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.

A376830 Numbers k such that tau(k + tau(k)) = tau(k) + tau(tau(k)), where tau = A000005.

Original entry on oeis.org

1, 13, 19, 31, 37, 53, 67, 83, 88, 89, 109, 113, 127, 131, 139, 152, 157, 181, 190, 199, 211, 225, 233, 251, 257, 263, 276, 286, 293, 307, 317, 337, 344, 353, 379, 389, 401, 406, 409, 443, 449, 467, 479, 487, 491, 499, 503, 509, 536, 541, 557, 563, 571, 577, 587, 612, 631, 642, 647, 653, 658, 677
Offset: 1

Views

Author

Robert Israel, Oct 06 2024

Keywords

Examples

			a(9) = 88 is a term because tau(88) = 8, tau(8) = 4 and tau(88 + 8) = tau(96) = 12 = 8 + 4.

Links

Crossrefs

Cf. A000005, A376831, A376843, A376844, A376848, A376849, A376851. Includes A115093.

Programs

  • Maple
    filter:= proc(k) uses numtheory; local s;
 s:= tau(k);
 tau(k+s) = s + tau(s)
end proc:
select(filter, [$1..1000]);
  • Mathematica
    Select[Range[680],DivisorSigma[0,#+DivisorSigma[0,#]]==DivisorSigma[0,#]+DivisorSigma[0,DivisorSigma[0,#]] &] (* Stefano Spezia, Oct 06 2024 *)

A376848 Numbers k such that phi(k + phi(k)) = phi(k) + phi(phi(k)), where phi = A000010.

Original entry on oeis.org

2, 5, 9, 10, 14, 18, 20, 27, 28, 36, 38, 40, 46, 54, 56, 72, 76, 78, 80, 81, 92, 108, 112, 144, 152, 156, 160, 162, 184, 216, 224, 234, 243, 258, 288, 294, 304, 312, 320, 324, 368, 432, 438, 448, 468, 486, 516, 526, 570, 576, 588, 608, 609, 624, 640, 648, 702, 718, 729, 736, 754, 774, 864, 876
Offset: 1

Views

Author

Robert Israel, Oct 06 2024

Keywords

Comments

If k is in A126955, then 8*k + 6 is a term.

Examples

			a(4) = 10 is a term because phi(10) = 4, phi(4) = 2, and phi(10 + 4) = phi(14) = 6 = 4 + 2.

Links

Crossrefs

Cf. A000010, A126955, A376830, A376831, A376843, A376844, A376849, A376851.

Programs

  • Maple
    filter:= proc(k) uses numtheory; local s;
 s:= phi(k);
 phi(k+s) = s + phi(s)
end proc:
select(filter, [$1..1000]);
  • Mathematica
    Select[Range[880], EulerPhi[ #+EulerPhi[#]]==EulerPhi[#]+EulerPhi[EulerPhi[#]] &] (* Stefano Spezia, Oct 07 2024 *)

A376849 Numbers k such that psi(k + psi(k)) = psi(k) + psi(psi(k)), where psi = A002322.

Original entry on oeis.org

2, 5, 10, 34, 80, 111, 196, 204, 222, 328, 351, 646, 654, 704, 837, 876, 935, 943, 969, 1053, 1100, 1140, 1220, 1224, 1372, 1408, 1526, 1824, 1864, 2368, 2496, 2511, 2715, 2816, 2842, 3159, 3294, 3528, 3648, 3672, 4332, 4473, 4600, 4736, 4992, 5500, 5632, 6325, 6528, 7296, 7412, 7512, 7533, 7832, 7959
Offset: 1

Views

Author

Robert Israel, Oct 06 2024

Keywords

Examples

			a(3) = 10 is a term because psi(10) = 4, psi(4) = 2, and psi(10 + 4) = psi(14) = 6 = 4 + 2.

Links

Crossrefs

Cf. A002322, A376830, A376831, A376843, A376844, A376848, A376851.

Programs

  • Maple
    filter:= proc(k) uses numtheory; local s;
 s:= lambda(k);
 lambda(k+s) = s + lambda(s)
end proc:
select(filter, [$1..10000]);
  • Mathematica
    Select[Range[8000], CarmichaelLambda[ #+CarmichaelLambda[#]] == CarmichaelLambda[#]+CarmichaelLambda[CarmichaelLambda[#]] &] (* Stefano Spezia, Oct 07 2024 *)

A376851 Numbers k such that sopfr(k + sopfr(k)) = sopfr(k) + sopfr(sopfr(k)), where sopfr = A001414.

Original entry on oeis.org

1, 2, 56, 98, 102, 198, 402, 611, 780, 981, 1230, 1275, 1377, 2288, 3685, 4030, 6600, 8851, 9282, 11371, 11607, 13680, 15390, 15862, 16445, 20916, 21266, 21867, 22606, 27504, 27538, 29282, 30685, 31832, 32724, 34153, 34293, 35672, 38805, 38874, 39886, 43706, 44253, 44772, 45408, 47742, 48032
Offset: 1

Views

Author

Robert Israel, Oct 06 2024

Keywords

Examples

			a(4) = 98 is a term because sopfr(98) = 2 + 2*7 = 16, sopfr(16) = 4 * 2 = 8, and sopfr(98 + 16) = sopfr(114) = 2 + 3 + 19 = 24 = 16 + 8.

Links

Crossrefs

Cf. A001414, A376830, A376831, A376843, A376844, A376848, A376849.

Programs

  • Maple
    sopfr:= proc(k) option remember; local t;
  add(t[1]*t[2],t=ifactors(k)[2])
end proc:
filter:= proc(k) local s;
 s:= sopfr(k);
 sopfr(k+s) = s + sopfr(s)
end proc:
select(filter, [$1..10^5]);
  • Mathematica
    f[n_] := Plus @@ Times @@@ FactorInteger@ n; Select[Range[48400], f[#+f[#]]==f[#]+f[f[#]]&] (* James C. McMahon, Oct 09 2024 *)

A376831 Numbers k such that sigma(k + sigma(k)) = sigma(k) + sigma(sigma(k)), where sigma = A000203.

Original entry on oeis.org

1986, 58920, 88092, 111276, 201588, 662160, 1103076, 1573536, 1671056, 1887900, 3172434, 4507152, 4575124, 8105188, 10971936, 42273728, 56886840
Offset: 1

Views

Author

Robert Israel, Oct 05 2024

Keywords

Examples

			a(3) =  88092 is a term because sigma(88092) = 222768, sigma(222768) = 812448 and sigma(88092 + 222768) = sigma(310860) = 1035216 = 222768 + 812448.

Crossrefs

Cf. A000203, A376830, A376843, A376844, A376848, A376849, A376851.

Programs

  • Maple
    filter:= proc(k) uses numtheory; local s;
 s:= sigma(k);
 sigma(k+s) = s + sigma(s)
end proc:
select(filter, [$1..10^7]);

A376844 Numbers k such that (k + k')' = k' + k'', where ' denotes the arithmetic derivative A003415.

Original entry on oeis.org

1, 2, 16, 98, 108, 275, 598, 729, 2419, 3503, 5324, 11304, 12500, 35937, 50498, 51179, 58339, 76302, 84375, 107830, 141775, 209663, 324480, 588800, 618434, 1090123, 1532188, 2190240, 2615251, 3175699, 3294172, 3974400, 4159375, 6795761, 8403500, 8831250, 9765625, 10342269, 14784120, 15714364, 16056320
Offset: 1

Views

Author

Robert Israel, Oct 06 2024

Keywords

Examples

			a(4) = 98 is a term because 98' = 77, 77' = 18 and (98  + 77)' = 175' = 95 = 77 + 18.

Crossrefs

Cf. A003415, A376830, A376831, A376843, A376848, A376849, A376851.

Programs

  • Maple
    deriv:= proc(n) local t; add(n*t[2]/t[1], t = ifactors(n)[2]) end proc:
filter:= proc(n) local s; s:= deriv(n); deriv(n+s) = s + deriv(s) end proc:
select(filter, [$1..10^6]);
Showing 1-6 of 6 results.

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

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