cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A176266 Binomial(prime(n),s)/prime(n) where s is the sum of the decimal digits of n.

Original entry on oeis.org

1, 1, 2, 5, 42, 132, 1144, 3978, 35530, 1, 15, 210, 2470, 22386, 228459, 2908360, 37584261, 284291205, 3701413144, 35, 852, 19019, 349812, 6529292, 132435472, 2000945100, 24366118700, 328386663605, 3520256293710, 2072, 81375, 2271776, 59988852, 1227434238, 33401522154, 584134601050, 11919696387170, 234924043375476, 3924875235562164, 208335
Offset: 1

Views

Author

Michel Lagneau, Dec 07 2010

Keywords

Comments

For n = 10^p, a(n) = 1.

Examples

			a(5) = 42 because prime(5) = 11, s = 5,  binomial(11,5)/11 = 462/11=42.
a(16)=2908360 because prime(16)=53, s=7, binomial(53,7)/53 =154143080/53 = 2908360.

Crossrefs

Cf. A075872.

Programs

  • Maple
    A176266 := proc(n) binomial(ithprime(n),A007953(n))/ithprime(n) ; end proc:
seq(A176266(n),n=1..20) ;
  • Mathematica
    Table[Binomial[Prime[n],Total[IntegerDigits[n]]]/Prime[n],{n,40}] (* Harvey P. Dale, Oct 25 2020 *)
  • Sage
    A176266 = lambda n: binomial(nth_prime(n), sum(n.digits()))/nth_prime(n) # D. S. McNeil, Dec 08 2010

Formula

a(n) = A007318( A000040(n), A007953(n))/A000040(n).
a(n) = A060604(n)/A000040(n), n<10.

A064813 a(n) = binomial(composite(n), n), where composite = A002808, composite numbers.

Original entry on oeis.org

4, 15, 56, 126, 252, 924, 3432, 6435, 11440, 43758, 167960, 293930, 497420, 1961256, 3268760, 5311735, 8436285, 13123110, 54627300, 225792840, 354817320, 548354040, 834451800, 1251677700, 5414950296, 8122425444, 12033222880
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 23 2001

Keywords

Links

Crossrefs

Cf. A002808, A060604.

Programs

  • PARI
    j=0; i=4; while(i<100, while(isprime(i),i=i+1); j=j+1; print(binomial(i,j)); i=i+1)
  • PARI
    nextComp(n)= { if (!isprime(n), return(n)); return(n + 1) } { c=3; for (n=1, 200, c=nextComp(c + 1); write("b064813.txt", n, " ", binomial(c, n)) ) } \\ Harry J. Smith, Sep 27 2009

Extensions

More terms from Matthew Conroy, Oct 30 2001

A075872 a(n) = binomial(prime(n),n)/prime(n) where prime(n) = n-th prime.

Original entry on oeis.org

1, 1, 2, 5, 42, 132, 1144, 3978, 35530, 690690, 2731365, 50067108, 429757960, 1822766520, 15991836267, 280086337895, 4703540164785, 21512315482350, 360471372561300, 3174207914954076, 14859478810664136, 248599618581498860, 2209822117125283440, 36246606227404101045
Offset: 1

Views

Author

Lekraj Beedassy, Oct 16 2002

Keywords

Comments

A prime p divides all the entries (binomial coefficients) in the p-th row of Pascal's triangle.

Crossrefs

Cf. A060604, A000040.

Programs

  • Maple
    seq(binomial(ithprime(n),n)/ithprime(n),n=1..30);
  • Mathematica
    f[n_]:=Module[{pn=Prime[n]},Binomial[pn,n]/pn]
f/@Range[30]  (* Harvey P. Dale, Feb 25 2011 *)
  • PARI
    a(n) = my(p=prime(n)); binomial(p, n)/p; \\ Michel Marcus, Jul 15 2022

Formula

a(n) = A060604(n)/A000040(n).

Extensions

More terms from Emeric Deutsch, Mar 04 2004

A081384 Numbers k such that binomial(prime(k), k) is divisible by k^2.

Original entry on oeis.org

1, 60, 624, 663, 1309, 1638, 1950, 2145, 2394, 2470, 2508, 3640, 4186, 5016, 5187, 5928, 6138, 6417, 6479, 7395, 7800, 7920, 7956, 8372, 8415, 8580, 8645, 9350, 10340, 10582, 11220, 11466, 11484, 12540, 12825, 12920, 13325, 13680, 13833, 14620, 14911, 15708
Offset: 1

Views

Author

Labos Elemer, Mar 26 2003

Keywords

Links

Crossrefs

Cf. A060604.

Programs

  • Mathematica
    Do[s=Binomial[Prime[n], n]; If[IntegerQ[s/n^2], Print[n]], {n, 1, 10000}]
Select[Range[10000],Divisible[Binomial[Prime[#],#],#^2]&] (* Harvey P. Dale, Mar 10 2013 *)
  • PARI
    is(k) = !(binomial(prime(k), k) % k^2); \\ Amiram Eldar, Oct 04 2024

Extensions

More terms from Harvey P. Dale, Mar 10 2013

A117927 a(n) = binomial(s(n), n) where s(n) = n-th semiprime.

Original entry on oeis.org

4, 15, 84, 210, 2002, 5005, 116280, 319770, 2042975, 5311735, 193536720, 548354040, 1476337800, 9669554100, 25140840660, 991493848554, 6499270398159, 27900908274925, 280576272201225, 1210269541711230, 3342649210440540, 34315056105966195, 227068876035237600, 2305629203680278840
Offset: 1

Views

Author

Jonathan Vos Post, May 03 2006

Keywords

Comments

Semiprime analog of A060604.

Examples

			a(1) = binomial(4,1) = 4.
a(2) = binomial(6,2) = 15.
a(3) = binomial(9,3) = 84.
a(4) = binomial(10,4) = 210.
a(5) = binomial(14,5) = 2002.
a(6) = binomial(15,6) = 5005.
a(7) = binomial(21,7) = 116280.
a(8) = binomial(22,8) = 319770.
a(9) = binomial(25,9) = 2042975.
a(10) = binomial(26,10) = 5311735.

Links

Crossrefs

Cf. A001358, A060604.

Programs

  • Mathematica
    Binomial[First[#],Last[#]]&/@With[{sprs=Select[Range[70],PrimeOmega[#] == 2&]},Thread[{sprs,Range[Length[sprs]]}]] (* Harvey P. Dale, Aug 10 2012 *)
  • PARI
    s(n) = my(k=1, nb=0); while (nb != n, if (bigomega(k)==2, nb++); if (nb != n, k++)); k; \\ A001358
a(n) = binomial(s(n), n); \\ Michel Marcus, Jul 15 2022

Formula

a(n) = binomial(A001358(n), n).

Extensions

More terms from Michel Marcus, Jul 15 2022

A365628 a(n) = binomial(primorial(n), n).

Original entry on oeis.org

1, 2, 15, 4060, 78738660, 545754554499462, 1018081517447240182211275, 1793004475784081302284255717158418120, 1943305407393725342965469143054357602760779899437185, 3772316402417100592416011698371929155605067111502494326520988270728160
Offset: 0

Views

Author

Darío Clavijo, Sep 13 2023

Keywords

Crossrefs

Cf. A002110, A007318, A014062, A060604, A086687.

Programs

  • PARI
    a(n) = binomial(vecprod(primes(n)), n); \\ Michel Marcus, Sep 14 2023
  • Python
    from sympy import binomial, primorial
a = lambda n: binomial(primorial(n), n)
print([a(n) for n in range(1,10)])

Formula

a(n) = binomial(A002110(n), n).
Showing 1-6 of 6 results.

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