A053232 -id:A053232 - cp's OEIS frontend

A053230 First differences between numbers k for which sigma(k) < sigma(k+1).

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 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, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2

Offset: 1

Asher Auel, Jan 10 2000

It seems that the expansion consists of only {1,2,3,4}.
The first exception is a(10010491) = 6, corresponding to the gap from 20021153 to 20021159. - Charles R Greathouse IV, Mar 09 2014
The asymptotic mean of this sequence is 2 (Erdős, 1936). - Amiram Eldar, Mar 19 2021

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Paul Erdős, On a problem of Chowla and some related problems, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 32, No. 4 (1936), pp. 530-540; alternative link.

Cf. A000203, A053224, A053231, A053232, A053233, A053234, A053235, A053236, A053237, A053238.

a053230 n = a053230_list !! (n-1)
a053230_list = zipWith (-) (tail a053224_list) a053224_list
-- Reinhard Zumkeller, May 07 2012

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

Differences[Select[Range[250],DivisorSigma[1,#]Harvey P. Dale, Apr 30 2011 *)

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

a(n) = A053224(n+1) - A053224(n).

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

Asher Auel, Jan 10 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

import Data.List (elemIndices)
a053233 n = a053233_list !! (n-1)
a053233_list = map (+ 1) $ elemIndices 2 a053230_list
-- Reinhard Zumkeller, May 07 2012

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);

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

Asher Auel, Jan 10 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

import Data.List (elemIndices)
a053234 n = a053234_list !! (n-1)
a053234_list = map (+ 1) $ elemIndices 1 a053230_list
-- Reinhard Zumkeller, May 07 2012

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

Asher Auel, Jan 10 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

import Data.List (elemIndices)
a053235 n = a053235_list !! (n-1)
a053235_list = map (+ 1) $ elemIndices 3 a053230_list
-- Reinhard Zumkeller, May 07 2012

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);

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

Asher Auel, Jan 10 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

import Data.List (elemIndices)
a053236 n = a053236_list !! (n-1)
a053236_list = map (+ 1) $ elemIndices 4 a053230_list
-- Reinhard Zumkeller, May 07 2012

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);

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

Asher Auel, Jan 10 2000

nonn

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

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);

The expansion of A053230 excluding 2's. a(n) = A053230(A053232(n)).

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

Asher Auel, Jan 10 2000

nonn

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

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

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);

cp's OEIS Frontend

A053230 First differences between numbers k for which sigma(k) < sigma(k+1).

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

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

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

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Maple

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

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Maple

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

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Maple

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

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Maple