A014688 -id:A014688 - cp's OEIS frontend

A306627 Primes of the form (p + prime(p))/2 with prime p.

43, 53, 79, 97, 199, 257, 347, 1103, 1187, 1303, 1367, 1753, 2029, 2351, 2593, 2647, 3181, 3253, 4663, 4787, 4903, 4919, 5573, 6091, 7079, 7331, 7369, 7457, 8101, 8527, 8543, 10159, 11549, 12647, 13151, 14699, 15031, 15559, 15679, 15881, 15889, 16661, 17099, 17881, 18251, 18959, 19219, 20399, 20431

Offset: 1

Zak Seidov, Mar 01 2019

nonn

Indices k of p's are 8, 9, 11, 12, 19, 23, 28, 62, 64, 68, 71, 85, 95, 104, 114, 115, 131, 134, 175, 178, 181, 182, 200, 216, 240, 246, 247, 250, 266, 276, 277, 316, 346, 372, 382.

And some of k's themselves are prime: 11, 19, 23, 71, 131, 181, 277, 421, 541, 601, 673, 751, 881, 937. See link to Fernandez.

			43 = (p + prime(p))/2 for p = 19 (prime),
53 = (p + prime(p))/2 for p = 23 (prime).

Neil Fernandez, An order of primeness.

Cf. A014688 (n-th prime + n).

lista(nn) = forprime(p=3,nn,k=(prime(p)+p)/2;if(isprime(k),print1(k, ", "))) \\ Jinyuan Wang, Mar 01 2019

A349779 Pairs of integers i,j such that i+prime(i) and j+prime(j) are a pair of amicable numbers.

Original entry on oeis.org

41, 51, 99525, 104283, 280899, 295869, 18378754, 21937204, 52084243, 53107499, 163148785, 166346021, 179162279, 183252051, 212063283, 244033955, 3731366783, 4226663091, 7134801326, 7818930716, 10469380661, 12074408463, 12445587194, 12667334246, 16055012737, 17258948163

Offset: 1

Michel Marcus, Nov 30 2021

nonn

			41 + prime(41) = 41 + 179 = 220, 51 + prime(51) = 51 + 233 = 284, and (220, 284) is an amicable pair.

Juan Luis Varona, On the Solution of the Equation n = a*k + b*p_k by Means of an Iterative Method, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.5.

Cf. A002025, A002046, A014688, A259180.

from sympy import nextprime, divisor_sigma
d = dict()
j, p = 0, 0
while True:
    j, p = j+1, nextprime(p)
    n = j+p
    a = divisor_sigma(n)-n
    d[(n, a)] = j
    if (a, n) in d:
        print(d[(a, n)], j) # Martin Ehrenstein, Dec 02 2021

a(7)-a(8) from Amiram Eldar, Nov 30 2021
a(9)-a(16) from Michel Marcus, Dec 01 2021
a(17)-a(26) from Martin Ehrenstein, Dec 03 2021

A365591 Numbers k such that Sum_{i=1..k} prime(i) + i is prime.

Original entry on oeis.org

1, 5, 8, 17, 28, 33, 40, 41, 49, 52, 64, 65, 69, 77, 92, 93, 108, 109, 120, 121, 136, 137, 140, 144, 165, 200, 201, 204, 225, 229, 265, 269, 272, 280, 292, 312, 325, 332, 337, 344, 356, 361, 369, 376, 388, 457, 464, 473, 480, 529, 541, 548, 553, 556, 573, 577

Offset: 1

Saish S. Kambali, Sep 10 2023

Numbers k such that A000217(k) + A007504(k) is prime. - Robert Israel, Sep 10 2023

			2+1 = 3, which is prime, so 1 is a term.
2+1 + 3+2 + 5+3 + 7+4 + 11+5 = 43, which is prime, so 5 is a term.

Cf. A000217, A007504, A014688.

P:= [seq(ithprime(i),i=1..1000)]:
S:= ListTools:-PartialSums(P):
select(i -> isprime(S[i]+i*(i+1)/2),[$1..1000]); # Robert Israel, Sep 10 2023

With[{m = 600}, Position[Accumulate[Range[m] + Prime[Range[m]]], ?PrimeQ] // Flatten] (* _Amiram Eldar, Sep 10 2023 *)

isok(k) = isprime(sum(i=1, k, i+prime(i))); \\ Michel Marcus, Sep 14 2023

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A306627 Primes of the form (p + prime(p))/2 with prime p.

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A349779 Pairs of integers i,j such that i+prime(i) and j+prime(j) are a pair of amicable numbers.

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A365591 Numbers k such that Sum_{i=1..k} prime(i) + i is prime.

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