A061067 -id:A061067 - 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

A064402 Numbers n such that prime(n)+n is a prime, where prime(n) denotes the n-th prime number.

Original entry on oeis.org

1, 2, 4, 6, 18, 22, 24, 26, 32, 34, 42, 48, 66, 70, 72, 82, 92, 96, 98, 100, 102, 104, 106, 108, 114, 116, 126, 130, 144, 150, 152, 158, 172, 180, 200, 202, 204, 206, 218, 222, 228, 236, 270, 282, 290, 300, 312, 322, 324, 328, 330, 350, 352, 356, 362, 378, 384

Offset: 1

Robert G. Wilson v, Sep 28 2001

nonn

a(n) = order among the primes of A061067(n).

Except for the first one all terms are even. Conjecture: First differences include all even integers. - Zak Seidov, Nov 10 2013

Harry J. Smith, Table of n, a(n) for n=1..1000

Cf. A061067, A061068, A064269.

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

Select[ Range[ 400 ], PrimeQ[ Prime[ # ] + # ] & ]

{ n=0; for (m=1, 10^9, if (isprime(prime(m) + m), write("b064402.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 13 2009

a(n) = A061068(n) - A061067(n-1).

A014688(a(n)) = A061068(n). - Zak Seidov, Nov 10 2013

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 08 2007

A077510 Numbers k such that k + pi(k) is a prime.

Original entry on oeis.org

2, 3, 7, 9, 12, 13, 21, 28, 32, 36, 45, 52, 55, 57, 61, 65, 70, 76, 79, 81, 84, 86, 89, 101, 104, 110, 119, 121, 131, 135, 139, 145, 147, 155, 160, 162, 172, 181, 185, 187, 195, 205, 216, 222, 223, 228, 231, 253, 258, 262, 273, 278, 286, 288, 292, 297, 305, 310

Offset: 1

Amarnath Murthy, Nov 08 2002

nonn

Conjecture: for k > 5, prime(n) <= k < prime(n+1) <= k + pi(k), i.e., the smallest prime greater than k is <= k + pi(k). Equality holds for k = 7.

			21 is a member as 21 + pi(21) = 21 + 8 = 29 is a prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A061067, A076757.

Select[Range[350],PrimeQ[#+PrimePi[#]]&] (* Harvey P. Dale, Nov 19 2011 *)

for(n=1,200,if(isprime(n+primepi(n)),print1(n,", "))) \\ Derek Orr, Jun 22 2015

pi=0; p=2; forprime(q=3,1e3, pi++; for(n=p,q-1, if(isprime(n+pi), print1(n", "))); p=q) \\ Charles R Greathouse IV, Jun 23 2015

More terms from David Garber, Nov 10 2002

A100916 Sum of a semiprime and its semiprime index is a new semiprime.

Original entry on oeis.org

10, 15, 25, 34, 46, 51, 55, 57, 69, 86, 91, 95, 106, 119, 121, 133, 141, 145, 155, 161, 166, 217, 218, 226, 247, 249, 253, 262, 274, 291, 298, 299, 302, 305, 341, 358, 365, 382, 407, 413, 445, 446, 481, 485, 501, 515, 533, 538, 543, 551, 559, 614, 623, 626

Offset: 1

Ray Chandler, Nov 26 2004

This is the semiprime analog of A061067.

			a(1) = 10 because 10 = semiprime(4) and semiprime(4) + 4 = 14 is
semiprime.
a(2) = 15 because 15 = semiprime(6) and semiprime(6) + 6 = 21 is
semiprime.

Eric Weisstein, World of Mathematics, Semiprime.

Cf. A001358, A061067, A100493, A100466, A100467, A100915.

Module[{sp=Select[Range[1000],PrimeOmega[#]==2&],len},len=Length[sp];Select[ Thread[{sp,Range[len]}],PrimeOmega[Total[#]]==2&]][[All,1]] (* Harvey P. Dale, Jan 13 2019 *)

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

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]

A254462 Primes prime(n) such that prime(n) + 5*n is also prime.

Original entry on oeis.org

2, 3, 13, 19, 29, 37, 43, 61, 113, 151, 163, 173, 223, 229, 239, 251, 311, 317, 337, 359, 373, 397, 409, 433, 503, 601, 647, 659, 673, 683, 757, 821, 857, 863, 887, 941, 1061, 1097, 1109, 1123, 1213, 1249, 1291, 1307, 1373, 1423, 1439, 1493, 1511, 1531, 1559

Offset: 1

Vincenzo Librandi, Feb 04 2015

nonn

			prime(2)=3 is in the sequence because 3+5*2 = 13 is prime.
prime(6)=13 is in the sequence because 13+5*6 = 43 is prime.

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

Cf. A061067, A076300, A231232, A231383.

[NthPrime(n): n in [1..300] | IsPrime(NthPrime(n)+5*n)];

P:= select(isprime, [2,seq(i,i=3..10000,2)]):
P[select(i -> isprime(P[i]+5*i),[$1..nops(P)])]; # Robert Israel, Aug 01 2024

Prime[Select[Range[300], PrimeQ[Prime[#] + 5 #] &]]

A227420 Primes p such that p - pi(p) is also prime.

Original entry on oeis.org

5, 7, 13, 19, 29, 43, 53, 61, 107, 113, 181, 193, 229, 251, 317, 337, 383, 433, 463, 491, 601, 827, 857, 887, 997, 1033, 1061, 1163, 1193, 1307, 1373, 1531, 1693, 1699, 1721, 1789, 1811, 1831, 1931, 2003, 2029, 2267, 2339, 2347, 2383, 2411, 2423, 2531, 2579, 2617

Offset: 1

Zak Seidov, Sep 16 2013

nonn

Note that pi(p) are all even, except for the first term. Differs from A101324.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A061067, A101324.

5 = A000040(3) and 5 - 3 = 2 prime, 43 = A000040(14) and 43 - 14 = 29 prime.

fQ[p_] := PrimeQ[p - PrimePi[p]]; Select[ Prime@ Range@ 400, fQ] (* Robert G. Wilson v, Dec 19 2014 *)

is(n)=isprime(n) && isprime(n-primepi(n)) \\ Charles R Greathouse IV, Sep 16 2013

v=primes(10^4); for(i=1,#v,if(isprime(v[i]-i),print1(v[i]", "))) \\ Charles R Greathouse IV, Sep 16 2013

A254665 Primes prime(n) such that prime(n) + 7*n is also prime.

Original entry on oeis.org

3, 71, 79, 89, 101, 199, 271, 281, 293, 349, 359, 433, 463, 479, 569, 577, 641, 659, 701, 743, 769, 787, 809, 839, 863, 911, 953, 1013, 1033, 1049, 1109, 1181, 1249, 1277, 1321, 1361, 1399, 1429, 1451, 1459, 1481, 1511, 1549, 1571, 1627, 1693, 1733, 1759, 1889

Offset: 1

Vincenzo Librandi, Feb 04 2015

nonn

			prime(2)=3 is in the sequence because 3+7*2 = 17 is prime.
prime(20)=71 is in the sequence because 71+7*20 = 211 is prime.

Cf. A061067, A231232, A231383, A254462.

[NthPrime(n): n in [1..300] | IsPrime(NthPrime(n)+7*n)];

Prime[Select[Range[300], PrimeQ[Prime[#] + 7# ]&]]

A254672 Primes prime(n) such that prime(n) + 6*n is also prime.

Original entry on oeis.org

5, 7, 11, 17, 19, 29, 31, 37, 43, 47, 53, 67, 71, 73, 79, 89, 101, 109, 113, 127, 149, 151, 157, 167, 181, 191, 193, 197, 227, 257, 263, 271, 277, 281, 331, 347, 349, 379, 383, 431, 433, 449, 467, 479, 499, 509, 521, 523, 547, 563, 569, 571, 577, 587, 619, 631

Offset: 1

Vincenzo Librandi, Feb 05 2015

nonn

			prime(5) = 11 is in the sequence because 11 + 6*5 = 41 is prime.
prime(8) = 19 is in the sequence because 19 + 6*8 = 67 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A061067, A231232, A231383, A254462, A254665, A254673.

[NthPrime(n): n in [1..200] | IsPrime(NthPrime(n)+6*n)]

Prime[Select[Range[150], PrimeQ[Prime[#] + 6 #] &]]

A254673 Primes prime(n) such that prime(n) + 4*n is also prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 47, 59, 71, 73, 79, 97, 103, 113, 127, 137, 181, 199, 251, 263, 271, 281, 293, 331, 359, 367, 397, 419, 433, 443, 449, 457, 463, 487, 503, 523, 541, 571, 607, 613, 617, 631, 653, 709, 719, 751, 761, 773, 829, 839, 877, 881, 953, 967, 971

Offset: 1

Vincenzo Librandi, Feb 05 2015

nonn

			prime(4)=7 is in the sequence because 7+4*4 = 23 is prime.
prime(6)=13 is in the sequence because 13+4*6 = 37 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A061067, A231232, A231383, A254462, A254665, A254672.

[NthPrime(n): n in [1..200] | IsPrime(NthPrime(n)+4*n)]

Prime[Select[Range[180], PrimeQ[Prime[#] + 4 #] &]]

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

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A064402 Numbers n such that prime(n)+n is a prime, where prime(n) denotes the n-th prime number.

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A077510 Numbers k such that k + pi(k) is a prime.

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A100916 Sum of a semiprime and its semiprime index is a new semiprime.

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

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A254462 Primes prime(n) such that prime(n) + 5*n is also prime.

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A227420 Primes p such that p - pi(p) is also prime.

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