A051168 -id:A051168 - cp's OEIS frontend

A059863 a(n) = Product_{i=3..n} (prime(i)-4).

1, 1, 1, 3, 21, 189, 2457, 36855, 700245, 17506125, 472665375, 15597957375, 577124422875, 22507852492125, 967837657161375, 47424045200907375, 2608322486049905625, 148674381704844620625, 9366486047405211099375, 627554565176149143658125, 43301264997154290912410625

Offset: 1

Labos Elemer, Feb 28 2001

nonn

See A059862 for references.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.

C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Cf. A049296, A002110, A005867, A000847, A022008, A051160-A051168, A048298, A059861-A059865.

a(n) = prod(i=3, n, prime(i)-4); \\ Michel Marcus, Aug 25 2019

More terms from Michel Marcus, Aug 25 2019

A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

Original entry on oeis.org

1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, A8, A1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954

G. C. Greubel, Table of n, a(n) for n = 1..350
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly,66 (1959), 375-384.

Cf. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.
Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.
Cf. A059864, A059865.

[n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023

Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)

a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017

def A059864(n): return product(nth_prime(j) -5 for j in range(4,n+1))
[A059864(n) for n in range(1,31)] # G. C. Greubel, Feb 02 2023

A179968 'AT(n,k)' triangle read by rows. AT(n,k) is the number of aperiodic k-compositions of n.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 4, 6, 4, 0, 1, 4, 9, 8, 5, 0, 1, 6, 15, 20, 15, 6, 0, 1, 6, 21, 32, 35, 18, 7, 0, 1, 8, 27, 56, 70, 54, 28, 8, 0, 1, 8, 36, 80, 125, 120, 84, 32, 9, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 0

Offset: 1

John P. McSorley, Aug 03 2010

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n.
A k-composition is aperiodic (primitive) if its period is k, or if it is not the concatenation of at least two smaller compositions.
Let AT(n,k) denote the number of aperiodic k-compositions of n.
This sequence is the 'AT(n,k)' triangle read by rows.
If we form the row sum sequence of the 'AT(n, k)' triangle above we get sequence A056278, except for the first term.
If we form the triangle which counts the aperiodic k-compositions of n up to cyclic equivalence, ATE(n, k), and place an extra 0 at the end of each row of the 'ATE(n, k)' triangle, we get sequence A051168 (Lyndon words).

			The triangle begins
  1
  1, 0
  1, 2,  0
  1, 2,  3,  0
  1, 4,  6,  4,   0
  1, 4,  9,  8,   5,   0
  1, 6, 15, 20,  15,   6,  0
  1, 6, 21, 32,  35,  18,  7,  0
  1, 8, 27, 56,  70,  54, 28,  8, 0
  1, 8, 36, 80, 125, 120, 84, 32, 9, 0
For example, row 8 is 1, 6, 21, 32, 35, 18, 7, 0.
We have AT(8,2)=6 because there are 6 aperiodic 2-compositions of 8, namely: 17, 71, 26, 62, 35, 53. The remaining 2-composition of 8 is 44, it is not aperiodic.
We have AT(8,3)=21 because all 21 3-compositions of 8 are aperiodic.

John P. McSorley: Counting k-compositions of n with palindromic and related structures. Preprint, 2010. [Apparently unpublished as of Mar 27 2017]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.

Cf. A051168, A056278.

T[n_, k_]:=(k/n) * Sum[MoebiusMu[d] Binomial[n/d, k/d], {d, Divisors[GCD[n, k]]}]; Flatten[Table[T[n, k], {n, 11}, {k, n}]] (* Indranil Ghosh, Mar 27 2017, after the formula by Andrew Howroyd *)

T(n,k) = k * A051168(n,k). - Andrew Howroyd, Mar 26 2017
T(n,k) = (k/n) * Sum_{d | gcd(n,k)} mu(d) * binomial(n/d,k/d). - Andrew Howroyd, Mar 26 2017
G.f.: Sum_{k>=1}mu(k)y^k A(x^k)/(1 - y^k A(x^k)) where mu(k) is the Moebius Mu function and A(x) = x/(1-x). - Geoffrey Critzer, Aug 05 2022

a(56)-a(66) from Andrew Howroyd, Mar 26 2017

A263318 Number of aperiodic necklaces (Lyndon words) with 9 black beads and n white beads.

Original entry on oeis.org

0, 1, 5, 18, 55, 143, 333, 715, 1430, 2700, 4862, 8398, 13995, 22610, 35530, 54477, 81719, 120175, 173583, 246675, 345345, 476901, 650325, 876525, 1168695, 1542684, 2017356, 2615085, 3362260, 4289780, 5433714, 6835972, 8544965, 10616463, 13114465, 16112057

Offset: 0

Criel Merino, Pedro Antonio, Oct 14 2015

nonn

A row of triangle A051168.

Pedro Antonio, Table of n, a(n) for n = 0..100
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Index entries for sequences related to Lyndon words
Index entries for linear recurrences with constant coefficients, signature (6, -15, 23, -33, 51, -64, 63, -63, 64, -51, 33, -23, 15, -6, 1).

Cf. A001840, A006918, A011795, A011796, A011797, A051168.

CoefficientList[Series[(x (x^4 - x^3 + 3*x^2 - x + 1))/((x^2 + x + 1)^3 (1 - x)^9), {x, 0, 40}], x] (* Wesley Ivan Hurt, Oct 15 2015 *)
CoefficientList[Series[((-1+x^3)^-3-(-1+x)^-9)/9,{x,0,40}],x] (* Herbert Kociemba, Oct 16 2016 *)
LinearRecurrence[{6,-15,23,-33,51,-64,63,-63,64,-51,33,-23,15,-6,1},{0,1,5,18,55,143,333,715,1430,2700,4862,8398,13995,22610,35530},40] (* Harvey P. Dale, Feb 10 2023 *)

a(n)= (1/(n+9))*sumdiv(gcd(n+9,9), d, moebius(d)*binomial( (n+9)/d , 9/d )); \\ Michel Marcus, Oct 14 2015

from sympy import mobius, binomial, gcd, divisors
print([sum(mobius(d) * binomial((n + 9)//d, 9//d) for d in divisors(gcd(n + 9, 9))) // (n + 9) for n in range(51)]) # Indranil Ghosh, Mar 26 2017

a(n) = (1/(n+9))*Sum_{d divides gcd(n+9,9)} mu(d)*binomial((n+9)/d, 9/d).
G.f.: (x*(x^4-x^3+3*x^2-x+1))/((x^2+x+1)^3*(1-x)^9).
G.f.: ((-1+x^3)^-3-(-1+x)^-9)/9. - Herbert Kociemba, Oct 16 2016

More terms from Michel Marcus, Oct 14 2015

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A059863 a(n) = Product_{i=3..n} (prime(i)-4).

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A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

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A179968 'AT(n,k)' triangle read by rows. AT(n,k) is the number of aperiodic k-compositions of n.

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A263318 Number of aperiodic necklaces (Lyndon words) with 9 black beads and n white beads.

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