A109723 -id:A109723 - cp's OEIS frontend

A109722 Sum of first 2n primes.

0, 5, 17, 41, 77, 129, 197, 281, 381, 501, 639, 791, 963, 1161, 1371, 1593, 1851, 2127, 2427, 2747, 3087, 3447, 3831, 4227, 4661, 5117, 5589, 6081, 6601, 7141, 7699, 8275, 8893, 9523, 10191, 10887, 11599, 12339, 13101, 13887, 14697, 15537, 16401, 17283

Offset: 0

Giovanni Teofilatto, Aug 10 2005

Bisection of A007504.

Ray Chandler, Table of n, a(n) for n = 0..10000
Romeo Meštrović, Curious conjectures on the distribution of primes among the sums of the first 2n primes, arXiv:1804.04198 [math.NT], 2018.

Cf. A007504, A109723, A109724, A109725, A109726.

f[n_] := Sum[Prime[k], {k, n}]; Table[f[2n], {n, 0, 43}]
Join[{0},With[{nn=100},Take[Accumulate[Prime[Range[nn]]],{2,nn,2}]]] (* Harvey P. Dale, Dec 20 2021 *)

a(n) = A007504(2n).

Edited and extended by Ray Chandler, Aug 11 2005

A270828 a(n) = (Sum_{k=1..2n-1} prime(k)) mod prime(n).

Original entry on oeis.org

0, 1, 3, 2, 1, 4, 0, 5, 3, 17, 30, 23, 35, 17, 23, 24, 41, 19, 38, 3, 54, 4, 44, 77, 38, 98, 62, 25, 3, 73, 108, 67, 27, 124, 108, 66, 34, 4, 130, 102, 80, 40, 32, 169, 132, 78, 79, 128, 75, 5, 215, 227, 189, 243, 255, 259, 261, 193, 197, 162, 98, 148, 9, 281, 213, 194, 87, 109, 261, 171

Offset: 1

Altug Alkan, Mar 23 2016

nonn

a(n) = 0 for n = 1, 7, 100. Are there any other values?

No other zero up to n=200000. - Michel Marcus, Jan 31 2019

			a(2) = 1 because (2 + 3 + 5) mod 3 = 1.
a(7) = 0 because (2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41) mod 17 = 238 mod 17 = 0.

Michel Marcus, Table of n, a(n) for n = 1..10000

Cf. A000040, A007504, A109723.

Table[Mod[Sum[Prime@ k, {k, 2 n - 1}], Prime@ n], {n, 70}] (* Michael De Vlieger, Mar 24 2016 *)

for(n=1, 1e2, print1(sum(k=1, 2*n-1, prime(k)) % prime(n), ", "));

lista(nn) = {my(s=0, p=1); for (n=1, nn, p = nextprime(p+1); s += p; print1(s % prime(n), ", "); p = nextprime(p+1); s += p;);} \\ Michel Marcus, Jan 31 2019

a(n) = A007504(2*n-1) mod A000040(n).

A321178 One-half the sum of the first 2n + 1 primes.

Original entry on oeis.org

1, 5, 14, 29, 50, 80, 119, 164, 220, 284, 356, 437, 530, 632, 740, 860, 994, 1138, 1292, 1457, 1633, 1819, 2014, 2219, 2444, 2675, 2915, 3169, 3435, 3709, 3991, 4291, 4603, 4927, 5269, 5620, 5983, 6359, 6745, 7144, 7558, 7984, 8420, 8866, 9325, 9790, 10273

Offset: 0

Martin Michael Musatov, Oct 29 2018

nonn

Ray Chandler, Table of n, a(n) for n = 0..10000

Cf. A000040, A001223, A007504, A109723.

a(n) = A109723(n)/2 = A007504(2n+1)/2.

A383938 a(n) is the least positive integer k such that b(2*j) is prime for 1 <= j <= n but not prime for j = n+1, where b(1) = k and b(m+1) = b(m) + prime(m) for m >= 1.

Original entry on oeis.org

2, 5, 21, 129, 69, 1, 51, 23991, 171, 1371, 3, 322141431, 1431357020859

Offset: 0

Om S. M. Yadav, Aug 18 2025

Similar to A227547, primes are added in successive manner except that here the sequence breaks if an even-indexed term is not prime and considers preceding even-indexed prime as the last term of the sequence. For example, a(2) = 21 [21, 23, 26, 31, 38, 49] but since 49 is not prime, last two terms (38 and 49) are omitted leaving 31 as last term in the sequence.

a(12) is the last term, because b(j) is always divisible by 11 for some j in {2, 4, 6, 8, 10, 14, 16, 18, 22, 24, 26}. - Pontus von Brömssen, Aug 19 2025

			a(n) = k, b(m+1) = b(m) + prime(m); b(1) = k
For n = 0, a(0) = 2; b(m+1) = b(m) + prime(m): [2]
For n = 1, a(1) = 5; b(m+1) = b(m) + prime(m): [5, 7(5+2)]
For n = 2, a(2) = 21; b(m+1) = b(m) + prime(m): [21, 23(21+2), 26(23+3), 31(26+5)]
For n = 3, a(3) = 129; b(m+1) = b(m) + prime(m): [129, 131(129+2), 134(131+3), 139(134+5), 146(139+7), 157(146+11)]
For n = 4, a(4) = 69; b(m+1) = b(m) + prime(m): [69, 71(69+2), 74(71+3), 79(74+5), 86(79+7), 97(86+11), 110(97+13), 127(110+17)]
For n = 5, a(5) = 1; b(m+1) = b(m) + prime(m): [1, 3(1+2), 6(3+3), 11(6+5), 18(11+7), 29(18+11), 42(29+13), 59(42+17), 78(59+19), 101(78+23)]
For a(n), even-indexed term is prime. e.g. for a(3) = 129 [129, 131, 134, 139, 146, 157], even indexed terms 131, 139, 157 are primes.

Cf. A014284, A092163, A109723, A227547.

a(n) = my(vp=concat(2, vector(n+1, i, sum(k=1, 2*i+1, prime(k)))), v=concat(vector(n, i, 1), 0), k=1); while (apply(ispseudoprime, vector(n+1, i, vp[i]+k)) != v, k++); k; \\ Michel Marcus, Aug 19 2025

a(11) from Michel Marcus, Aug 19 2025

a(12) from Pontus von Brömssen, Aug 19 2025

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A109722 Sum of first 2n primes.

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A270828 a(n) = (Sum_{k=1..2n-1} prime(k)) mod prime(n).

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A321178 One-half the sum of the first 2n + 1 primes.

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A383938 a(n) is the least positive integer k such that b(2*j) is prime for 1 <= j <= n but not prime for j = n+1, where b(1) = k and b(m+1) = b(m) + prime(m) for m >= 1.

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