A061068 -id:A061068 - cp's OEIS frontend

A128996 Intersection of A061068 and A064270.

3, 11, 19, 79, 683, 733, 971, 1433, 1453, 2531, 3181, 3931, 4027, 4111, 4153, 4943, 6397, 6491, 6653, 6673, 6883, 8521, 8641, 8969, 10463, 10477, 10667, 11383, 11411, 11587, 12527, 13229, 15749, 16631, 17971, 21757, 21929, 24767, 27031, 28859

Offset: 1

Zak Seidov, Apr 30 2007

nonn

Primes which are equal to (some prime plus its subscript) and also to (some other prime minus its subscript). Primes of the form p(m)+m and p(n)-n, p(k) = k-th prime.

			3=p(1)+1=2+1 and 3=p(4)-4=7-4 (that is m=1, n=4),
11=p(4)+4=7+4 and 11=p(8)-8=19-8 (m=4, n=8),
19=p(6)+6=13+6 and 19=p(10)-10=29-10 (m=6, n=10),
79=p(18)+18=61+18 and 79=p(28)-28=107-28 (m=18, n=28),
683=p(106)+106=577+106 and 683=p(144)-144=827-144 (m=106, n=144).

Cf. A061068, A064270.

p=p(m)+m=p(n)-n for some m and some n>m.

A014688 a(n) = n-th prime + n.

Original entry on oeis.org

3, 5, 8, 11, 16, 19, 24, 27, 32, 39, 42, 49, 54, 57, 62, 69, 76, 79, 86, 91, 94, 101, 106, 113, 122, 127, 130, 135, 138, 143, 158, 163, 170, 173, 184, 187, 194, 201, 206, 213, 220, 223, 234, 237, 242, 245, 258, 271, 276, 279, 284, 291, 294, 305, 312, 319, 326

Offset: 1

Mohammad K. Azarian

Conjecture: this sequence contains an infinite number of primes (A061068), yet contains 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
Does an n exist such that n*prime(n)/(n+prime(n)) is an integer? - Ctibor O. Zizka, Mar 04 2008. The answer to Zizka's question is easily seen to be No: such an integer k would be positive and less than prime(n), but then k*(n + prime(n)) = prime(n)*n would be impossible. - Robert Israel, Apr 20 2015
Complement of A064427. - Jaroslav Krizek, Oct 28 2009
According to a theorem of Lu and Deng (see LINKS), there exists at least one prime number p such that a(n)-n < p <= a(n); equivalently pi(a(n)) - pi(a(n)-n) >= 1 (see A332086). For example, prime number 3 is in the range of (2,3], 5 in (3,5], 7 in (5,8], and 29 & 31 in (23,32]. - Ya-Ping Lu, Sep 02 2020

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.
Carlos Rivera, Puzzle 821. Prime numbers and complementary sequences, The Prime Puzzles and Problems Connection.
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. A000040, A078916, A093570, A093571, A076556, A061068, A332086.

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

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

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

A139025 This is to A014688 as A014688 to A000027, see comments for definition.

Original entry on oeis.org

4, 7, 14, 23, 84, 107, 120, 135, 172, 183, 234, 283, 396, 433, 446, 519, 588, 617, 638, 661, 680, 695, 706, 725, 758, 783, 854, 891, 1000, 1043, 1064, 1119, 1226, 1283, 1458, 1469, 1490, 1521, 1618, 1661, 1708, 1765, 2046, 2157, 2224, 2333, 2428, 2507, 2516

Offset: 1

Zak Seidov, Apr 07 2008

nonn

Take some initial sequence s1 = a(1), a(2),...
then for new sequence s2 = b(1), b(2),.. we define
b(n) = n + (n-th prime in s1).
If s1 = A000027 then we clearly get A014688.
If s1 = A014688 = 3,5,8,11,16,19,24,27,32,39,42,49,54,57,62,69,76,79,86,91,94
then b(1) = 1 + 3 (because 3 is the first prime in s1)
b(2) = 2 + 5 (because 5 is the 2nd prime in s1)
b(3) = 3 + 11 (because 11 is the 3rd prime in s1)
b(4) = 4 + 19 (because 19 is the 4th prime in s1)
b(5) = 5 + 79 (because 79 is the 5th prime in s1),
resulting sequence is A139025
Repeating the same procedure we have next sequences:
A139026: 8,25,110,287,438,623,668,1291,2342,2813,3790,3863,4230,4663,4828,6377,7468
A139027: 1292,3865,4666,8973,13936,50339,57266,67597,72316,85343,110934,132941,147990
A139028:270240,375255,635282,1000695,2039428,2602013,3398274,3748771,4300120
A139029:43448724,59672019,102128690,113904945,145135734,169755139

Cf. A000027, A014688, A061068, A139026-A139029.

A139025(n)=n+A061068(n)

A072063 Smallest prime of form prime(n)+k*n, k>0.

Original entry on oeis.org

3, 5, 11, 11, 31, 19, 31, 43, 41, 59, 53, 61, 67, 71, 107, 101, 127, 79, 181, 131, 157, 101, 313, 113, 197, 127, 157, 163, 167, 173, 251, 163, 269, 173, 359, 223, 379, 239, 401, 293, 1163, 223, 277, 281, 467, 337, 587, 271, 521, 379, 641, 499

Offset: 1

Reinhard Zumkeller, Jun 12 2002

nonn

According to Dirichlet's theorem primes of form prime(n)+k*n exist for all n, as gcd(n, prime(n))=1.

Nontrivial least prime == prime(n) (mod n).

			n=3, prime(3)=5: 5+1*3=8 is not prime, but 5+2*3=11, therefore a(3)=11 and A072064(3)=2.

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

Cf. A034694, A000040, A061068, A072064.

Cf. A090471, A090472, A090473.

f:= proc(n) local p,k,k0;
  p:= ithprime(n);
  if n::odd then k0:= 2 else k0:= 1 fi;
  for k from k0 by k0 do
    if isprime(p+k*n) then return p+k*n fi
  od:
end proc:
f(1):= 3:
map(f, [$1..100]); # Robert Israel, Nov 27 2023

sp[n_]:=Module[{p=Prime[n],k=1},While[!PrimeQ[p+k*n],k++];p+k*n]; Array[ sp,60] (* Harvey P. Dale, Apr 19 2019 *)

a072063(n) = {my (p=prime(n), j); for (k=1, oo, if(isprime(j=p+k*n), return(j)))}; \\ Hugo Pfoertner, Nov 27 2023

A100466 Semiprimes of special form: sum of an integer k and the k-th semiprime.

Original entry on oeis.org

14, 21, 34, 46, 62, 69, 74, 77, 93, 115, 122, 129, 141, 158, 161, 177, 187, 194, 206, 215, 221, 289, 291, 302, 326, 329, 334, 346, 361, 382, 391, 393, 398, 403, 451, 471, 481, 502, 535, 543, 581, 583, 629, 635, 655, 674, 698, 706, 713, 723, 734, 802, 813

Offset: 1

Jonathan Vos Post, Nov 20 2004

This is the semiprime analog of A061068.

			a(3) = 34 because 34 is the third semiprime appearing in A100493.

Eric Weisstein, World of Mathematics, Semiprime.

Cf. A001358, A061068, A100493, A100467, A100915, A100916.

a(n) = A100915(n) + A100916(n) = A100915(n) + A001358(A100915(n)).

Edited, corrected and extended by Ray Chandler, Nov 26 2004

A139026 This is to A139025 as A139025 to A014688, see A139025 for details.

Original entry on oeis.org

8, 25, 110, 287, 438, 623, 668, 1291, 2342, 2813, 3790, 3863, 4230, 4663, 4828, 6377, 7468, 8969, 9122, 9759, 10202, 11505, 12804, 13931, 15078, 15765, 16360, 16475, 16858, 18179, 18950, 19171, 19574, 19761, 19962, 20885, 22040, 24981, 25406

Offset: 1

Zak Seidov, Apr 07 2008

nonn

Notice that a(n)-n is always prime by definition, e.g.,

a(1) - 1 = 7, a(2) - 2 = 23, a(3) - 3 = 107, etc.

Cf. A000027, A014688, A061068, A139025, A139027, A139028, A139029.

cp's OEIS Frontend

A128996 Intersection of A061068 and A064270.

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A014688 a(n) = n-th prime + n.

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

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A139025 This is to A014688 as A014688 to A000027, see comments for definition.

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