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 A073051 Least k such that Sum_{i=1..k} (prime(i) + prime(i+2) - 2*prime(i+1)) = 2n + 1. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 LINKS Table of n, a(n) for n=1..43. Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers. FORMULA a(n) = A038664(n) - 1. - Filip Zaludek, Nov 19 2016 EXAMPLE a(3) = 8 because 1+0+2-2+2-2+2+2 = 5 and (5+1)/2 = 3. 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 PROG (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 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. Sequence in context: A093537 A373400 A180621 * A183930 A183922 A340493 Adjacent sequences: A073048 A073049 A073050 * A073052 A073053 A073054 KEYWORD nonn,changed AUTHOR Robert G. Wilson v, Aug 15 2002 STATUS approved

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Last modified June 13 17:32 EDT 2024. Contains 373391 sequences. (Running on oeis4.)