A073051 Least k such that Sum_{i=1..k} (prime(i) + prime(i+2) - 2*prime(i+1)) = 2n + 1.

1, 3, 8, 23, 33, 45, 29, 281, 98, 153, 188, 262, 366, 428, 589, 737, 216, 1182, 3301, 2190, 1878, 1830, 7969, 3076, 3426, 2224, 3792, 8027, 4611, 4521, 3643, 8687, 14861, 12541, 15782, 3384, 34201, 19025, 17005, 44772, 23282, 38589, 14356

Offset: 1

Author

Robert G. Wilson v, Aug 15 2002

Keywords

• nonn

Comments

Also, least k such that 2n = A001223(k-1) = prime(k+1) - prime(k), where prime(k) = A001223(n). - Alexander Adamchuk, Jul 30 2006

Also the least number k>0 such that the k-th maximal run of composite numbers has length 2n-1. For example, the 8th such run (24,25,26,27,28) is the first of length 2(3)-1, so a(3) = 8. Also positions of first appearances in A176246 (A046933 without first term). - Gus Wiseman, Jun 12 2024

Examples

			a(3) = 8 because 1+0+2-2+2-2+2+2 = 5 and (5+1)/2 = 3.
		

Links

• Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Crossrefs

Cf. A000230, A001223.

Position of first appearance of 2n+1 in A176246.

For nonsquarefree runs we have a bisection of A373199.

A000040 lists the primes, first differences A001223.

A002808 lists the composite numbers, differences A073783, sums A053767.

A046933 counts composite numbers between primes.

A065855 counts composite numbers up to n.

Cf. A005381, A027833, A038664, A045881, A068780, A174965, A371201, A373403.

Programs

Mathematica

NextPrim[n_Integer] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {50}]; s = 0; k = 1; p = 0; q = 2; r = 3; While[k < 10^6, p = q; q = r; r = NextPrim[q]; s = s + p + r - 2q; If[s < 101 && a[[(s + 1)/2]] == 0, a[[(s + 1)/2]] = k]; k++ ]; a

PARI

a001223(n) = prime(n+1) - prime(n);
a(n) = {my(k = 1); while(2*n != A001223(k+1), k++); k;} \\ Michel Marcus, Nov 20 2016

Formula

a(n) = A038664(n) - 1. - Filip Zaludek, Nov 19 2016