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 1-9 of 9 results.

A053233 Numbers n such that A053230(n) = 2.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72, 73, 76, 77, 78, 79, 80, 81, 82, 83, 85
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Links

Crossrefs

Cf. A000203, A053224, A053230, A053231, A053232, A053234, A053235, A053236, A053237.

Programs

  • Haskell
    import Data.List (elemIndices)
a053233 n = a053233_list !! (n-1)
a053233_list = map (+ 1) $ elemIndices 2 a053230_list
-- Reinhard Zumkeller, May 07 2012
  • Maple
    with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] = 2,i,print( )), i=1..100);
  • Mathematica
    Position[Differences@ Select[Range[170], Less @@ DivisorSigma[1, # + {0, 1}] &], 2][[All, 1]] (* Michael De Vlieger, Nov 19 2019 *)

A053234 Numbers n such that A053230(n) = 1.

Original entry on oeis.org

1, 2, 31, 32, 38, 39, 67, 68, 74, 75, 98, 99, 128, 129, 157, 197, 198, 201, 228, 229, 240, 241, 247, 248, 262, 277, 278, 283, 284, 307, 308, 313, 314, 332, 333, 339, 340, 349, 369, 370, 382, 383, 386, 400, 401, 413, 414, 430, 431, 459, 460, 475, 489, 490, 502
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Links

Crossrefs

Cf. A000203, A053224, A053230, A053231, A053232, A053233, A053235, A053236, A053237.

Programs

  • Haskell
    import Data.List (elemIndices)
a053234 n = a053234_list !! (n-1)
a053234_list = map (+ 1) $ elemIndices 1 a053230_list
-- Reinhard Zumkeller, May 07 2012
  • Maple
    with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] = 2,i,print( )), i=1..1000);

A053235 Numbers n such that A053230(n) = 3.

Original entry on oeis.org

158, 202, 263, 350, 387, 476, 567, 582, 701, 790, 879, 894, 926, 999, 1103, 1236, 1282, 1403, 1418, 1501, 1523, 1646, 1661, 1737, 1831, 1847, 1953, 2059, 2074, 2149, 2185, 2237, 2265, 2370, 2505, 2563, 2683, 2729, 2873, 2894, 2909, 3032, 3107, 3127
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Links

Crossrefs

Cf. A000203, A053224, A053230, A053231, A053232, A053233, A053234, A053236, A053237.

Programs

  • Haskell
    import Data.List (elemIndices)
a053235 n = a053235_list !! (n-1)
a053235_list = map (+ 1) $ elemIndices 3 a053230_list
-- Reinhard Zumkeller, May 07 2012
  • Maple
    with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] = 3,i,print( )), i=1..1000);
  • Mathematica
    Position[Differences@ Select[Range[10^4], Less @@ DivisorSigma[1, # + {0, 1}] &], 3][[All, 1]] (* Michael De Vlieger, Nov 19 2019 *)

A053236 Numbers n such that A053230(n) = 4.

Original entry on oeis.org

23, 54, 59, 84, 114, 138, 149, 172, 177, 232, 257, 281, 293, 311, 355, 392, 417, 422, 434, 445, 481, 506, 561, 596, 601, 644, 656, 686, 715, 745, 763, 775, 798, 809, 853, 864, 944, 955, 979, 984, 1013, 1018, 1061, 1072, 1140, 1164, 1187, 1192, 1222, 1227
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Links

Crossrefs

Cf. A000203, A053224, A053230, A053231, A053232, A053233, A053234, A053235, A053237.

Programs

  • Haskell
    import Data.List (elemIndices)
a053236 n = a053236_list !! (n-1)
a053236_list = map (+ 1) $ elemIndices 4 a053230_list
-- Reinhard Zumkeller, May 07 2012
  • Maple
    with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..8000)];
seq( `if`(f[i+1] - f[i] = 4,i,print( )), i=1..3000);

A053232 Numbers for which values not equal to 2 occur in the expansion of A053230.

Original entry on oeis.org

1, 2, 23, 31, 32, 38, 39, 54, 59, 67, 68, 74, 75, 84, 98, 99, 114, 128, 129, 138, 149, 157, 158, 172, 177, 197, 198, 201, 202, 228, 229, 232, 240, 241, 247, 248, 257, 262, 263, 277, 278, 281, 283, 284, 293, 307, 308, 311, 313, 314, 332, 333, 339, 340, 349, 350
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Crossrefs

Cf. A000203, A053224, A053230, A053231, A053233, A053234, A053235, A053236, A053237.

Programs

  • Maple
    with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] <> 2,i,print( )), i=1..100);

A053237 Numbers n such that both A053230(n) and A053230(n+1) = 1.

Original entry on oeis.org

1, 31, 38, 67, 74, 98, 128, 197, 228, 240, 247, 277, 283, 307, 313, 332, 339, 369, 382, 400, 413, 430, 459, 489, 502, 520, 551, 609, 622, 646, 664, 729, 759, 771, 823, 830, 843, 908, 915, 940, 969, 1038, 1057, 1086, 1117, 1124, 1148, 1206, 1247, 1266, 1290
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Comments

Pairs of consecutive 1's occur uncommonly often in A053230.

Crossrefs

Cf. A000203, A053224, A053230, A053231, A053232, A053233, A053234, A053235, A053236.

Programs

  • Maple
    with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq(`if`((f[i+2]-f[i+1]=1) and (f[i+1]-f[i]=1),i,print( )), i=1..1500);

A053224 Numbers k for which sigma(k) < sigma(k+1).

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 62, 63, 65, 67, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131
Offset: 1

Views

Author

Asher Auel, Jan 06 2000

Keywords

Comments

The asymptotic density of this sequence is 1/2 (Erdős, 1936). - Amiram Eldar, Mar 19 2021

Links

Crossrefs

Cf. A000203, A053222, A053226, A053230 (first differences).

Programs

  • Haskell
    import Data.List (elemIndices)
a053224 n = a053224_list !! (n-1)
a053224_list = map (+ 1) $ elemIndices True $
   zipWith (<) a000203_list $ tail a000203_list
-- Reinhard Zumkeller, May 07 2012
  • Maple
    with(numtheory): seq( `if`(sigma(i) < sigma(i+1),i,[][]), i=1..134);
  • Mathematica
    Select[Range[150], DivisorSigma[1, #] < DivisorSigma[1, # + 1] &] (* Carl Najafi, Aug 16 2011 *)
  • PARI
    is(n)=sigma(n)Charles R Greathouse IV, Mar 09 2014

A053238 First differences between numbers k for which sigma(k) > sigma(k+1).

Original entry on oeis.org

2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Comments

It seems that the expansion consists of only {1,2,3,4}.
The first exception is a(18360922) = 6, corresponding to the gap from 36721680 to 36721686. - Charles R Greathouse IV, Mar 09 2014
The asymptotic mean of this sequence is 2 (Erdős, 1936). - Amiram Eldar, Mar 19 2021

Links

Crossrefs

Cf. A000203, A053226, A053230, A053239, A053240, A053241, A053242, A053243, A053244, A053245.

Programs

  • Haskell
    a053238 n = a053238_list !! (n-1)
a053238_list = zipWith (-) (tail a053226_list) a053226_list
-- Reinhard Zumkeller, Oct 16 2011
  • Maple
    with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),i,print( )), i=1..5000)];
seq( f[i+1] - f[i], i=1..2000);
  • Mathematica
    Differences[Select[Range[250],DivisorSigma[1,#]>DivisorSigma [1,#+1]&]]  (* Harvey P. Dale, Apr 22 2011 *)
Differences[Flatten[Position[Partition[DivisorSigma[1,Range[300]],2,1],?(#[[1]]>#[[2]]&),1,Heads->False]]] (* _Harvey P. Dale, Oct 18 2020 *)
  • PARI
    last=ls=1; for(n=2,200,ns=sigma(n+1); if(ls<=ns,ls=ns; next); ls=ns; print1(n-last", ");last=n) \\ Charles R Greathouse IV, Mar 09 2014

Formula

a(n) = A053226(n+1) - A053226(n).

A053231 First differences between n for which sigma(n) < sigma(n+1), which are not 2.

Original entry on oeis.org

1, 1, 4, 1, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 4, 4, 1, 3, 4, 4, 1, 1, 1, 3, 1, 1, 4, 1, 1, 1, 1, 4, 1, 3, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 1, 3, 4, 1, 1, 1, 1, 4, 4, 1, 1, 4, 4, 1, 1, 1, 3, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 4, 1, 3, 1, 3, 4, 4, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1
Offset: 1

Views

Author

Asher Auel, Jan 10 2000

Keywords

Crossrefs

Cf. A000203, A053224, A053230, A053232, A053233, A053234, A053235, A053236, A053237.

Programs

  • Maple
    with(numtheory): f := [seq( `if`((sigma(i+1) > sigma(i)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] <> 2,f[i+1] - f[i],print( )), i=1..2000);

Formula

The expansion of A053230 excluding 2's. a(n) = A053230(A053232(n)).
Showing 1-9 of 9 results.

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