A185382 Sum_{j=1..n-1} P(n)-P(j), where P(j) = A065091(j) is the j-th odd prime.

0, 2, 6, 18, 26, 46, 58, 86, 134, 152, 212, 256, 280, 332, 416, 506, 538, 640, 712, 750, 870, 954, 1086, 1270, 1366, 1416, 1520, 1574, 1686, 2092, 2212, 2398, 2462, 2792, 2860, 3070, 3286, 3434, 3662, 3896, 3976, 4386, 4470, 4642, 4730, 5270

Offset

1,2

Comments

It appears 1/3 of a(n) values are divisible by 3 (as measured up to n = 8000). Almost all of these cases occur consecutively (i.e., in "runs"). The sizes of these runs, including runs of 1, in the first 250 primes are given by this sequence: {2, 4, 1, 1, 2, 4, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 2, 3, 3, 9, 1, ..} with two runs up to 12 in length occurring in the first 5000 primes. - Richard R. Forberg, Mar 26 2015

a(n+1) == a(n) (mod 3) iff n == 0 (mod 3) or P(n+1) == P(n) (mod 3); this should have asymptotic probability 2/3, and explains some of the above comment. - Robert Israel, Mar 26 2015

Links

Robert Israel, Table of n, a(n) for n = 1..10000 (corrected by Ray Chandler, Jan 19 2019)

Formula

a(n) = (n-1)*A065091(n) - A071148(n-1) = (n-1)*prime(n+1) - sum_{1 < k <= n} prime(k). [Corrected and extended by M. F. Hasler, May 02 2015]

a(n) = A206803(n) - A206803(n-1).

a(n) = Sum_{j=1..n-1} j*A001223(j+1). - Robert Israel, Mar 26 2015

Example

a(4)=(11-3)+(11-5)+(11-7)=18.

Maple

N:= 1000: # to get terms for all odd primes <= N
P:= select(isprime, [seq(2*i+1, i=1..floor((N-1)/2))]):
Q:= ListTools[PartialSums](P):
seq(n*P[n]-Q[n], n=2..nops(P)); # Robert Israel, Mar 26 2015

Mathematica

s[k_] := Prime[k + 1]; p[n_] := Sum[s[k], {k, 1, n}]; c[n_] := n*s[n] - p[n]; Table[c[n], {n, 2, 100}]

Prog

(PARI) A185382(n)=(n-1)*prime(n+1)-sum(k=2, n-1, prime(k)) \\ M. F. Hasler, May 02 2015

Crossrefs

Cf. A001223, A152535, A206802 (a(n)/2), A206803 (= partial sums of this), A206804, A206817.

Sequence in context: A220142 A032649 A237989 * A257065 A066286 A324541

Adjacent sequences: A185379 A185380 A185381 * A185383 A185384 A185385

Keyword

nonn

Author

Clark Kimberling, Feb 13 2012

Extensions

Edited and a(1)=0 prefixed by M. F. Hasler, May 02 2015

Status

approved