(9)).

Links

Crossrefs

Cf. A025192, A290002.

Programs

  • Maple
    psi:= proc(n)  n*mul((1+1/i[1]), i=ifactors(n)[2]) end:
A290002:= select(psi @ numtheory:-phi = numtheory:-phi @ psi, {$1..3000}):
sort(convert(A290002 minus map(`*`,A290002,2), list));
  • Mathematica
    f[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors@ n}]; With[{s = Select[Range[3000], f[EulerPhi@ #] == EulerPhi[f@ #] &]}, Select[s, FreeQ[s, #/2] &]] (* Michael De Vlieger, Sep 06 2017 *)

A291682 Numbers k such that phi(psi(phi(k))) = psi(phi(psi(k))).

Original entry on oeis.org

1, 11, 19, 23, 25, 31, 41, 47, 59, 67, 71, 77, 79, 89, 95, 101, 109, 121, 127, 131, 137, 139, 143, 149, 155, 161, 175, 181, 191, 199, 287, 299, 311, 319, 323, 325, 329, 335, 341, 379, 383, 395, 407, 409, 413, 419, 439, 461, 463, 475, 479, 491, 497, 527, 529, 533, 539, 545, 569, 599, 611, 623, 635
Offset: 1

Views

Author

Altug Alkan, Sep 04 2017

Keywords

Comments

Prime terms are 11, 19, 23, 31, 41, 47, 59, 67, 71, 79, 89, 101, 109, 127, 131, ...
Up to 10^9, twin prime pairs in this sequence are (137, 139), (461, 463), (1019, 1021), (1427, 1429), (2969, 2971), (4229, 4231).

Examples

			11 is a term because phi(psi(phi(11))) = psi(phi(psi(11))).

Links

Crossrefs

Cf. A000010, A001615, A067160, A290002.

Programs

  • Mathematica
    psi[n_] := If[n < 1, 0, n Sum[MoebiusMu[d]^2/d, {d, Divisors@n}]]; fQ[n_] := EulerPhi[psi[EulerPhi[n]]] == psi[EulerPhi[psi[n]]]; Select[Range@635, fQ] (* Robert G. Wilson v, Sep 23 2017 *)
  • PARI
    a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = a001615(eulerphi(a001615(n)))==eulerphi(a001615(eulerphi(n))); \\ after Charles R Greathouse IV at A001615

A330701 Numbers k such that psi(phi(k)) = 2 * phi(psi(k)), where psi(k) is the Dedekind psi function (A001615) and phi(k) is the Euler totient function (A000010).

Original entry on oeis.org

26, 39, 45, 51, 52, 58, 74, 82, 98, 104, 111, 116, 135, 142, 146, 147, 148, 164, 178, 195, 196, 208, 219, 232, 284, 286, 292, 296, 328, 356, 357, 386, 392, 405, 406, 416, 435, 464, 495, 555, 561, 568, 572, 574, 579, 584, 585, 592, 598, 615, 622, 638, 646, 650
Offset: 1

Views

Author

Amiram Eldar, Dec 26 2019

Keywords

Comments

Sandor proved that this sequence is infinite.

Examples

			26 is in the sequence since psi(phi(26)) = psi(12) = 24, and 2 * phi(psi(26)) = 2 * phi(42) = 2 * 12 = 24.

Links

Crossrefs

Cf. A000010, A001615, A290002.

Programs

  • Mathematica
    psi[1] = 1; psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); Select[Range[1000], psi[EulerPhi[#]] == 2 * EulerPhi[psi[#]] &]

A292005 Composite numbers k such that psi(k - phi(k)) = phi(psi(k) - k).

Original entry on oeis.org

35, 65, 77, 161, 185, 209, 221, 335, 341, 371, 377, 437, 485, 515, 611, 671, 707, 731, 767, 779, 851, 899, 917, 965, 1007, 1067, 1115, 1157, 1211, 1247, 1271, 1337, 1385, 1397, 1529, 1535, 1577, 1631, 1691, 1781, 1817, 1841, 1991, 2117, 2171, 2201, 2285, 2291, 2327
Offset: 1

Views

Author

Altug Alkan, Sep 07 2017

Keywords

Comments

10217383 = 11*19^2*31*83 is the smallest term that is not squarefree.
A176875 is a subsequence.

Examples

			77 = 7*11 is a term because 77 - phi(77) = 17, psi(77) - 77 = 19 and phi(19) = psi(17).

Crossrefs

Cf. A000010, A001615, A176875, A290002.

Programs

  • PARI
    a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = !isprime(n) && a001615(n-eulerphi(n))==eulerphi(a001615(n)-n); \\ after Charles R Greathouse IV at A001615

A292048 Squarefree numbers n such that psi(phi(n)) = phi(psi(n) - n).

Original entry on oeis.org

2, 30, 35070, 36570, 43230, 159810, 224610, 331170, 525630, 1039890, 1094730, 1290810, 1656930, 1770510, 2139990, 4878390, 5110710, 5996310, 6052530, 6127890, 7493430, 9918930, 10146570, 12171810, 12551370, 13821870, 21398370, 23282130, 25587030, 30223830, 31317510, 31364970
Offset: 1

Views

Author

Altug Alkan, Sep 08 2017

Keywords

Examples

			30 = 2*3*5 is a term because psi(phi(30)) = phi(psi(30)-30).
60 = 30*2 is not a term because it is not a squarefree number.

Crossrefs

Cf. A000010, A001615, A290002.

Programs

  • PARI
    a001615(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1));
isok(n) = issquarefree(n) && a001615(eulerphi(n))==eulerphi(a001615(n)-n); \\ after Charles R Greathouse IV at A001615
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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