login
Binomial coefficients formed from consecutive primes: a(n) = binomial( prime(n+1), prime(n) ).
15

%I #53 Feb 16 2024 10:13:27

%S 3,10,21,330,78,2380,171,8855,475020,465,2324784,101270,903,178365,

%T 22957480,45057474,1830,99795696,971635,2628,277962685,1837620,

%U 581106988,144520208820,4082925,5253,5160610,5886,6438740

%N Binomial coefficients formed from consecutive primes: a(n) = binomial( prime(n+1), prime(n) ).

%C Conjecture: for each value of n > 1, if a(n+1) has the same number of digits as a(n) and a(n+1) > a(n), then prime(n+2) - prime(n+1) = prime(n+1) - prime(n). This conjecture has been verified for all n < 3*10^7. - _Ahmad J. Masad_, Oct 08 2019

%H Michel Marcus, <a href="/A058077/b058077.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A058077/a058077.png">Colored logarithmic scatterplot of the first 100000 terms</a> (where the color is function of A001223(n))

%F a(n) = binomial(A000040(n+1), A001223(n)).

%e n=6: a(6) = C(p(7),p(6)) = C(17,13) = 57120/24 = 2380.

%t Table[Binomial[Prime[n+1],Prime[n]],{n,1,20}] (* _Vaclav Kotesovec_, Nov 13 2014 *)

%Y Cf. A000040, A001223.

%Y Cf. A037293, A066526, A080911, A277341.

%K nonn

%O 1,1

%A _Labos Elemer_, Nov 13 2000

%E Offset corrected by _Vaclav Kotesovec_, Nov 13 2014