A280098 -id:A280098 - cp's OEIS frontend

A183010 a(n) = 24*n - 1.

-1, 23, 47, 71, 95, 119, 143, 167, 191, 215, 239, 263, 287, 311, 335, 359, 383, 407, 431, 455, 479, 503, 527, 551, 575, 599, 623, 647, 671, 695, 719, 743, 767, 791, 815, 839, 863, 887, 911, 935, 959, 983, 1007, 1031, 1055, 1079, 1103, 1127, 1151, 1175, 1199

Offset: 0

Omar E. Pol, Jan 21 2011

a(n) is also the denominator of the finite algebraic formula for the number of partitions of n, if n >= 1. The formula is  p(n) = Tr(n)/(24*n - 1), n >= 1. See theorem 1.1 of the Bruinier-Ono paper in the link. For the numerators see A183011.
It appears that a(n) is also the denominator of the coefficient of the third term in the n-th Bruinier-Ono "partition polynomial" H_n(x). See the Bruinier-Ono paper, chapter 5 "Examples". For the numerators see A183007. - Omar E. Pol, Jul 13 2011
Also exponents in the formula q^(-1) + q^23 + 2*q^47 + 3*q^71 + 5*q^95 + 7*q^119 + 11*q^143 + 15*q^167 + ... in which the coefficients are the partition numbers (see A000041, Example section). - Omar E. Pol, Feb 27 2013

			G.f. = -1 + 23*x + 47*x^2 + 71*x^3 + 95*x^4 + 119*x^5 + 143*x^6 + 167*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. H. Bruinier and K. Ono, Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms
A. Dabholkar, S. Murthy, and D. Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms, arXiv:1208.4074 [hep-th], 2012-2014, see p. 46.
H. Gupta, Congruent properties of sigma(n), Math. Student 13 (1945) 25-29.
E. Larson and L. Rolen, Integrality properties of the CM-values of certain weak Maass forms, arXiv:1107.4114 [math.NT], 2011.
K. Ono, Congruences for the Andrews spt-function, (see 2.1 Producing modular forms)
W. Sierpinski, Elementary Theory of numbers, Monografie Mathematyczne, vol. 42 (1964) chapt 4, p. 168.
Leo Tavares, Illustration: Star Pairs
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A000041, A000203, A008606, A134517 (subset of primes), A183009, A183011, A187206, A280097 (sum of divisors), A280098.

Cf. A008594.

[24*n-1: n in [0..50]]; // G. C. Greubel, Aug 14 2018

Range[23, 2000, 24] (* Vladimir Joseph Stephan Orlovsky, Jun 14 2011 *)
(24*Range[0,50])-1 (* Harvey P. Dale, Mar 28 2015 *)

a(n)=24*n-1 \\ Charles R Greathouse IV, Jun 14 2011

a(n) = A008606(n) - 1.
a(1)=23, a(2)=47, a(n) = 2*a(n-1) - a(n-2). - Vincenzo Librandi, Jan 23 2011
a(n) = A183011(n)/A000041(n). - Omar E. Pol, Jul 14 2011
24 * A280098(n) = A000203(a(n)) if n>0. - Michael Somos, Dec 25 2016
E.g.f.: (24*x-1)*exp(x). - G. C. Greubel, Aug 14 2018
G.f.: (-1 + 25*x)/(1-x)^2. - Wolfdieter Lang, Dec 10 2021
a(n) = 2*A008594(n) - 1. - Leo Tavares, Jun 06 2023

A280097 Sum of the divisors of 24*n - 1.

Original entry on oeis.org

24, 48, 72, 120, 144, 168, 168, 192, 264, 240, 264, 336, 312, 408, 360, 384, 456, 432, 672, 480, 504, 576, 600, 744, 600, 720, 648, 744, 840, 720, 744, 840, 912, 984, 840, 864, 888, 912, 1296, 1104, 984, 1080, 1032, 1272, 1176, 1104, 1368, 1152, 1488, 1320, 1224, 1320, 1344, 1824, 1320

Offset: 1

Omar E. Pol, Dec 25 2016

All terms are multiples of 24 [Gupta, Sierpinski]. - Vincenzo Librandi, Apr 07 2011

Note that 24n - 1 is also the denominator of the Bruinier-Ono finite algebraic formula for the number of partitions of n (Cf. A183010).

			For n = 5 we have that 24*5 - 1 = 119, and the sum of the divisors of 119 is 1 + 7 + 17 + 119 = 144, so a(5) = 144.

Hansraj Gupta, Congruent properties of sigma(n), Math. Student 13 (1945), 25-29; entire issue.
Wacław Sierpiński, Elementary Theory of numbers, Monografie Mathematyczne, Vol. 42 (1964), chapter 4, p. 168.

Cf. A000203, A008606, A183010, A280098.

DivisorSigma[1,24*Range[60]-1] (* Harvey P. Dale, Jan 25 2024 *)

a(n) = sigma(24*n - 1); \\ Amiram Eldar, Jan 09 2025

a(n) = A000203(A183010(n)).

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = 4*Pi^2/3 = 13.159472... . - Amiram Eldar, Mar 28 2024

A291900 Sum of the divisors of 24*n - 1, divided by 24, minus n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 2, 0, 3, 0, 0, 2, 0, 9, 0, 0, 2, 2, 7, 0, 4, 0, 3, 6, 0, 0, 3, 5, 7, 0, 0, 0, 0, 15, 6, 0, 3, 0, 9, 4, 0, 10, 0, 13, 5, 0, 3, 3, 22, 0, 4, 0, 5, 12, 0, 19, 0, 0, 13, 0, 0, 0, 10, 14, 4, 6, 7, 5, 19, 11, 0, 0, 0, 16, 5, 4, 12, 8, 28, 0, 0, 0, 0, 35, 6, 4, 0, 5, 32, 4, 18, 8, 0, 31, 0

Offset: 1

Omar E. Pol, Nov 02 2017

The indices of the zeros give A131210.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000203, A008606, A086463, A131210, A134517, A183010, A183011, A280097, A280098, A294614.

a[n_] := DivisorSigma[1, 24 n - 1]/24 - n; Array[a, 90] (* Robert G. Wilson v, Nov 04 2017 *)

a(n) = sigma(24*n-1)/24 - n; \\ Michel Marcus, Nov 04 2017

a(n) = sigma(24*n-1)/24 - n = A000203(A183010(n))/24 - n = A280097(n)/24 - n = A280098(n) - n.

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 - 1/2 = 0.048311... . - Amiram Eldar, Mar 28 2024

A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

Original entry on oeis.org

1, 2, 4, 4, 5, 6, 7, 10, 9, 12, 11, 14, 16, 14, 15, 16, 20, 22, 19, 20, 21, 22, 31, 28, 28, 26, 30, 34, 29, 30, 36, 32, 40, 38, 35, 36, 37, 56, 39, 40, 41, 42, 52, 48, 57, 50, 47, 62, 49, 50, 56, 60, 64, 54, 55, 62, 57, 70, 68, 60, 66, 62, 76, 70, 70, 76

Offset: 1

M. F. Hasler, Dec 08 2017

Robert G. Wilson v observes in A280098 that {1, 3, 4, 6, 8, 12, 24} seem to be the only positive integers k such that sigma(kn-1)/k is an integer for all n > 0.

Muniru A Asiru, Table of n, a(n) for n = 1..100000

Cf. A280098 (analog for k = 24), A097723 (analog for k = 4), A033686 (analog for k = 3), A000203 (sigma, also the  analog for k = 1).
The analog for k = 8 is A258835, up to the offset.
The analog for k = 6 is A098098 (up to the offset), a signed variant of this and the preceding one is A258831.
Cf. A086463.

sequence := List([1..10^5], n-> Sigma(12 *n-1)/12); # Muniru A Asiru, Dec 28 2017

with(numtheory):
seq(sigma(12*n-1)/12, n=1..10^3); # Muniru A Asiru, Dec 28 2017

Array[DivisorSigma[1, 12 # - 1]/12 &, 66] (* Michael De Vlieger, Dec 08 2017 *)

PARI
```
vector(90,n,sigma(12*n-1)/12)
```

Sum_{k=1..n} a(k) = c * n^2 + O(n*log(n)), where c = Pi^2/18 = 0.548311... (A086463). - Amiram Eldar, Mar 28 2024

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A183010 a(n) = 24*n - 1.

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A280097 Sum of the divisors of 24*n - 1.

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A291900 Sum of the divisors of 24*n - 1, divided by 24, minus n.

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A295012 a(n) = sigma(12n - 1)/12, where sigma = sum of divisors (A000203).

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