A053226 -id:A053226 - cp's OEIS frontend

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

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

Asher Auel, Jan 06 2000

nonn

The asymptotic density of this sequence is 1/2 (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, A053222, A053226, A053230 (first differences).

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

with(numtheory): seq( `if`(sigma(i) < sigma(i+1),i,[][]), i=1..134);

Select[Range[150], DivisorSigma[1, #] < DivisorSigma[1, # + 1] &] (* Carl Najafi, Aug 16 2011 *)

is(n)=sigma(n)Charles R Greathouse IV, Mar 09 2014

A053222 First differences of sigma(n).

Original entry on oeis.org

2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8, 23, -43, 60, -76, 58

Offset: 1

Asher Auel, Jan 06 2000

a(A002961(n)) = 0. - Reinhard Zumkeller, Dec 28 2011

Considering the values |a(n)| <= 100 for n < 10^13, we notice that some odd values do not appear within that range, namely 9, 17, 25, 27, 33, 37, 39, 45, 47, 49, 51, 55, 57, 59, 69, 71, 77, 81, 83, 87, 89, 91, 95, 97, and 99. All the other absolute values <= 100 appear for n < 3600, with the exception of a(1159742043) = 62. - Giovanni Resta, Jun 26 2017

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A000203, A053223, A053225, A053227, A053224, A053226.

List([1..70], n -> Sigma(n+1)-Sigma(n)); # Muniru A Asiru, Feb 14 2018

a053222 n = a053222_list !! (n-1)
a053222_list = zipWith (-) (tail a000203_list) a000203_list
-- Reinhard Zumkeller, Oct 16 2011

[DivisorSigma(1, n+1) - DivisorSigma(1,n): n in [1..100]]; // G. C. Greubel, Sep 03 2018

A053222 := proc(n)
numtheory[sigma](n+1)-numtheory[sigma](n) ;
end proc: # R. J. Mathar, Jul 08 2013

DivisorSigma[1, Range[100]] // Differences (* Jean-François Alcover, Jan 26 2018 *)

a(n)=sigma(n+1)-sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

a(n) = A000203(n+1) - A000203(n).
G.f.:  2*(x-1)/(Q(0) -  2*x^2 + 2*x), where Q(k)= (2*x^(k+2) - x - 1)*k - 1 - 2*x + 3*x^(k+2) - x*(k+3)*(k+1)*(1-x^(k+2))^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 16 2013
G.f.: -1 + (1 - x)*Sum_{k>=1} k*x^(k-1)/(1 - x^k). - Ilya Gutkovskiy, Jan 29 2017

A340793 Sequence whose partial sums give A000203.

Original entry on oeis.org

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

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

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

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

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A053222 First differences of sigma(n).

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A340793 Sequence whose partial sums give A000203.

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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.