A008316 -id:A008316 - cp's OEIS frontend

A349115 a(n) = 8^n * P(n, 3*n), where P(n, x) is n-th Legendre polynomial.

1, 24, 3424, 926208, 369378816, 194988441600, 128184980586496, 100904418485993472, 92542260511611682816, 96909547417109671182336, 114095278582299648325582848, 149184455262733048487847395328, 214496285274348399077675463868416, 336346643957900669242934177071890432

Offset: 0

Vaclav Kotesovec, Nov 08 2021

nonn

In general, for k>=1, P(n, k*n) ~ 2^n * k^n * n^(n - 1/2) / sqrt(Pi).

Eric Weisstein's World of Mathematics, Legendre Polynomial.
Wikipedia, Legendre polynomials.

Cf. A008316, A110129, A349113, A349114.

Table[8^n*LegendreP[n, 3*n], {n, 0, 15}]

a(n) = 8^n*pollegendre(n, 3*n); \\ Michel Marcus, Nov 08 2021

a(n) ~ 2^(4*n) * 3^n * n^(n - 1/2) / sqrt(Pi).

A330203 Composite numbers k such that D(k) == 3 (mod k), where D(k) is the k-th central Delannoy number (A001850).

Original entry on oeis.org

10, 15, 50, 370, 2418, 4371, 5341, 8430, 20535, 25338, 26958, 278674, 1194649, 4304445, 11984885, 12327121, 20746461, 27585010, 72363853, 79501818

Offset: 1

Amiram Eldar, Dec 05 2019

Equivalently, composite numbers k such that P(k, 3) == 3 (mod k), where P(k, 3) = D(k) is the k-th Legendre polynomial evaluated at 3.

P(p, 3) == 3 (mod p) for all primes p. This is a special case of Schur congruences, named after Issai Schur, first published by his student Hildegard Ille in her Ph.D. thesis in 1924, and proven by Wahab in 1952. This sequence consists of the composite numbers for which the congruence holds.

			10 is in the sequence since it is composite and D(10) = 8097453 == 3 (mod 10).

Hildegard Ille, Zur Irreduzibilität der Kugelfunktionen, Jahrbuch der Dissertationen der Universität Berlin, (1924).
Peter S. Landweber, Elliptic Curves and Modular Forms in Algebraic Topology: Proceedings of a Conference held at the Institute for Advanced Study, Princeton, Sept. 15-17, 1986, Springer, 2006. See pp. 74-76.

Jean-Paul Allouche and Guentcho Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol 214 (2000), pp. 21-49.
S. K. Chatterjea, On Congruence Properties of Legendre Polynomials, Mathematics Magazine, Vol. 34, No. 6 (1961), pp. 329-336.
Sen-Peng Eu, Shu-Chung Liu, and Yeong-Nan Yeh, On the Congruences of Some Combinatorial Numbers, Studies in Applied Mathematics, Vol. 116, No. 2 (2006), pp. 135-144.
J. H. Wahab, New cases of irreducibility for Legendre polynomials, Duke Mathematical Journal, Vol. 19 (1952), pp. 165-176.

Cf. A001850, A008316.

Select[Range[2500], CompositeQ[#] && Divisible[LegendreP[#, 3] - 3, #] &]

a, b = 1, 1
for n in range(1, 10000):
    a, b = b, ((6*n-3)*b - (n-1)*a)//n
    if (b%n == 3) and (not Integer(n).is_prime()): print(n)  # Robin Visser, Aug 08 2023

a(18) from Robin Visser, Aug 08 2023

a(19)-a(20) from Robin Visser, Sep 11 2023

A349113 a(n) = 8^n * P(3*n, n), where P(n, x) is n-th Legendre polynomial.

Original entry on oeis.org

1, 8, 40636, 748832256, 37759888297756, 4086692369433395200, 815254385427670754825764, 270587150855247020644760551424, 138859707622050969870951620062449436, 104286590422721059977069662227099300134912, 109828573459404650800550127862919905133973562480

Offset: 0

Vaclav Kotesovec, Nov 08 2021

nonn

In general, for k>=1, P(k*n, n) ~ 2^(k*n) * n^(k*n) / sqrt(k*Pi*n).

Eric Weisstein's World of Mathematics, Legendre Polynomial.
Wikipedia, Legendre polynomials.

Cf. A008316, A110129, A349077, A349115.

Table[8^n*LegendreP[3*n, n], {n, 0, 12}]

a(n) = 8^n*pollegendre(3*n, n); \\ Michel Marcus, Nov 08 2021

a(n) ~ 2^(6*n) * n^(3*n - 1/2) / sqrt(3*Pi).

A349114 a(n) = 4^n * P(n, 2*n), where P(n, x) is n-th Legendre polynomial.

Original entry on oeis.org

1, 8, 376, 33984, 4526176, 797459200, 174910868224, 45926958135296, 14047764722238976, 4905641267399503872, 1925859774286175997952, 839619968812285810868224, 402496047174560754869846016, 210424519428145503482634174464, 119148510992477432889126160826368

Offset: 0

Vaclav Kotesovec, Nov 08 2021

nonn

Eric Weisstein's World of Mathematics, Legendre Polynomial.
Wikipedia, Legendre polynomials.

Cf. A008316, A110129, A349077, A349115.

Table[4^n*LegendreP[n, 2*n], {n, 0, 16}]

a(n) = 4^n*pollegendre(n, 2*n); \\ Michel Marcus, Nov 08 2021

a(n) ~ 2^(4*n) * n^(n - 1/2) / sqrt(Pi).

A171484 Beukers integral int(int( -log(xy) / (1-xy) * P_n(2x-1) P_n(2y-1) ,x=0..1,y=0..1)) = (A(n) + B(n)zeta(3)) / A003418(n)^3. This sequence gives negated values of A(n).

Original entry on oeis.org

0, 12, 1404, 750372, 137096340, 425299945236, 11144361386340, 104074481089949004, 23323094579273069340, 18031967628526215059268, 525443267415363230379732, 20671296686851400981142679500

Offset: 0

Max Alekseyev, Dec 09 2009

nonn

Values of B(n) are given in A171485. P_n(x) are the Legendre Polynomials (see A008316) defined by n!*P_n(x) = (d/dx)^n (x^n*(1-x)^n).

F. Beukers, A note on the irrationality of zeta(2) and zeta(3), Bull. London Math. Soc. 11 (1979) 268-272.
Wikipedia, Apéry's theorem

Cf. A104684.

A171485 Beukers integral Integral_{y = 0..1} Integral_{x = 0..1} -log(xy) / (1-xy) * P_n(2x-1) P_n(2y-1) dx dy = (A(n) + B(n)zeta(3)) / A003418(n)^3. This sequence gives the values of B(n).

Original entry on oeis.org

2, 10, 1168, 624240, 114051456, 353810160000, 9271076400000, 86580328116240000, 19402654331894400000, 15000926812307614080000, 437120128035736887168000, 17196604114594832318160000000, 514325437537328572480262784000, 34134351456507030556755674947200000

Offset: 0

Max Alekseyev, Dec 09 2009

Values of A(n) are given in A171484. P_n(x) are the Legendre Polynomials defined by n!*P_n(x) = (d/dx)^n (x^n*(1-x)^n), see A008316.

F. Beukers, A note on the irrationality of zeta(2) and zeta(3), Bull. London Math. Soc., Vol. 11, No. 3 (1979), 268-272.
Wikipedia, Apéry's theorem

Cf. A002117, A003418, A005259, A104684.

Cf. A008316, A171484.

seq( 2 * lcm(seq(i, i = 1..n))^3 * add(binomial(n,k)^2*binomial(n+k,k)^2, k = 0..n), n = 0..20); # Peter Bala, Aug 01 2025

Join[{2}, Table[2*(LCM @@ Range[n])^3 * HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1, 1, 1}, 1], {n, 1, 20}]] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = 2 * A003418(n)^3 * A005259(n). - Peter Bala, Aug 01 2025

A330204 Composite numbers k such that P(k, 5) == 5 (mod k), where P(k, 5) = A006442(k) is the k-th Legendre polynomial evaluated at 5.

Original entry on oeis.org

4, 15, 35, 165, 255, 615, 1815, 1876, 2636, 2948, 5380, 5565, 11235, 28545, 288380, 903644, 1807995, 2486165, 2674060, 10538572, 11791595, 14145121, 28558415, 45153277, 45682751

Offset: 1

Amiram Eldar, Dec 05 2019

P(p, 5) == 5 (mod p) for all primes p. This is a special case of Schur congruences (see A330203 for references). This sequence consists of the composite numbers for which the congruence holds.

			4 is in the sequence since it is composite and P(4, 5) = 2641 == 5 (mod 4).

Cf. A006442, A008316, A330203.

Select[Range[3000], CompositeQ[#] && Divisible[LegendreP[#, 5] - 5, #] &]

isok(k) = Mod(subst(pollegendre(k), x, 5), k) == 5;
forcomposite (k=1, 10000, if (isok(k), print1(k, ", "))); \\ Michel Marcus, Dec 06 2019

a, b = 1, 5
for n in range(2, 10000):
    a, b = b, ((10*n-5)*b - (n-1)*a)//n
    if (b%n == 5%n) and (not Integer(n).is_prime()): print(n)  # Robin Visser, Aug 17 2023

a(22)-a(23) from Robin Visser, Aug 17 2023

a(24)-a(25) from Robin Visser, Sep 11 2023

A330205 Composite numbers k such that P(k, 7) == 7 (mod k), where P(k, 7) = A084768(k) is the k-th Legendre polynomial evaluated at 7.

Original entry on oeis.org

6, 15, 21, 22, 105, 119, 231, 426, 483, 1290, 1939, 4429, 4450, 4578, 10609, 12999, 14118, 16899, 23262, 26733, 37401, 39858, 82194, 108345, 121335, 127434, 302253, 380757, 724647, 836437, 840147, 1078270, 1522677, 2007411, 15009050, 28913991

Offset: 1

Amiram Eldar, Dec 05 2019

P(p, 7) == 7 (mod p) for all primes p. This is a special case of Schur congruences (see A330203 for references). This sequence consists of the composite numbers for which the congruence holds.

			6 is in the sequence since it is composite and P(6, 7) = 1651609 == 7 (mod 6).

Cf. A008316, A084768, A330203.

Select[Range[2000], CompositeQ[#] && Divisible[LegendreP[#, 7] - 7, #] &]

isok(k) = Mod(subst(pollegendre(k), x, 7), k) == 7;
forcomposite (k=1, 10000, if (isok(k), print1(k, ", "))); \\ Michel Marcus, Dec 06 2019

a, b = 1, 7
for n in range(2, 10000):
    a, b = b, ((14*n-7)*b - (n-1)*a)//n
    if (b%n == 7%n) and (not Integer(n).is_prime()): print(n)  # Robin Visser, Aug 18 2023

a(35)-a(36) from Robin Visser, Aug 18 2023

cp's OEIS Frontend

A349115 a(n) = 8^n * P(n, 3*n), where P(n, x) is n-th Legendre polynomial.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A330203 Composite numbers k such that D(k) == 3 (mod k), where D(k) is the k-th central Delannoy number (A001850).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Sage

Extensions

A349113 a(n) = 8^n * P(3*n, n), where P(n, x) is n-th Legendre polynomial.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349114 a(n) = 4^n * P(n, 2*n), where P(n, x) is n-th Legendre polynomial.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A171484 Beukers integral int(int( -log(x*y) / (1-x*y) * P_n(2*x-1) * P_n(2*y-1) ,x=0..1,y=0..1)) = (A(n) + B(n)*zeta(3)) / A003418(n)^3. This sequence gives negated values of A(n).

Views

Author

Keywords

Comments

Links

Crossrefs

A171485 Beukers integral Integral_{y = 0..1} Integral_{x = 0..1} -log(x*y) / (1-x*y) * P_n(2*x-1) * P_n(2*y-1) dx dy = (A(n) + B(n)*zeta(3)) / A003418(n)^3. This sequence gives the values of B(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A330204 Composite numbers k such that P(k, 5) == 5 (mod k), where P(k, 5) = A006442(k) is the k-th Legendre polynomial evaluated at 5.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Sage

Extensions

A330205 Composite numbers k such that P(k, 7) == 7 (mod k), where P(k, 7) = A084768(k) is the k-th Legendre polynomial evaluated at 7.

Views

Author

Keywords

A171484 Beukers integral int(int( -log(xy) / (1-xy) * P_n(2x-1) P_n(2y-1) ,x=0..1,y=0..1)) = (A(n) + B(n)zeta(3)) / A003418(n)^3. This sequence gives negated values of A(n).

A171485 Beukers integral Integral_{y = 0..1} Integral_{x = 0..1} -log(xy) / (1-xy) * P_n(2x-1) P_n(2y-1) dx dy = (A(n) + B(n)zeta(3)) / A003418(n)^3. This sequence gives the values of B(n).