A061068 -id:A061068 - cp's OEIS frontend

A231432 Primes p such that abs(p - 3*k) is also prime, where p is the k-th prime.

3, 7, 13, 19, 31, 41, 47, 53, 61, 71, 79, 89, 101, 107, 113, 139, 151, 173, 193, 199, 223, 229, 239, 251, 271, 281, 293, 349, 373, 397, 433, 457, 463, 521, 541, 557, 569, 593, 601, 613, 619, 641, 647, 673, 683, 743, 787, 809, 839, 911, 941, 953, 971, 1013, 1049

Offset: 1

K. D. Bajpai, Nov 09 2013

			The first prime, 2, is not a term since |2-3*1| = 1.
The second prime, 3, is a term, since |3-2*3| = 3 is a prime.
a(11) = 79 which is the 22nd prime, prime(22)-3*22 = 79-66 = 13 which is also prime.
a(15) = 113 which is the 30th prime, prime(30)-3*30 = 113-90 = 23 which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..6400

Cf. A061068 (primes: prime(m) plus its subscript).
Cf. A064402 (numbers n: prime(n)+n is prime).
Cf. A231232 (primes p : p+2*k is also prime).
Cf. A231383 (primes p : p+3*k is also prime).

KD := proc() local a, b;  a:= ithprime(n); b:= abs(a-3*n); if isprime(b) then RETURN (a); fi; end: seq(KD(), n=1..500);

KD = Select[Table[{Prime[n], Prime[n] - 3*n}, {n, 200}], PrimeQ[#[[2]]] &]; Transpose[KD][[1]]
Select[Table[{k,Prime[k]},{k,200}],PrimeQ[Abs[#[[2]]-3#[[1]]]]&][[;;,2]] (* Harvey P. Dale, Jul 14 2024 *)

k=0;forprime(p=2,1e3,if(isprime(abs(p-k++*3)), print1(p", "))) \\ Charles R Greathouse IV, Mar 11 2014

A239743 Primes of the form prime(k)^3 + k.

Original entry on oeis.org

29, 347, 2203, 704993, 2248123, 2685653, 3442987, 81182821, 86938393, 95444081, 230346509, 1064332433, 1298596753, 1829276767, 2202074113, 3449796071, 4306879141, 6740558659, 8205739219, 9649993009, 11650768499, 14225261327, 15124198183, 19968681493, 21415471837

Offset: 1

K. D. Bajpai, Mar 26 2014

nonn

			29 is in the sequence because prime(2)^3 + 2 = 29, which is prime.
347 is in the sequence because prime(4)^3 + 4 = 347, which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..7000

Cf. A000040 (prime numbers).
Cf. A184935 (primes: k^2 + prime(k)).
Cf. A188831 (primes: k^2 - prime(k)).
Cf. A229203 (primes: k^3 - prime(k)).
Cf. A061068 (primes: prime(k) + k)
Cf. A212304 (primes: prime(k)^2 + k).

[q: k in [1..1000] | IsPrime(q) where q is NthPrime(k)^3+k]; // Bruno Berselli, Mar 26 2014

KD := proc() local a,b; a:= ithprime(n); b:=a^3+n; if isprime(b) then RETURN (b); fi; end: seq(KD(), n=1..1000);

Select[Table[Prime[k]^3 + k, {k, 1000}], PrimeQ]

s=[]; for(k=1, 1000, n=prime(k)^3+k; if(isprime(n), s=concat(s, n))); s \\ Colin Barker, Mar 26 2014

A254867 Numbers n such that prime(n) + n and prime(n) + n^2 are prime.

Original entry on oeis.org

1, 2, 4, 22, 66, 96, 106, 144, 180, 222, 324, 378, 466, 492, 604, 742, 760, 778, 784, 960, 984, 990, 994, 1050, 1150, 1162, 1186, 1248, 1302, 1308, 1356, 1360, 1380, 1744, 1830, 1866, 1870, 1956, 2052, 2070, 2112, 2182, 2212, 2380, 2470, 2556, 2586, 2638, 2676, 2760, 2766

Offset: 1

Zak Seidov, Feb 09 2015

nonn

			a(4) = 22 = A064402(6): prime(22) = 79, 79 + {22, 22^2} = {101, 563} both prime.

Subsequence of A064402. Cf. A000040, A014688, A061067, A061068.

A254867:=n->`if`(isprime(ithprime(n)+n) and isprime(ithprime(n)+n^2), n, NULL): seq(A254867(n), n=1..10^4); # Wesley Ivan Hurt, Jan 16 2017

Select[Range[1000], PrimeQ[Prime[#] + #] && PrimeQ[Prime[#] + #^2] &] (* Alonso del Arte, Feb 09 2015 *)
Select[Range[3000],AllTrue[Prime[#]+{#,#^2},PrimeQ]&] (* Harvey P. Dale, Jan 17 2023 *)

A084294 Number of primes in the interval [prime(n),n+prime(n)].

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 3, 2, 3, 3, 3, 4, 4, 3, 4, 4, 5, 5, 5, 5, 4, 5, 5, 7, 6, 6, 5, 5, 5, 5, 7, 7, 7, 7, 8, 7, 8, 9, 8, 8, 7, 7, 9, 8, 9, 8, 9, 11, 10, 10, 11, 10, 10, 9, 10, 11, 10, 9, 9, 9, 8, 10, 11, 11, 10, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 14, 15, 14, 13

Offset: 1

Labos Elemer, May 27 2003

nonn

Cf. A000040, A000720, A061067, A061068, A084295.

t[x_] := Table[w, {w, Prime[x], x+Prime[x]}] Table[Count[PrimeQ[t[n]], True], {n, 1, 128}] (* or *) Table[PrimePi[n+Prime[n]]-n+1, {n, 1, 25}];

a(n) = primepi(n+prime(n)) - n + 1; \\ Michel Marcus, Aug 28 2019

a(n) = Pi(n+prime(n)) - n + 1 = A000720(n+A000040(n)) - n + 1. [corrected by Michel Marcus, Aug 28 2019]

A260376 Values of n such that prime(prime(n)) + n is a prime number.

Original entry on oeis.org

2, 6, 16, 20, 30, 42, 78, 90, 92, 102, 108, 118, 124, 126, 128, 132, 136, 150, 160, 166, 180, 182, 186, 194, 196, 202, 206, 210, 214, 230, 270, 280, 294, 348, 356, 358, 366, 370, 374, 378, 418, 436, 456, 458, 472, 474, 478, 492, 522, 546, 556, 560, 570, 584, 586, 590

Offset: 1

Altug Alkan, Aug 29 2015

Inspired by A061068.
Obviously, sequence only can include even numbers.
Initial primes of the form prime(prime(n)) + n are 7, 47, 257, 373, 647, 1129, 2797, 3389, 3499, 4129, 4447.

			For n=2, prime(prime(2)) + 2 = prime(3) + 2 = 7. 7 is a prime number.

Cf. A006450.

[n: n in [1..600] | IsPrime(NthPrime(NthPrime(n))+n)]; // Vincenzo Librandi, Aug 30 2015

Select[Range[600], PrimeQ[Prime[Prime[#]] + #] &] (* Vincenzo Librandi, Aug 30 2015 *)

for(n=1, 1e3, if(isprime(prime(prime(n))+n), print1(n", ")))

use ntheory ":all"; my @p = grep { is_prime(nth_prime(nth_prime($))+$) } 1..10000; say join ", ", @p; # Dana Jacobsen, Sep 02 2015

cp's OEIS Frontend

A231432 Primes p such that abs(p - 3*k) is also prime, where p is the k-th prime.

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A239743 Primes of the form prime(k)^3 + k.

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A254867 Numbers n such that prime(n) + n and prime(n) + n^2 are prime.

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A084294 Number of primes in the interval [prime(n),n+prime(n)].

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A260376 Values of n such that prime(prime(n)) + n is a prime number.

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