A064402 -id:A064402 - cp's OEIS frontend

A061068 Primes which are the sum of a prime and its subscript.

3, 5, 11, 19, 79, 101, 113, 127, 163, 173, 223, 271, 383, 419, 431, 503, 571, 599, 619, 641, 659, 673, 683, 701, 733, 757, 827, 863, 971, 1013, 1033, 1087, 1193, 1249, 1423, 1433, 1453, 1483, 1579, 1621, 1667, 1723, 2003, 2113, 2179, 2287, 2381, 2459, 2467

Offset: 1

Labos Elemer, May 28 2001

nonn

a(n) = A061067(n-1) + A064402(n). - Leroy Quet, Jun 30 2006
This sequence is the intersection of A014688 with the set of primes. Conjecture: this sequence is infinite, yet derives from arbitrarily long "prime deserts" such as the 11 composites in A014688 between a(6) = 19 and a(18) = 79 and the 17 composites in A014688 between a(48) = 271 and a(66) = 383. - Jonathan Vos Post, Nov 22 2004
Primes not of the form n + pi(n-1). - Thomas Ordowski, Sep 21 2013
Except for the first pair (3, 5) no two consecutive primes are terms of the sequence. - Zak Seidov, Nov 10 2013

			5th term is 79=61+18=prime(18)+18.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A061067, A064402.

Select[Range[500], PrimeQ[Prime[ # ] + # ] &] + Prime[Select[Range[500], PrimeQ[Prime[ # ] + # ] &]] (* Stefan Steinerberger, Jul 21 2006 *)

{ n=0; m=0; forprime (p=2, 109567, if (isprime(p + m++), write("b061068.txt", n++, " ", p + m)) ) } \\ Harry J. Smith, Jul 17 2009

Edited by N. J. A. Sloane, Apr 29 2007

Definition clarified by Jonathan Sondow, Jul 12 2012

A061067 m-th prime prime(m) is included iff prime(m) + m is also prime.

Original entry on oeis.org

2, 3, 7, 13, 61, 79, 89, 101, 131, 139, 181, 223, 317, 349, 359, 421, 479, 503, 521, 541, 557, 569, 577, 593, 619, 641, 701, 733, 827, 863, 881, 929, 1021, 1069, 1223, 1231, 1249, 1277, 1361, 1399, 1439, 1487, 1733, 1831, 1889, 1987, 2069, 2137, 2143, 2203

Offset: 0

Labos Elemer, May 28 2001

nonn

			5th term here is 61 = prime(18) and 61 + 18 = 79.

Harry J. Smith, Table of n, a(n) for n = 0..1000

Cf. A064402, A061068, A227420.

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+ n)]; // Vincenzo Librandi, Jan 19 2015

Prime[Select[Range[500], PrimeQ[Prime[ # ] + # ] &]] (* Stefan Steinerberger, Jul 21 2006 *)
Select[Prime[Range[400]],PrimeQ[#+PrimePi[#]]&] (* Harvey P. Dale, Oct 03 2016 *)

{ n=-1; m=0; forprime (p=2, 109597, if (isprime(p + m++), write("b061067.txt", n++, " ", p)) ) } \\ Harry J. Smith, Jul 17 2009

a(n) + A064402(n+1) = A061068(n+1). [corrected by Martin Fuller]

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A231232 Primes p = prime(k) such that p + 2*k is prime.

Original entry on oeis.org

3, 5, 17, 23, 31, 37, 41, 43, 61, 89, 103, 107, 109, 113, 151, 163, 191, 193, 241, 251, 257, 269, 281, 307, 311, 313, 317, 359, 373, 409, 433, 463, 487, 557, 563, 593, 601, 607, 643, 647, 691, 701, 761, 787, 811, 823, 857, 863, 907, 911, 953, 977, 1019, 1033

Offset: 1

K. D. Bajpai, Nov 06 2013

			31 = prime(11) is a term: prime(11) + 2*11 = 31 + 22 = 53 is also prime.
89 = prime(24) is a term: prime(24) + 2*24 = 89 + 48 = 137 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5800

Cf. A061068 (primes: prime(m) plus its subscript).
Cf. A064402 (numbers n: prime(n)+n is prime).
Subsequence of A364877.

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+2*n)]; // Vincenzo Librandi, Jan 19 2015

KD := proc() local a,b;  a:= ithprime(n); b:= a+2*n; if isprime(b) then RETURN (a); fi; end: seq(KD(),n=1..500);

t = Select[Table[{Prime[n], Prime[n] + 2*n}, {n, 200}], PrimeQ[#[[2]]] &]; Transpose[t][[1]] (* T. D. Noe, Nov 06 2013 *)

is(n)=isprime(n+2*primepi(n)) && isprime(n) \\ Charles R Greathouse IV, Aug 25 2014

Name edited by David A. Corneth, Sep 07 2023

A076297 Numbers k such that prime(k) + s*k is prime, s=2.

Original entry on oeis.org

2, 3, 7, 9, 11, 12, 13, 14, 18, 24, 27, 28, 29, 30, 36, 38, 43, 44, 53, 54, 55, 57, 60, 63, 64, 65, 66, 72, 74, 80, 84, 90, 93, 102, 103, 108, 110, 111, 117, 118, 125, 126, 135, 138, 141, 143, 148, 150, 155, 156, 162, 165, 171, 174, 180, 183, 186, 188, 190, 198

Offset: 1

Zak Seidov, Oct 05 2002

nonn

See also A064402 (s=1), A076298 (s=3), A076299 (s=4), A076300 (s=5).

			3 is OK because p(3) + 2*3 = 5 + 6 = 11 is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A064402, A076298, A076299, A076300.

[n: n in [0..500]| IsPrime(NthPrime(n) +2*n)]; // G. C. Greubel, May 04 2018

Select[Range[200],PrimeQ[Prime[#]+2#]&] (* Harvey P. Dale, Sep 01 2015 *)

select(x->isprime(x), vector(500, n, prime(n) + 2*n), 1) \\ G. C. Greubel, May 04 2018

A100915 Numbers n such that n plus n-th semiprime is semiprime.

Original entry on oeis.org

4, 6, 9, 12, 16, 18, 19, 20, 24, 29, 31, 34, 35, 39, 40, 44, 46, 49, 51, 54, 55, 72, 73, 76, 79, 80, 81, 84, 87, 91, 93, 94, 96, 98, 110, 113, 116, 120, 128, 130, 136, 137, 148, 150, 154, 159, 165, 168, 170, 172, 175, 188, 190, 191, 199, 200, 206, 215, 217, 220, 230

Offset: 1

Ray Chandler, Nov 26 2004

This is the semiprime analog of A064402.

			a(3) = 9 because 9 + semiprime(9) = 9 + 25 = 34 is semiprime.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Eric Weisstein, World of Mathematics, Semiprime.

Cf. A001358, A064402, A100493, A100466, A100467, A100916.

With[{c=Select[Range[1000],PrimeOmega[#]==2&]},Transpose[Select[Thread[ {c,Range[ Length[c]]}], PrimeOmega[Total[#]]==2&]][[2]]] (* Harvey P. Dale, Oct 25 2011 *)

a(n) = A100466(n) - A100916(n) = A100466(n) - A001358(A100915(n)).

A231383 Primes p such that p + 3*k is also prime, where p is k-th prime.

Original entry on oeis.org

2, 7, 13, 19, 29, 37, 53, 71, 101, 107, 131, 139, 163, 173, 181, 199, 223, 229, 263, 281, 293, 311, 337, 397, 443, 463, 491, 557, 569, 659, 673, 719, 733, 787, 809, 827, 839, 857, 953, 983, 1013, 1069, 1091, 1109, 1151, 1223, 1249, 1283, 1307, 1451, 1493, 1549

Offset: 1

K. D. Bajpai, Nov 08 2013

nonn

			a(5)= 29 which is 10th prime.  prime(10)+3*10= 29+30= 59 which is also prime.
a(7)= 53 which is 16th prime.  prime(16)+3*16= 53+48= 101 which is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5100

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).

[NthPrime(n): n in [1..250] | IsPrime(NthPrime(n)+3*n)]; // Vincenzo Librandi, Jan 19 2015

KD := proc() local a, b;  a:= ithprime(n); b:= 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]]

is(n)=isprime(n) && isprime(n+3*primepi(n)) \\ Charles R Greathouse IV, Nov 08 2013

A076300 Numbers k such that prime(k) + s*k is prime, s=5.

Original entry on oeis.org

1, 2, 6, 8, 10, 12, 14, 18, 30, 36, 38, 40, 48, 50, 52, 54, 64, 66, 68, 72, 74, 78, 80, 84, 96, 110, 118, 120, 122, 124, 134, 142, 148, 150, 154, 160, 178, 184, 186, 188, 198, 204, 210, 214, 220, 224, 228, 238, 240, 242, 246, 250, 252, 254, 258, 260, 268, 270

Offset: 1

Zak Seidov, Oct 05 2002

nonn

			2 is in the sequence because p(2) + 5*2 = 3 + 10 = 13 is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A064402 (s=1), A076297 (s=2), A076298 (s=3), A076299 (s=4).

[n: n in [0..500]| IsPrime(NthPrime(n) +5*n)]; // Vincenzo Librandi, Apr 06 2011

Select[Range[300],PrimeQ[Prime[#]+5#]&] (* Harvey P. Dale, Nov 27 2013 *)

select(x->isprime(x), vector(500, n, prime(n) + 5*n), 1) \\ Michel Marcus, Jan 15 2015

isok(n) = isprime(prime(n) + 5*n); \\ Michel Marcus, May 05 2018

A076298 Numbers k such that prime(k) + s*k is prime, s=3.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 16, 20, 26, 28, 32, 34, 38, 40, 42, 46, 48, 50, 56, 60, 62, 64, 68, 78, 86, 90, 94, 102, 104, 120, 122, 128, 130, 138, 140, 144, 146, 148, 162, 166, 170, 180, 182, 186, 190, 200, 204, 208, 214, 230, 238, 244, 246, 250, 252, 254, 260, 270, 282

Offset: 1

Zak Seidov, Oct 05 2002

nonn

See also A064402 (s=1), A076297 (s=2), A076299 (s=4), A076300 (s=5).

			4 is OK because p(4) + 3*4 = 7 + 12 = 19 is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A064402, A076297, A076299, A076300.

[n: n in [0..500]| IsPrime(NthPrime(n) +3*n)]; // G. C. Greubel, May 04 2018

Select[Range[300],PrimeQ[3#+Prime[#]]&] (* Harvey P. Dale, Sep 06 2012 *)

select(x->isprime(x), vector(500, n, prime(n) + 3*n), 1) \\ G. C. Greubel, May 04 2018

A076299 Numbers k such that prime(k) + s*k is prime, s=4.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 15, 17, 20, 21, 22, 25, 27, 30, 31, 33, 42, 46, 54, 56, 58, 60, 62, 67, 72, 73, 78, 81, 84, 86, 87, 88, 90, 93, 96, 99, 100, 105, 111, 112, 113, 115, 119, 127, 128, 133, 135, 137, 145, 146, 151, 152, 162, 163, 164, 165, 168, 170, 172, 173, 176, 177

Offset: 1

Zak Seidov, Oct 05 2002

nonn

See also A064402 (s=1), A076297 (s=2), A076298 (s=3), A076300 (s=5).

			4 is OK because p(4) + 4*4 = 7 + 16 = 23 is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A064402, A076297, A076298, A076300.

[n: n in [0..500]| IsPrime(NthPrime(n) +4*n)]; // G. C. Greubel, May 04 2018

Select[Range[500], PrimeQ[Prime[#] + 4 #] &] (* G. C. Greubel, May 04 2018 *)

select(x->isprime(x), vector(500, n, prime(n) + 4*n), 1) \\ G. C. Greubel, May 04 2018

A186102 Smallest prime p such that p == n (mod prime(n)).

Original entry on oeis.org

3, 2, 3, 11, 5, 19, 7, 103, 101, 97, 11, 197, 13, 229, 109, 281, 17, 79, 19, 233, 167, 101, 23, 113, 607, 127, 233, 349, 29, 821, 31, 163, 307, 173, 631, 1093, 37, 853, 373, 1597, 41, 223, 43, 1009, 439, 643, 47, 271, 503, 2111, 983, 769, 53, 1811, 569, 2423

Offset: 1

Zak Seidov, Feb 12 2011

nonn

a(n) = n iff n is prime.

			Eighth prime is 19, and 103 is the smallest prime p such that p mod 19 is 8. Therefore a(8) = 103.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A061067, A061068, A064402, A076297, A076298, A076299, A076300.

Cf. A000040, A260416.

a186102 n = f a000040_list where
   f (q:qs) = if (q - n) `mod` (a000040 n) == 0 then q else f qs
-- Reinhard Zumkeller, Aug 21 2015

Aux:=function(n); q:=NthPrime(n); p:=2; while p mod q ne n do p:=NextPrime(p); end while; return p; end function; [ Aux(n): n in [1..70] ]; // Klaus Brockhaus, Feb 12 2011

k=200;Table[p=Prime[n];m=n;While[!PrimeQ[m],m=m+p];m,{n,k}]; (* For the first k terms. Zak Seidov, Dec 13 2013 *)
Flatten[With[{prs=Prime[Range[500]]},Table[Select[prs,Mod[#,Prime[n]] == n&,1],{n,60}]]] (* Harvey P. Dale, Mar 30 2012 *)

def A186102(n): np = nth_prime(n); return next(p for p in Primes() if p % np == n) # [D. S. McNeil, Feb 13 2011]

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A061068 Primes which are the sum of a prime and its subscript.

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A061067 m-th prime prime(m) is included iff prime(m) + m is also prime.

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A231232 Primes p = prime(k) such that p + 2*k is prime.

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A076297 Numbers k such that prime(k) + s*k is prime, s=2.

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A100915 Numbers n such that n plus n-th semiprime is semiprime.

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A231383 Primes p such that p + 3*k is also prime, where p is k-th prime.

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A076300 Numbers k such that prime(k) + s*k is prime, s=5.

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