Lucian Craciun - cp's OEIS frontend

A320467 Two-column table read by rows: The Mayan 260-day Tzolkin cycle, with day names replaced by numbers.

1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 1, 14, 2, 15, 3, 16, 4, 17, 5, 18, 6, 19, 7, 20, 8, 1, 9, 2, 10, 3, 11, 4, 12, 5, 13, 6, 1, 7, 2, 8, 3, 9, 4, 10, 5, 11, 6, 12, 7, 13, 8, 14, 9, 15, 10, 16, 11, 17, 12, 18

Offset: 1

Lucian Craciun, Oct 13 2018

Day 1 of year 1 of the Mayan Long Count calendar (0.0.1.0.1) coincides with the first day of the Tzolkin cycle (1,1). Two Tzolkin cycles before that date, there was a new moon.

			The first pair, (1,1), represents 1 Imix; the second pair, (2,2), represents 2 Ik; the thirteenth pair, (13,13), represents 13 Ben; the fourteenth pair, (1,14), represents 1 Ix; the fifteenth pair, (2,15), represents 2 Men; etc.

Lucian Craciun, Table of n, a(n) for n = 1..520
John Walker, Calendar Converter.
Wikipedia, Tzolk'in.
Wikipedia, Mesoamerican Long Count calendar.

Cf. A081244, A215146.

For[{A := {}, k := 0}, k < 260, k++, A = Append[A, {1 + Mod[k, 13], 1 + Mod[k, 20]}]]; Flatten[A]
a[n_]:=(Mod[(n-1)/2, 13] + 1)*Mod[n, 2]+(Mod[n/2-1, 20] + 1)*(1-Mod[n, 2]); Array[a, 260] (* Stefano Spezia, Dec 07 2018 *)

a(2n-1) = ((n - 1) mod 13) + 1.
a(2n) = ((n - 1) mod 20) + 1.
a(n) = ((n - 1)/2 mod 13 + 1)*(n mod 2) + ((n/2 - 1) mod 20 + 1)*(1 - (n mod 2)). - Stefano Spezia, Dec 07 2018

A283074 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^5.

Original entry on oeis.org

1, 84331608790, 94482127740, 164273806200, 438726722148, 541278246600, 549361342530, 808172086449, 912226745430, 959218287720, 1017676553985, 1017868271175, 1078659050256, 1286556180525, 1418394308100, 1475851476960, 1489765799610, 1535790227400, 1562434592400, 1642639268270

Offset: 1

Lucian Craciun, Feb 28 2017

nonn

Equivalently, numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^5.

The asymptotic density of this sequence is 2.83248121476... * 10^(-10) (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

			The central binomial coefficient C(2*84331608790,84331608790) is divisible by 84331608790^5.

Giovanni Resta, Table of n, a(n) for n = 1..123 (terms < 10^13).
Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.

Cf. A000108, A000984, A014847, A121943, A282163, A282346, A283073, A282672.

a(3)-a(20) from Giovanni Resta, Mar 03 2017

A283073 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^4.

Original entry on oeis.org

1, 227736432, 338956200, 386160984, 482213160, 544508118, 548823405, 715592220, 726922482, 731987190, 1427877360, 1448431600, 1467104760, 1490842353, 1491241258, 1504640335, 1646570115, 1852712100, 1923506200, 1923927460, 1924947570, 2056580995, 2064409413

Offset: 1

Lucian Craciun, Feb 28 2017

nonn

Equivalently, numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^4.

The asymptotic density of this sequence is 1.330129946... * 10^(-7) (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

			The central binomial coefficient C(2*227736432,227736432) is divisible by 227736432^4.

Giovanni Resta, Table of n, a(n) for n = 1..1002
Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.
Wikipedia Mathematics Reference Desk, n^4 Divides Central Binomial Coefficient.

Cf. A000108, A000984, A014847, A121943, A282163, A282346, A283074, A282672.

A283073:={}; k:=4; For[n:=1, n<=10^9, n++, {f=FactorInteger[n], For[j:=1, j<=Length[f], j++, {b=True, If[Sum[Floor[2n/f[[j, 1]]^i]-2 Floor[n/f[[j, 1]]^i], {i, 1, Length[IntegerDigits[2n, f[[j, 1]]]]}]A283073=Append[A283073, n]]}] (* Legendre's formula for drastic time reduction *)

a(11)-a(22) from Giovanni Resta, Feb 28 2017

A282529 Decimal expansion of Product_{n>=2} (1-1/n!).

Original entry on oeis.org

3, 9, 5, 3, 3, 8, 5, 6, 7, 3, 6, 7, 4, 4, 5, 5, 6, 6, 0, 3, 2, 3, 5, 6, 2, 0, 0, 4, 3, 1, 1, 8, 0, 6, 1, 3, 0, 2, 6, 6, 5, 6, 4, 8, 6, 7, 5, 8, 7, 2, 0, 1, 3, 2, 5, 0, 1, 1, 9, 7, 3, 9, 0, 1, 5, 1, 7, 4, 1, 6, 2, 0, 3, 8, 4, 7, 7, 5, 6, 1, 2, 6, 7, 0, 9, 2, 0, 0, 7, 7, 9, 3, 8, 9, 1, 0, 2, 2, 7, 0, 1, 1, 5, 8, 6, 8, 5, 4, 9, 6, 4, 0, 8, 4, 6, 7

Offset: 0

Lucian Craciun, Feb 22 2017

Conjectured to be irrational, transcendental and normal, none have been shown.

			0.39533856736744556603235620043118061302665648675872...

Lucian Craciun, Table of n, a(n) for n = 0..9900
StackExchange, Closed form for Product_{n>=2} (1-1/n!), Nov 11 2016.
Wolfram Alpha, Product_{n>=2} (1-1/n!).

Cf. A238695, A247551.

evalf(product(1-1/n!, n = 2..infinity), 100); # Lucian Craciun, Feb 22 2017

RealDigits[Product[1-1/n!, {n, 2, 70}], 10, 100][[1]] (* Robert G. Wilson v, Mar 19 2014 *)

prodinf(n=3, 1-1/gamma(n)) \\ Charles R Greathouse IV, Nov 13 2014

Equals 1/A247551. - Amiram Eldar, Sep 15 2023

A282346 Least number m > 1 such that the central binomial coefficient C(2m,m) is divisible by m^n.

Original entry on oeis.org

2, 924, 154836, 227736432, 84331608790, 1138842118714300

Offset: 1

Lucian Craciun and Robert G. Wilson v, Feb 12 2017

Equivalently, least number m > 1 such that the m-th Catalan number C(2m,m)/(m+1) is divisible by m^n. - Lucian Craciun, Mar 01 2017
a(6) <= 4380346834858680. - David A. Corneth, Mar 04 2017
a(7) <= 2404760413443713325. - Giovanni Resta, Mar 16 2017

Wikipedia, Kummer's theorem.

Cf. A000108, A000984, A014847, A121943, A282163, A283073, A283074, A282672.

f[n_] := Block[{k = 2}, While[Mod[Binomial[2k, k], k^n] > 0, k++]; k]

a(4)-a(5) from Giovanni Resta, Feb 23 2017

a(6) from Giovanni Resta, Mar 15 2017

A282163 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^3.

Original entry on oeis.org

1, 154836, 985320, 1108536, 1113959, 1492260, 1576696, 1632708, 1649238, 1684540, 1805570, 1988008, 2508792, 2548810, 2550408, 2659260, 2698740, 2746590, 2995122, 3074552, 3286710, 3330795, 3538458, 3574200, 3730155, 4039932, 4160240, 4318548, 4374370, 4426695, 4523985

Offset: 1

Lucian Craciun, Feb 07 2017

nonn

Equivalently, numbers k such that the k-th Catalan number C(2*k,k)/(k+1) is divisible by k^3. - Lucian Craciun, Feb 09 2017

The asymptotic density of this sequence is 0.000031511777... (Ford and Konyagin, 2021). - Amiram Eldar, Jan 26 2021

			The central binomial coefficient C(2*154836,154836) is divisible by 154836^3.

Lucian Craciun, Table of n, a(n) for n = 1..16000 (corrected and extended by Giovanni Resta)
Kevin Ford and Sergei Konyagin, Divisibility of the central binomial coefficient binomial(2n, n), Trans. Amer. Math. Soc., Vol. 374, No. 2 (2021), pp. 923-953; arXiv preprint, arXiv:1909.03903 [math.NT], 2019-2020.
Wikipedia Mathematics Reference Desk, n^4 Divides Central Binomial Coefficient.

Cf. A000108, A000984, A014847, A121943, A282346, A283073, A283074, A282672.

A282163 := proc (n, m) local a, cbc, k; a := {}; cbc := binomial(2*n, n); for k from n+1 to m do cbc := cbc*(4-2/k); if type(cbc/k^3, integer) then a := `union`(a, {k}) end if end do; a end proc; A282163(0, 10^6)

Select[Table[n, {n, 10^6}], IntegerQ[Binomial[2#, #]/#^3] &] (* for small n *)
n := 0; m := 10^6; A282163 := {}; cbc := Binomial[2n, n]; For[k := n+1, k <= m, k++, {cbc *= 4-2/k, If[IntegerQ[cbc/k^3], A282163 = Append[A282163, k]]}] (* for large m *)
A282163:={}; k:=3; For[n:=1, n<=10^6, n++, {f=FactorInteger[n], For[j:=1, j<=Length[f], j++, {b=True, If[Sum[Floor[2n/f[[j, 1]]^i]-2 Floor[n/f[[j, 1]]^i], {i, 1, Length[IntegerDigits[2n, f[[j, 1]]]]}]A282163=Append[A282163, n]]}] (* Legendre's formula for drastic time reduction, Lucian Craciun, Feb 28 2017; optimized by Lucian Craciun, Mar 02 2017 *)

cp's OEIS Frontend

User: Lucian Craciun

A320467 Two-column table read by rows: The Mayan 260-day Tzolkin cycle, with day names replaced by numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A283074 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A283073 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A282529 Decimal expansion of Product_{n>=2} (1-1/n!).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A282346 Least number m > 1 such that the central binomial coefficient C(2m,m) is divisible by m^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A282163 Numbers k such that the central binomial coefficient C(2*k,k) is divisible by k^3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica