A053223 -id:A053223 - 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

A053228 n for which second differences of sigma(n) are positive.

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132

Offset: 1

Asher Auel, Jan 10 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000203, A053223, A053229.

with(numtheory): seq(`if`(sigma(i+2)-sigma(i+1)>(sigma(i+1)-sigma(i)),i,print( )), i=1..142);

Select[Range[150], DivisorSigma[1, #+2] - DivisorSigma[1, #+1] > (DivisorSigma[1, # + 1] - DivisorSigma[1, #]) &] (* G. C. Greubel, Sep 03 2018 *)
Position[Differences[#,2]&/@Partition[DivisorSigma[1,Range[140]],3,1],?(Positive[ #]&)][[All,1]] (* _Harvey P. Dale, Jan 18 2023 *)

A053229 Numbers n for which second differences of sigma(n) are negative.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 104, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129

Offset: 1

Asher Auel, Dec 10 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A000203, A053223, A053228.

with(numtheory): seq(`if`(sigma(i+2)-sigma(i+1)<(sigma(i+1)-sigma(i)),i,print( )),i=1..133);

Select[Range[150], DivisorSigma[1, # + 2] - DivisorSigma[1, # + 1] < (DivisorSigma[1, # + 1] - DivisorSigma[1, #]) &] (* G. C. Greubel, Sep 03 2018 *)

A053247 Second differences of sigma(n) that are positive.

Original entry on oeis.org

2, 7, 11, 7, 22, 24, 7, 34, 41, 14, 48, 40, 18, 68, 71, 21, 49, 75, 38, 102, 92, 100, 103, 47, 110, 96, 50, 138, 140, 15, 103, 134, 78, 195, 161, 6, 116, 194, 70, 182, 140, 72, 234, 178, 56, 156, 227, 76, 229, 218, 226, 276, 160, 260, 162, 26, 252, 280, 74, 224

Offset: 1

Asher Auel, Jan 11 2000

nonn

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

Cf. A000203, A053223, A053248.

with (numtheory): f := [seq( sigma(i+2) - sigma(i+1) - (sigma(i+1) - sigma(i)), i=1..200)];
seq(`if`(f[i] > 0,f[i],print( )), i=1..200);

Select[Differences[DivisorSigma[1,Range[150]],2],#>0&] (* Harvey P. Dale, May 10 2012 *)

a(n) = A053223(A053228(n))

A053246 First differences of chowla(n).

Original entry on oeis.org

0, 0, 2, -2, 5, -5, 6, -3, 4, -7, 15, -15, 9, -1, 6, -14, 20, -20, 21, -11, 3, -13, 35, -30, 10, -3, 15, -27, 41, -41, 30, -16, 5, -7, 42, -54, 21, -5, 33, -49, 53, -53, 39, -7, -7, -25, 75, -68, 35, -22, 25, -45, 65, -49, 47, -41, 9, -31, 107, -107, 33, 7, 22, -44, 59, -77, 57, -31

Offset: 1

Asher Auel, Jan 10 2000

sign

Second differences give A053223, for n>1.

If the first term is changed to 1, this is also the first differences of A001065. - N. J. A. Sloane, Jan 17 2023

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A048050, A053222, A053223.

Cf. also A001065.

[0] cat [DivisorSigma(1,n+1) - DivisorSigma(1,n) - 1: n in [2..100]]; // G. C. Greubel, Sep 03 2018

with(numtheory): seq( sigma(i+1) - sigma(i) - 1, i=2..100); # for n>1

Chowlan[n_] := If[n == 1, 0, DivisorSigma[1, n] - n - 1]; Table[Chowlan[n + 1] - Chowlan[n], {n, 1, 100}] (* G. C. Greubel, Sep 03 2018 *)
Differences[Join[{0},Table[DivisorSigma[1,n]-n-1,{n,2,100}]]] (* Harvey P. Dale, Dec 19 2022 *)

concat([0], vector(100, n, n++; sigma(n+1) - sigma(n) -1)) \\ G. C. Greubel, Sep 03 2018

a(n) = A053222(n) - 1, for n>1

A053248 Second differences of sigma(n) that are negative.

Original entry on oeis.org

1, 4, 10, 9, 11, 30, 10, 20, 40, 32, 16, 65, 13, 42, 82, 46, 12, 96, 26, 82, 106, 46, 18, 143, 57, 70, 114, 88, 40, 214, 26, 66, 136, 88, 120, 244, 30, 60, 160, 171, 47, 256, 36, 150, 266, 96, 40, 286, 88, 176, 226, 124, 12, 24, 342, 170, 230, 222, 94, 34, 443, 71

Offset: 1

Asher Auel, Jan 11 2000

nonn

Cf. A000203, A053223, A053247.

with (numtheory): f := [seq( sigma(i+2) - sigma(i+1) - (sigma(i+1) - sigma(i)), i=1..200)];
seq(`if`(f[i] < 0,f[i],print( )), i=1..200);

Abs[Select[Differences[DivisorSigma[1,Range[200]],2],Negative]] (* Harvey P. Dale, Aug 13 2015 *)

a(n) = A053223(A053229(n)).

A358395 Odd numbers k such that sigma(k) + sigma(k+2) > 2*sigma(k+1); odd terms in A053228.

Original entry on oeis.org

1125, 1573, 1953, 2205, 2385, 3465, 5185, 5353, 5773, 6433, 6613, 6825, 7245, 7425, 7665, 7693, 8505, 8925, 9133, 9205, 9405, 9945, 10393, 10773, 11473, 11653, 12285, 12493, 12705, 13473, 13585, 13725, 14025, 15013, 15145, 15433, 16065, 16245, 16905, 17253, 17325, 17953

Offset: 1

Jianing Song, Nov 13 2022

nonn

Odd numbers k such that A053223(k) > 0.
Terms congruent to 5 modulo 6 exist but must be very large: for example A053223(670173643268502741420822977335461337017377351999597045900203591953125) = 1311786588705365455963902347308393766941056366825184647502989937872.
A number m coprime to 2 and 3 such that sigma(m)/m >= 3 (m = A358412(3) = A358413(2) = 5^4*7^3*11^2*13^2*17*...*157 ~ 5.16404*10^66 is the smallest such number; see the link from Mercurial, the Spectre) produces a family of infinitely many terms congruent to 5 modulo 6 in this sequence, by Dirichlet's theorem on arithmetic progressions. Concretely, let k == 5 (mod 6), N(t) = t*k*(k+2) + (k+1)/6 for t >= 0, then:
(i) If sigma(k)/k >= 3. If N(t) is prime and 6*N(t)+1 is composite, then sigma(6*N(t)-1) >= 3*(6*N(t)-1), sigma(6*N(t)) = 12*(N(t)+1) and sigma(6*N(t)+1) >= 1+sqrt(6*N(t)+1)+(6*N(t)+1), so A053223(6*N(t)-1) >= sqrt(6*N(t)+1) - 25 >= sqrt(k+2) - 25 > 0.
(ii) If sigma(k+2)/(k+2) >= 3. If N(t) is prime and 6*N(t)-1 is composite, then sigma(6*N(t)+1) >= 3*(6*N(t)+1), sigma(6*N(t)) = 12*(N(t)+1) and sigma(6*N(t)-1) >= 1+sqrt(6*N(t)-1)+(6*N(t)-1), so A053223(6*N(t)-1) >= sqrt(6*N(t)-1) - 21 >= sqrt(k) - 21 > 0.

			1125 is a term since sigma(1126) = 1692 is smaller than the average of sigma(1125) = 2028 and sigma(1127) = 1368.

Jianing Song, Table of n, a(n) for n = 1..20000
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.
Wikipedia, Dirichlet's theorem on arithmetic progressions

Cf. A053228, A053223, A000203 (sigma), A358396.

Cf. also A358412, A358413.

isA358395(n) = (n%2) && (sigma(n) + sigma(n+2) > 2*sigma(n+1))

A358396 Even numbers k such that sigma(k) + sigma(k+2) < 2*sigma(k+1); even terms in A053229.

Original entry on oeis.org

104, 134, 164, 314, 404, 494, 524, 554, 566, 584, 674, 692, 734, 764, 854, 944, 974, 1124, 1154, 1196, 1214, 1304, 1322, 1364, 1394, 1484, 1574, 1682, 1724, 1754, 1784, 1814, 1826, 1844, 1994, 2024, 2144, 2204, 2384, 2414, 2456, 2474, 2564, 2624, 2654, 2804, 2834, 3002

Offset: 1

Jianing Song, Nov 13 2022

nonn

Even numbers k such that A053223(k) < 0.

			104 is a term since sigma(105) = 192 is greater than the average of sigma(104) = 210 and sigma(106) = 162.

Jianing Song, Table of n, a(n) for n = 1..20000

Cf. A053229, A053223, A000203 (sigma), A358395.

isA358396(n) = !(n%2) && (sigma(n) + sigma(n+2) < 2*sigma(n+1))

A167206 Binomial transform of A164555.

Original entry on oeis.org

1, 2, 4, 7, 10, 11, 8, 1, -6, -5, 16, 89, -348, -7747, -58764, -301959, -1226902, -4249557, -13125130, -36998357, -95306260, -219609123, -430081728, -623477651, -457458788, -3070156979, -61496380490, -630601717145, -4635893019708, -27904927526379

Offset: 0

Paul Curtz, Oct 30 2009

sign

Binomial transform of the numerators of the Bernoulli number fractions A164555/A027642.

Cf. A053223.

A164555 := proc(n) if n = 1 then -numer(bernoulli(n)) ; else numer(bernoulli(n)) ; end if; end proc:
read("transforms") : a164555 := [seq(A164555(n),n=0..50)] : BINOMIAL(a164555) ; # R. J. Mathar, Oct 31 2009

Keyword:sign set, sequence extended by R. J. Mathar, Oct 31 2009

cp's OEIS Frontend

A053222 First differences of sigma(n).

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A340793 Sequence whose partial sums give A000203.

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A053228 n for which second differences of sigma(n) are positive.

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A053229 Numbers n for which second differences of sigma(n) are negative.

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A053247 Second differences of sigma(n) that are positive.

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A053246 First differences of chowla(n).

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A053248 Second differences of sigma(n) that are negative.

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A358395 Odd numbers k such that sigma(k) + sigma(k+2) > 2*sigma(k+1); odd terms in A053228.

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