A138383 - OEIS

%I #25 Jun 01 2024 20:07:47

%S 3,3,9,13,38,25,62,37,86,159,61,207,158,85,182,303,339,121,387,278,

%T 145,459,326,519,748,398,205,422,217,446,1687,518,807,277,1445,301,

%U 927,963,662,1023,1059,361,1865,385,782,397,2466,2610,902,457,926,1419,481,2465,1527

%N If prime(i) = i-th prime, a(n) = prime(n)+1 + prime(n)+2 + ... + prime(n+1). a(0) = 3 by convention.

%C First differences of A034953 for n > 0. - _Gionata Neri_, May 17 2015

%F a(n) = (prime(n+1)-prime(n))*(prime(n+1)+prime(n)+1)/2 for n >= 1. - _N. J. A. Sloane_, May 08 2008

%e 3 = 1 + 2;

%e 3 = 3;

%e 9 = 4 + 5;

%e 13 = 6 + 7;

%e 38 = 8 + 9 + 10 + 11;

%e ...

%p 3, seq((ithprime(n+1)-ithprime(n))*(ithprime(n+1)+ithprime(n)+1)/2, n=1..100); # _Robert Israel_, May 17 2015

%t Join[{3}, Table[(Prime[n+1] - Prime[n]) (Prime[n+1] + Prime[n] + 1)/2, {n, 60}]] (* _Vincenzo Librandi_, May 18 2015 *)

%t Join[{3},(#[[2]]-#[[1]]) (Total[#]+1)/2&/@Partition[Prime[Range[ 60]],2,1]] (* _Harvey P. Dale_, Oct 27 2020 *)

%o (Magma) [3] cat [(NthPrime(n+1) - NthPrime(n))*(NthPrime(n+1) + NthPrime(n)+1)/2: n in [1..60]]; // _Vincenzo Librandi_, May 18 2015

%o (Python)

%o from sympy import prime, nextprime

%o def A138383(n):

%o     if n == 0: return 3

%o     q = nextprime(p:=prime(n))

%o     return (q-p)*(p+q+1)>>1 # _Chai Wah Wu_, Jun 01 2024

%Y Cf. A000040.

%K nonn,easy

%O 0,1

%A _Odimar Fabeny_, May 08 2008