A206804 -id:A206804 - cp's OEIS frontend

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

Clark Kimberling, Feb 13 2012

nonn

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

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

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

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

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

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}]

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

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

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

A206803 Sum_{0A065091(j) is the j-th odd prime.

Original entry on oeis.org

2, 8, 26, 52, 98, 156, 242, 376, 528, 740, 996, 1276, 1608, 2024, 2530, 3068, 3708, 4420, 5170, 6040, 6994, 8080, 9350, 10716, 12132, 13652, 15226, 16912, 19004, 21216, 23614, 26076, 28868, 31728, 34798, 38084, 41518, 45180, 49076

Offset: 2

Clark Kimberling, Feb 13 2012

nonn

Partial sums of A185382.

Danny Rorabaugh, Table of n, a(n) for n = 2..10000

Cf. A065091, A185382, A206804, A206817.

s[k_] := Prime[k + 1]; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n]
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 100}]  (* A185382 *)
%/2  (* A206802 *)
Flatten[Table[t[n], {n, 2, 40}]]  (*  A206803 *)
%/2  (* A206804 *)

[sum([sum([nth_prime(k+1)-nth_prime(j+1) for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,41)] # Danny Rorabaugh, Apr 18 2015

A206802 a(n) = (1/2)*A185382(n).

Original entry on oeis.org

1, 3, 9, 13, 23, 29, 43, 67, 76, 106, 128, 140, 166, 208, 253, 269, 320, 356, 375, 435, 477, 543, 635, 683, 708, 760, 787, 843, 1046, 1106, 1199, 1231, 1396, 1430, 1535, 1643, 1717, 1831, 1948, 1988, 2193, 2235, 2321, 2365, 2635, 2911, 3005

Offset: 2

Clark Kimberling, Feb 13 2012

nonn

See A185382.

Cf. A185382.

s[k_] := Prime[k + 1]; t[1] = 0;
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n]
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 100}]  (* A185382 *)
%/2  (* A206802 *)
Flatten[Table[t[n], {n, 2, 40}]]  (*  A206803 *)
%/2  (* A206804 *)
Table[Sum[Prime[n + 1] - Prime[j + 1], {j, n - 1}]/2, {n, 2, 48}] (* Michael De Vlieger, Sep 13 2016 *)

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

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A206803 Sum_{0A065091(j) is the j-th odd prime.

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A206802 a(n) = (1/2)*A185382(n).

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