A085704 -id:A085704 - cp's OEIS frontend

A210497 a(n) = 2*prime(n+1) - prime(n).

4, 7, 9, 15, 15, 21, 21, 27, 35, 33, 43, 45, 45, 51, 59, 65, 63, 73, 75, 75, 85, 87, 95, 105, 105, 105, 111, 111, 117, 141, 135, 143, 141, 159, 153, 163, 169, 171, 179, 185, 183, 201, 195, 201, 201, 223, 235, 231, 231, 237, 245, 243, 261, 263, 269, 275, 273, 283

Offset: 1

Marco Piazzalunga, Jan 24 2013

The subsequence of multiples of 3 begins: 9, 15, 15, 21, 21, 27, 33, 45.
The subsequence of primes begins: 7, 43, 73, 163, 179, 223.
Some terms, like a(3)=15 or a(5)=21, are repeated twice, other terms, like a(23)=105, are repeated three times.

			a(2) = 7 because prime(3) = 5, prime(2) = 3, and 2 * 5 - 3 = 7.
a(3) = 9 because prime(4) = 7, prime(3) = 5, and 2 * 7 - 5 = 9.
a(4) = 15 because prime(5) = 11, prime(4) = 7, and 2 * 11 - 7 = 15.

Bruno Berselli and Zak Seidov, Table of n, a(n) for n = 1..10000 (a(1)-a(1000) from Bruno Berselli).

Cf. A001223, A062234, A085704 (subsequence).

[2*NextPrime(p)-p: p in PrimesUpTo(300)]; // Bruno Berselli, Jan 24 2013

Table[2 Prime[n + 1] - Prime[n], {n, 50}] (* Vincenzo Librandi, May 03 2015 *)
ListConvolve[{2, -1}, Prime[Range[100]]] (* Paolo Xausa, Oct 29 2024 *)

a(n)=my(p=prime(n));2*nextprime(p+1)-p \\ Charles R Greathouse IV, Jan 24 2013

from sympy import prime, nextprime
def A210497(n): return -(p:=prime(n))+(nextprime(p)<<1) # Chai Wah Wu, Oct 29 2024

a(n) ~ n log n. - Charles R Greathouse IV, Jan 24 2013

A215808 Primes of the form 2*prime(k) - prime(k+1).

Original entry on oeis.org

3, 3, 17, 41, 47, 67, 151, 167, 199, 227, 251, 257, 347, 367, 557, 587, 601, 607, 641, 647, 727, 941, 971, 1051, 1091, 1097, 1117, 1181, 1217, 1277, 1361, 1427, 1447, 1447, 1487, 1487, 1499, 1607, 1697, 1741, 1747, 1741, 1777, 1877, 1901, 2087, 2143, 2281

Offset: 1

Zak Seidov, Sep 06 2012

nonn

Corresponding values of k: 3, 4, 9, 15, 16, 21, 37, 40, 47, 51, 55, 56, 71, 74, 103 (A216075).

			k=3: 2*5-7=3, k=4: 2*7-11=3, k=9: 2*23-29=17.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A062234, A085704, A094104, A094105, A216075.

pr=Prime[Range[1000]]; s=Select[2*Most[pr]-Rest[pr],PrimeQ]
Select[2#[[1]]-#[[2]]&/@Partition[Prime[Range[500]],2,1],PrimeQ] (* Harvey P. Dale, Feb 25 2017 *)

A163981 a(n) is the smallest prime of the form prime(n+1)*k - prime(n), k >= 1, where prime(n) is the n-th prime.

Original entry on oeis.org

7, 2, 2, 37, 2, 89, 2, 73, 151, 2, 43, 127, 2, 239, 59, 419, 2, 73, 359, 2, 401, 419, 1163, 881, 307, 2, 967, 2, 569, 3697, 397, 691, 2, 457, 2, 163, 821, 839, 179, 1259, 2, 2111, 2, 1777, 2, 223, 3803, 3863, 2, 3499, 1201, 2, 2269, 263, 269, 1889, 2, 283, 1409, 2, 2647

Offset: 1

Leroy Quet, Aug 07 2009

nonn

a(n) = 2 if and only if n is in A029707. - Robert Israel, Jan 16 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A029707, A129919.

Contains A085704.

a := proc (n) local k: for k while isprime(ithprime(n+1)*k-ithprime(n)) = false do end do: ithprime(n+1)*k-ithprime(n) end proc: seq(a(n), n = 1 .. 65); # Emeric Deutsch, Aug 10 2009

a[n_] := Module[{p, q, r}, For[p = Prime[n]; q = Prime[n + 1]; k = 1, True, k++, If[PrimeQ[r = q k - p], Return[r]]]];
Array[a, 100] (* Jean-François Alcover, Aug 28 2020 *)

a(n) = my(k=1); while (!isprime(p=prime(n+1)*k - prime(n)), k++); p; \\ Michel Marcus, Jul 02 2021

from sympy import isprime, nextprime, prime
def a(n):
    pn = prime(n); pn1 = nextprime(pn); k = 1
    while not isprime(pn1*k - pn): k += 1
    return pn1*k - pn
print([a(n) for n in range(1, 62)]) # Michael S. Branicky, Jul 02 2021

Extended by Emeric Deutsch, Aug 10 2009

cp's OEIS Frontend

A210497 a(n) = 2*prime(n+1) - prime(n).

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A215808 Primes of the form 2*prime(k) - prime(k+1).

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A163981 a(n) is the smallest prime of the form prime(n+1)*k - prime(n), k >= 1, where prime(n) is the n-th prime.

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