A005733 -id:A005733

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A005735

Greatest k such that binomial(k,n) has fewer than n distinct prime factors.
(Formerly M2719)

+10
4

1, 3, 8, 14, 32, 62, 87, 169, 132, 367, 389, 510, 394, 512, 512, 1880, 1880, 1882, 2099, 1879, 1885, 2102, 3470, 3470, 4805, 4806, 4806, 3475, 4806, 4938, 4939, 5108, 5119, 6271, 5122, 5869, 10663, 10663, 10663, 7421, 10667, 10667, 10668, 11710, 11711

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Table 2 in Selmer's paper has a typo for n = 76. Selmer "cheats" to find a(n) for n>27. - T. D. Noe, Apr 05 2007

REFERENCES

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

LINKS

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.

MATHEMATICA

Join[{1}, Table[n=k; b=1; n0=Infinity; While[n++; b=b*n/(n-k); If[Length[FactorInteger[b]]<k, n0=n]; n<10*n0]; n0, {k, 2, 30}]] (* T. D. Noe, Apr 05 2007 *)

CROSSREFS

Cf. A005733, A129233.

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 26 2004

Edited by T. D. Noe, Apr 05 2007

STATUS

approved

A129233

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

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.

EXAMPLE

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.

MATHEMATICA

Join[{1}, Table[cnt=1; n=k; b=1; n0=Infinity; While[n++; b=b*n/(n-k); If[Length[FactorInteger[b]]<k, cnt=cnt+1; n0=n]; n<10*n0]; cnt, {k, 2, 20}]]

CROSSREFS

Cf. A005733, A005735.

KEYWORD

nonn

AUTHOR

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

STATUS

approved

A322158

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

+10
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

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

LINKS

Table of n, a(n) for n=2..28.

MATHEMATICA

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

PROG

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

CROSSREFS

Cf. A005733, A005735, A321852.

KEYWORD

nonn,more

AUTHOR

Zachary M Franco, Nov 27 2018

EXTENSIONS

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

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

STATUS

approved

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