A005735 -id:A005735 - cp's OEIS frontend

A005733 Least k such that binomial(k,n) has n or more distinct prime factors.

2, 4, 9, 10, 22, 26, 40, 50, 54, 55, 78, 115, 123, 154, 155, 209, 288, 220, 221, 292, 301, 378, 494, 494, 551, 715, 670, 786, 805, 803, 1079, 966, 1190, 1222, 1274, 1274, 1276, 1771, 1836, 1807, 1834, 2147, 2263, 2519, 2519, 3021, 3306, 3306, 3427, 3441, 3445

Offset: 1

N. J. A. Sloane

nonn

Table 3 in Selmer's paper has typos for n = 83, 100 and 117. - T. D. Noe, Apr 05 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
Ernst S. Selmer, On the number of prime divisors of a binomial coefficient, Math. Scand. 39 (1976), no. 2, 271-281.

Cf. A005735, A129233.

Table[n=k; b=1; While[n++; b=b*n/(n-k); Length[FactorInteger[b]]T. D. Noe, Apr 05 2007 *)
lk[n_]:=Module[{k=n+1},While[PrimeNu[Binomial[k,n]]Harvey P. Dale, May 13 2018 *)

Edited by T. D. Noe, Apr 05 2007

A129233 Number of integers k>=n such that binomial(k,n) has fewer than n distinct prime factors.

Original entry on oeis.org

1, 2, 6, 9, 20, 26, 43, 63, 75, 91, 130, 151, 185, 243, 279, 307, 383, 392, 488, 511, 595, 716, 904, 917, 1053, 1213, 1282, 1262, 1403, 1632, 1851, 1839, 1932, 2135, 2283, 2426, 2641, 2913, 3322, 3347, 3713, 3642, 4103, 4386, 4361, 4893, 5459

Offset: 1

T. D. Noe, Apr 05 2007, May 20 2007

nonn

This sequence, which is much smoother than the closely related A005735, is calculated using the same "cheat" as described in Selmer's paper. That is, after we seem to have found the largest k for a given n, we search up to 10k for binomial coefficients having fewer than n distinct prime factors.

			Consider n=3. The values of binomial(k,n) are 1,4,10,20,35,56,84,120 for k=3..10. Selmer shows that k=8 yields the largest value having fewer than 3 distinct prime factors. Factoring the other values shows that a(3)=6.

T. D. Noe, Table of n, a(n) for n=1..500
Ernst S. Selmer, On the number of prime divisors of a binomial coefficient. Math. Scand. 39 (1976), no. 2, 271-281.

Cf. A005733, A005735.

Join[{1}, Table[cnt=1; n=k; b=1; n0=Infinity; While[n++; b=b*n/(n-k); If[Length[FactorInteger[b]]

A322158 a(n) is the smallest m for which binomial(m,5) has exactly n distinct prime factors.

Original entry on oeis.org

6, 9, 11, 22, 25, 70, 78, 276, 497, 990, 1771, 8178, 20504, 44254, 181051, 416328, 1013728, 3383579, 8667726, 34332376, 122289552, 244215150, 969751302, 1865174676, 6648863728, 26888317326, 107132035803

Offset: 2

Zachary M Franco, Nov 27 2018

Binomial(m,5) is never prime, so the offset is 2.

Cf. A005733, A005735, A321852.

a[n_] := Module[{m = 5}, While[PrimeNu[Binomial[m, 5]] != n, m++]; m]; Array[a, 10, 2] (* Amiram Eldar, Nov 29 2018 *)

a(n) = for(m=5, oo, if(omega(binomial(m, 5))==n, return(m))) \\ Felix Fröhlich, Dec 01 2018

a(22)-a(23) from Chai Wah Wu, Dec 29 2018

a(24)-a(28) from Giovanni Resta, Jan 04 2019

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A005733 Least k such that binomial(k,n) has n or more distinct prime factors.

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A129233 Number of integers k>=n such that binomial(k,n) has fewer than n distinct prime factors.

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A322158 a(n) is the smallest m for which binomial(m,5) has exactly n distinct prime factors.

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