A134517 -id:A134517 - 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

A214360 Primes congruent to 23 modulo 3120613860.

Original entry on oeis.org

23, 3120613883, 6241227743, 9361841603, 12482455463, 15603069323, 18723683183, 21844297043, 24964910903, 28085524763, 34326752483, 43688594063, 62412277223, 115462712843, 124824554423, 156030693023, 159151306883, 171633762323, 180995603903, 196598673203

Offset: 1

Reinhard Zumkeller, Jul 13 2012

nonn

A211889(9) = 3120613860;

the first 10 terms constitute row 9 of triangle A211890, an arithmetic progression of 10 primes.

Cf. A010051.

Sequences of numbers congruent 23 modulo m: A134517 m=24, A141945 m=25, A140375 m=26, A141963 m=27, A141974 m=28, A141999 m=29, A132235 m=30, A142027 m=31, A142044 m=32, A142062 m=33, A142091 m=35, A142107 m=36, A142132 m=37, A142173 m=39, A142192 m=40, A142220 m=41, A142244 m=42, A142272 m=43, A142302 m=44, A142324 m=45, A142374 m=47, A142405 m=48, A142433 m=49, A142490 m=51, A142518 m=52, A142553 m=53, A142617 m=55, A142650 m=56, A142679 m=57, A142750 m=59, A142790 m=60, A142821 m=61, A142902 m=63, A142935 m=64, A140844 m=210.

a214360 n = a214360_list !! (n-1)
a214360_list = [x | k <- [0..], let x = 3120613860*k+23, a010051' x == 1]

select(isprime,[seq(23+i*3120613860,i=0..1000)]); # Robert Israel, Jun 07 2015

Select[Range[23, 2 10^11, 3120613860], PrimeQ] (* Vincenzo Librandi, Jun 07 2015 *)

is(n)=isprime(n) && n%3120613860==23 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 658414080n log n. - Charles R Greathouse IV, Jul 02 2016

A141376 Primes of the form -x^2 + 8xy + 8y^2 (as well as of the form 15x^2 + 24xy + 8*y^2).

Original entry on oeis.org

23, 47, 71, 167, 191, 239, 263, 311, 359, 383, 431, 479, 503, 599, 647, 719, 743, 839, 863, 887, 911, 983, 1031, 1103, 1151, 1223, 1319, 1367, 1439, 1487, 1511, 1559, 1583, 1607, 1823, 1847, 1871, 2039, 2063, 2087, 2111, 2207, 2351, 2399, 2423, 2447, 2543

Offset: 1

Laura Caballero Fernandez, Lourdes Calvo Moguer, Maria Josefa Cano Marquez, Oscar Jesus Falcon Ganfornina and Sergio Garrido Morales (oscfalgan(AT)yahoo.es), Jun 28 2008

nonn

Discriminant = +96.
Values of the quadratic form are {0, 8, 12, 15, 20, 23} mod 24, so this is a subsequence of A134517. - R. J. Mathar, Jul 30 2008
Is this the same sequence as A134517?
Substituting 2y = y' gives the quadratic form A141171, so these terms are a subsequence of the terms in A141171. - R. J. Mathar, Jun 10 2020

			a(2)=47 because we can write 47 = -1^2 + 8*1*2 + 8*2^2 (or 47 = 15*1^2 + 24*1*1 + 8*1^2).

Z. I. Borevich and I. R. Shafarevich, Number Theory.

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS: Index to related sequences, programs, references. OEIS wiki, June 2014.
D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.

Cf. A107003, A134517, A141171, A141373, A141375 (d = -96).

More terms from Arkadiusz Wesolowski, Jul 25 2012

A138905 a(n) is n-th prime == -1 (mod 6n).

Original entry on oeis.org

5, 23, 71, 167, 179, 431, 461, 863, 863, 839, 1583, 1511, 1949, 2099, 2339, 4127, 4283, 4751, 4673, 4919, 5669, 6599, 8693, 10079, 7349, 10607, 12149, 11087, 12527, 11159, 15809, 19583, 16829, 19583, 13859, 25703, 24197, 25307, 23633, 21839, 34439

Offset: 1

Zak Seidov, Apr 03 2008

nonn

			a(1) = 1st term in A007528 (Primes of form 6n-1)
a(2) = 2nd term in A068231 (Primes congruent to 11 (mod 12))
a(3) = 3rd term in A061242 (Primes of form 18n-1)
a(4) = 4th term in A134517 (Primes of form 24n-1)
a(5) = 5th term in A132236 (Primes congruent to 29 (mod 30))

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A007528, A061242, A068231, A093871, A132236, A134517, A138906, A138907.

a[n_]:=Module[{p=1,cnt=0},Until[cnt==n,If[Mod[Prime[p],6n]==6n-1,cnt++];p++];Prime[p-1]];Array[a,41] (* James C. McMahon, Jun 22 2025 *)

A195051 Number of divisors of 24*n - 1.

Original entry on oeis.org

2, 2, 2, 4, 4, 4, 2, 2, 4, 2, 2, 4, 2, 4, 2, 2, 4, 2, 8, 2, 2, 4, 4, 6, 2, 4, 2, 4, 4, 2, 2, 4, 4, 4, 2, 2, 2, 2, 8, 4, 2, 4, 2, 4, 4, 2, 6, 2, 6, 4, 2, 4, 4, 8, 2, 4, 2, 4, 4, 2, 8, 2, 2, 4, 2, 2, 2, 4, 4, 4, 4, 4, 4, 6, 4, 2, 2, 2, 4, 4, 4, 4, 4, 8, 2, 2

Offset: 1

Omar E. Pol, Jan 13 2012

Robert Israel, Table of n, a(n) for n = 1..10000
George E. Andrews, Frank G. Garvan, and Jie Liang, Self-conjugate vector partitions and the parity of the spt-function, Acta Arith., Vol. 158, No. 3 (2013), pp. 199-218; alternative link; author's link.

Cf. A000005, A001620, A092269, A134517, A183010, A183011, A195052, A195053.

List([1..100],n->Tau(24*n-1)); # Muniru A Asiru, Jun 27 2018

seq(numtheory:-tau(24*n-1),n=1..100); # Robert Israel, Jun 27 2018

Table[DivisorSigma[0, 24*n-1], {n, 100}] (* T. D. Noe, Jan 14 2012 *)

a(n) = numdiv(24*n-1); \\ Amiram Eldar, Dec 22 2023

a(n) = A000005(A183010(n)).
a(n) = 2 * A195052(n).
Sum_{k=1..n} a(k) ~ (n/3) * (log(n) + 2*gamma - 1 + 5*log(2) + 2*log(3)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 22 2023

A195052 Number of divisors of 24*n - 1 divided by 2.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 4, 1, 1, 2, 2, 3, 1, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 1, 1, 4, 2, 1, 2, 1, 2, 2, 1, 3, 1, 3, 2, 1, 2, 2, 4, 1, 2, 1, 2, 2, 1, 4, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 2, 1, 1, 1, 2, 2, 2, 2, 2, 4, 1, 1

Offset: 1

Omar E. Pol, Jan 13 2012

It appears that this sequence has the same parity as the spt function A092269 (See A195053). - Omar E. Pol, Jan 30 2012

George E. Andrews, Frank G. Garvan, and Jie Liang, Self-conjugate vector partitions and the parity of the spt-function, Acta Arith., Vol. 158, No. 3 (2013), pp. 199-218; alternative link; author's link.

Cf. A000005, A001620, A092269, A134517, A183010, A183011, A195051, A195053.

a[n_] := DivisorSigma[0, 24*n-1]/2; Array[a, 100] (* Amiram Eldar, Dec 22 2023 *)

a(n) = numdiv(24*n-1)/2; \\ Amiram Eldar, Dec 22 2023

a(n) = A000005(A183010(n))/2 = A195051(n)/2.

Sum_{k=1..n} a(k) ~ (n/6) * (log(n) + 2*gamma - 1 + 5*log(2) + 2*log(3)), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 22 2023

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

A139527 Numbers n such that numbers 24n+5 are primes.

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 8, 11, 12, 13, 16, 19, 21, 23, 27, 28, 29, 32, 33, 34, 39, 42, 44, 46, 49, 51, 53, 54, 57, 62, 67, 68, 71, 72, 78, 79, 81, 82, 83, 86, 89, 92, 93, 96, 97, 98, 99, 103, 106, 109, 112, 114, 116, 118, 119, 121, 123, 134, 141, 142, 144, 147, 148, 149, 153, 154

Offset: 1

Artur Jasinski, Apr 25 2008

nonn

Numbers n such that:
24n+1 is prime see A111174, primes 24n+1 see A107008
24n+5 is prime see A139527, primes 24n+5 see A107003
24n+7 is prime see A139483, primes 24n+7 see A107006
24n+11 is prime A139528, primes 24n+11 see A107007
24n+13 is prime see A139529, primes 24n+13 see A139530
24n+17 is prime see A139531, primes 24n+17 see A107181
24n+19 is prime see A139532, primes 24n+19 see A141373
24n+23 is prime see A131210, primes 24n+23 see A134517

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

a = {}; Do[If[PrimeQ[24 n + 5], AppendTo[a, n]], {n, 0, 200}]; a
Select[Table[(Prime[n]-5)/24,{n,800}],IntegerQ] (* Harvey P. Dale, Feb 25 2016 *)

A207374 Composites of the form 24n - 1.

Original entry on oeis.org

95, 119, 143, 215, 287, 335, 407, 455, 527, 551, 575, 623, 671, 695, 767, 791, 815, 935, 959, 1007, 1055, 1079, 1127, 1175, 1199, 1247, 1271, 1295, 1343, 1391, 1415, 1463, 1535, 1631, 1655, 1679, 1703, 1727, 1751, 1775, 1799, 1895, 1919, 1943, 1967, 1991, 2015

Offset: 1

Omar E. Pol, Feb 18 2012

nonn

Also denominators that are composite numbers A002808 in the Bruinier-Ono formula for the partition function (see A183010 and A183011).

The union of A134517 and this sequence gives A183010.

Cf. A008606, A134517, A183010, A183011.

Select[24*Range[250] - 1, ! PrimeQ[#] &] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2012 *)

A002808 INTERSECT A183010.

A256397 Primes congruent to {17, 23} mod 24.

Original entry on oeis.org

17, 23, 41, 47, 71, 89, 113, 137, 167, 191, 233, 239, 257, 263, 281, 311, 353, 359, 383, 401, 431, 449, 479, 503, 521, 569, 593, 599, 617, 641, 647, 719, 743, 761, 809, 839, 857, 863, 881, 887, 911, 929, 953, 977, 983, 1031, 1049, 1097, 1103, 1151, 1193

Offset: 1

Arkadiusz Wesolowski, Jun 03 2015

nonn

All these primes do not divide any number of the form 2^k + 3. Therefore, they are not in A256396.

Cf. A107181, A134517, A256396.

[p: p in PrimesUpTo(1193) | p mod 24 in {17, 23}];

Select[Prime@Range[196], MemberQ[{17, 23}, Mod[#, 24]] &]

select(p->my(k=p%24); k==17||k==23, primes(1000)) \\ Charles R Greathouse IV, Jun 03 2015

A107181 UNION A134517.

cp's OEIS Frontend

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

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A214360 Primes congruent to 23 modulo 3120613860.

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A141376 Primes of the form -x^2 + 8*x*y + 8*y^2 (as well as of the form 15*x^2 + 24*x*y + 8*y^2).

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A138905 a(n) is n-th prime == -1 (mod 6n).

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A195051 Number of divisors of 24*n - 1.

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A195052 Number of divisors of 24*n - 1 divided by 2.

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

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A141376 Primes of the form -x^2 + 8xy + 8y^2 (as well as of the form 15x^2 + 24xy + 8*y^2).