A053244 -id:A053244 - cp's OEIS frontend

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

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

Asher Auel, Jan 10 2000

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

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

a053238 n = a053238_list !! (n-1)
a053238_list = zipWith (-) (tail a053226_list) a053226_list
-- Reinhard Zumkeller, Oct 16 2011

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

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 *)

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

A053240 n for which values not equal to 2 occur in the expansion of A053238.

Original entry on oeis.org

5, 20, 21, 29, 34, 49, 50, 56, 57, 65, 70, 79, 80, 94, 99, 108, 109, 123, 132, 133, 145, 146, 154, 155, 170, 171, 177, 178, 198, 200, 201, 227, 230, 231, 239, 244, 253, 254, 259, 260, 274, 277, 278, 280, 289, 290, 304, 307, 308, 310, 327, 332, 340, 341, 347

Offset: 1

Asher Auel, Jan 10 2000

nonn

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

Cf. A000203, A053226, A053238, A053239, A053241 (complement), A053242, A053243, A053244, A053245.

import Data.List (findIndices)
a053240 n = a053240_list !! (n-1)
a053240_list = map (+ 1) $ findIndices (/= 2) a053238_list
-- Reinhard Zumkeller, Oct 16 2011

with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] <> 2,i,print( )), i=1..1000);

A053241 Numbers n such that A053238(n) = 2.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 85

Offset: 1

Asher Auel, Jan 10 2000

nonn

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

Cf. A000203, A053226, A053238, A053239, A053240 (complement), A053242, A053243, A053244, A053245.

import Data.List (elemIndices)
a053241 n = a053241_list !! (n-1)
a053241_list = map (+ 1) $ elemIndices 2 a053238_list
-- Reinhard Zumkeller, Oct 16 2011

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

A053242 Numbers n such that A053238(n) = 1.

Original entry on oeis.org

20, 21, 49, 50, 56, 57, 79, 80, 108, 109, 132, 133, 145, 146, 155, 170, 171, 177, 178, 201, 230, 231, 253, 254, 260, 277, 278, 289, 290, 307, 308, 341, 347, 348, 376, 382, 383, 405, 406, 412, 413, 424, 425, 437, 438, 467, 495, 496, 548, 549, 555, 570, 585

Offset: 1

Asher Auel, Jan 10 2000

nonn

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

Cf. A000203, A053226, A053238, A053239, A053240, A053241, A053243, A053244, A053245 (subsequence).

import Data.List (elemIndices)
a053242 n = a053242_list !! (n-1)
a053242_list = map (+ 1) $ elemIndices 1 a053238_list
-- Reinhard Zumkeller, Oct 16 2011

with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] = 2,i,print( )), i=1..1000);

A053245 Numbers k such that both A053238(k) and A053238(k+1) = 1.

Original entry on oeis.org

20, 49, 56, 79, 108, 132, 145, 170, 177, 230, 253, 277, 289, 307, 347, 382, 405, 412, 424, 437, 495, 548, 585, 592, 633, 645, 704, 734, 752, 764, 789, 802, 841, 854, 930, 943, 967, 974, 1005, 1012, 1053, 1066, 1130, 1154, 1179, 1186, 1216, 1223, 1264

Offset: 1

Asher Auel, Jan 10 2000

nonn

Pairs of consecutive 1's occur uncommonly often in A053238.

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

Cf. A000203, A053226, A053238, A053239, A053240, A053241, A053242, A053243, A053244.

a053245 n = a053245_list !! (n-1)
a053245_list = f a053242_list where
   f (x:x':xs) | x' == x+1 = x : f xs
               | otherwise = f (x':xs)
-- Reinhard Zumkeller, Oct 16 2011

with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),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);

A053239 First differences between n for which sigma(n) > sigma(n+1), which are not 2.

Original entry on oeis.org

4, 1, 1, 4, 4, 1, 1, 1, 1, 4, 4, 1, 1, 4, 4, 1, 1, 4, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 4, 3, 1, 4, 1, 1, 4, 4, 1, 1, 3, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 4, 4, 3, 1, 1, 1, 4, 4, 3, 1, 1, 1, 4, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 4, 3, 1, 4, 4, 1, 1, 4, 4, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 4, 4, 1, 1, 4, 1, 1, 4, 4, 3, 1, 3, 1

Offset: 1

Asher Auel, Jan 10 2000

nonn

Cf. A000203, A053226, A053238, A053240, A053241, A053242, A053243, A053244, A053245.

with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),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 A053238 excluding 2's. a(n) = A053238(A053240(n)).

A053243 Numbers n such that A053238(n) = 3.

Original entry on oeis.org

154, 200, 259, 340, 375, 466, 554, 569, 673, 688, 779, 869, 884, 912, 989, 1095, 1232, 1276, 1394, 1409, 1493, 1513, 1630, 1645, 1719, 1805, 1819, 1923, 2027, 2042, 2117, 2153, 2162, 2208, 2240, 2345, 2480, 2542, 2662, 2706, 2850, 2871, 2886, 3003, 3078

Offset: 1

Asher Auel, Jan 10 2000

nonn

Cf. A000203, A053226, A053238, A053239, A053240, A053241, A053242, A053244, A053245.

with(numtheory): f := [seq( `if`((sigma(i) > sigma(i+1)),i,print( )), i=1..5000)];
seq( `if`(f[i+1] - f[i] = 3,i,print( )), i=1..1000);

cp's OEIS Frontend

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

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A053240 n for which values not equal to 2 occur in the expansion of A053238.

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A053241 Numbers n such that A053238(n) = 2.

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A053242 Numbers n such that A053238(n) = 1.

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A053245 Numbers k such that both A053238(k) and A053238(k+1) = 1.

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Haskell

Maple

A053239 First differences between n for which sigma(n) > sigma(n+1), which are not 2.

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Programs

Maple

Formula

A053243 Numbers n such that A053238(n) = 3.

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Maple