A053225 -id:A053225 - cp's OEIS frontend

A053222 First differences of sigma(n).

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

A053227 First differences of sigma(n) that are negative.

Original entry on oeis.org

1, 4, 2, 6, 14, 13, 19, 10, 12, 29, 2, 26, 40, 15, 6, 53, 4, 48, 52, 6, 6, 24, 67, 21, 44, 48, 40, 30, 106, 43, 76, 30, 72, 121, 44, 88, 65, 42, 116, 12, 90, 122, 40, 24, 154, 15, 115, 112, 18, 30, 54, 170, 64, 134, 96, 28, 2, 36, 227, 18, 68, 184, 79, 120, 176, 132, 148

Offset: 1

Asher Auel, Jan 06 2000

nonn

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A053222, A053225.

with(numtheory): seq( `if`((sigma(i) > sigma(i+1)),(sigma(i) - sigma(i+1)),print( )), i=1..200);

Abs[Select[Differences[DivisorSigma[1,Range[200]]],#<0&]] (* Harvey P. Dale, Apr 17 2022 *)

a(n) = A053222(A053226(n))

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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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Maple

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A053227 First differences of sigma(n) that are negative.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula