A064269 -id:A064269 - cp's OEIS frontend

A234694 a(n) = |{0 < k < n: p = k + prime(n-k) and prime(p) - p + 1 are both prime}|.

0, 1, 0, 2, 1, 2, 1, 0, 0, 2, 2, 4, 1, 1, 2, 4, 2, 1, 1, 2, 3, 3, 2, 3, 1, 1, 1, 3, 5, 4, 3, 4, 3, 3, 3, 2, 4, 3, 2, 5, 4, 4, 4, 1, 1, 5, 4, 2, 1, 2, 5, 5, 2, 3, 4, 2, 3, 5, 7, 7, 6, 2, 5, 6, 2, 5, 4, 4, 7, 6, 6, 5, 4, 8, 7, 4, 5, 3, 5, 7, 3, 5, 4, 7, 6, 7, 2

Offset: 1

Zhi-Wei Sun, Dec 29 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 9. Also, for any integer n > 51 there is a positive integer k < n such that p = k + prime(n-k) and prime(p) + p + 1 are both prime.
(ii) If n > 9 (or n > 21), then there is a positive integer k < n such that m - 1 and prime(m) + m (or prime(m) - m, resp.) are both prime, where m = k + prime(n-k).
(iii) If n > 483, then for some 0 < k < n both prime(m) + m and prime(m) - m are prime, where m = k + prime(n-k).
(iv) If n > 3, then there is a positive integer k < n such that prime(k + prime(n-k)) + 2 is prime.
Clearly, part (i) of the conjecture implies that there are infinitely many primes p with prime(p) - p + 1 (or prime(p) + p + 1) also prime.
See A234695 for primes p with prime(p) - p + 1 also prime.

			a(5) = 1 since 2 + prime(3) = 7 and prime(7) - 6 = 11 are both prime.
a(25) = 1 since 20 + prime(5) = 31 and prime(31) - 30 = 97 are both prime.
a(27) = 1 since 18 + prime(9) = 41 and prime(41) - 40 = 139 are both prime.
a(45) = 1 since 6 + prime(39) = 173 and prime(173) - 172 = 859 are both prime.
a(49) = 1 since 26 + prime(23) = 109 and prime(109) - 108 = 491 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A014688, A014689, A014692, A064269, A064270, A232861, A233150, A233183, A233206, A233296, A234695.

f[n_,k_]:=k+Prime[n-k]
q[n_,k_]:=PrimeQ[f[n,k]]&&PrimeQ[Prime[f[n,k]]-f[n,k]+1]
a[n_]:=Sum[If[q[n,k],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

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

A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, kprime(k) - 1, kprime(k) + 1 all prime.

Original entry on oeis.org

22110, 23742, 128238, 275592, 346560, 1061910, 1281522, 1339002, 1378188, 1461600, 1850130, 2064150, 2354952, 2478270, 2523708, 2689260, 2694300, 3916638, 4422618, 4933530, 6179082, 6541080, 6641562, 6740478, 6759030, 7315812, 8484798, 8711010, 9133308, 9687720

Offset: 1

Zhi-Wei Sun, Dec 01 2013

nonn

Obviously, each term of the sequence is a multiple of 6.
Conjecture: (i) This sequence contains infinitely many terms.
(ii) Let P(x) be a non-constant integer-valued polynomial with positive leading coefficient. Then, there are infinitely many positive integers k with prime(k) - k in the range P(Z) = {P(m): m is an integer}, if and only if the degree of P(x) is at most 3. We may also replace prime(k) - k by prime(k) + k.

			a(1) = 22110 with the six numbers 22110 - 1 = 22109, 22110 + 1 = 22111, prime(22110) - 22110 = 228841, prime(22110) + 22110 = 273061, 22110*prime(22110) - 1 = 5548526609, 22110*prime(22110) + 1 = 5548526611 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..2000
Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014-2017.

Cf. A000040, A014574, A014689, A064269, A064270, A064370, A105645, A232443, A232463, A232548.

n=0
Do[If[PrimeQ[k-1]&&PrimeQ[k+1]&&PrimeQ[Prime[k]-k]&& PrimeQ[Prime[k]+k]&& PrimeQ[k*Prime[k]-1]&& PrimeQ[k*Prime[k]+1],n=n+1;Print[n," ",k]],{k,1,9700000}]

A064270 Primes of the form prime(k) - k; or primes arising in A014689.

Original entry on oeis.org

2, 3, 7, 11, 19, 29, 37, 43, 79, 83, 139, 149, 179, 197, 251, 269, 307, 349, 373, 397, 491, 683, 709, 733, 829, 859, 883, 971, 997, 1093, 1153, 1289, 1429, 1433, 1453, 1511, 1531, 1549, 1637, 1699, 1721, 1931, 1993, 1999, 2029, 2053, 2063, 2161, 2203

Offset: 1

Jason Earls, Sep 23 2001

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

Cf. A064269.

t={}; Do[If[PrimeQ[q=Prime[n]-n], AppendTo[t,q]], {n,378}]; t (* Jayanta Basu, May 14 2013 *)

j=[]; for(n=1,500, if(isprime(prime(n)-n), j=concat(j, prime(n)-n))); j

{ n=0; for (m=1, 10^9, if (isprime(prime(m) - m), write("b064270.txt", n++, " ", prime(m) - m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 11 2009

A238881 Number of odd primes p < 2n with prime(n(p+1)/2) + n*(p+1)/2 prime.

Original entry on oeis.org

0, 1, 1, 0, 0, 2, 1, 3, 1, 2, 2, 5, 2, 3, 2, 4, 3, 5, 2, 3, 2, 4, 1, 8, 4, 4, 4, 3, 2, 5, 2, 4, 4, 4, 4, 8, 2, 4, 2, 5, 4, 6, 3, 2, 6, 6, 3, 11, 6, 10, 4, 8, 2, 11, 4, 7, 4, 7, 2, 12, 4, 6, 2, 6, 3, 8, 3, 5, 8, 12, 6, 12, 4, 15, 8, 11, 5, 12, 2, 11

Offset: 1

Zhi-Wei Sun, Mar 06 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5, and a(n) = 1 only for n = 2, 3, 7, 9, 23. Moreover, for any r = 1,-1 and n > 5*(2+r) there is a positive integer k < n such that 2*k+r and prime(k*n)+k*n are both prime.
(ii) If n > 1 is not equal to 13, then prime(k*n) - k*n is prime for some k = 1, ..., n.
This conjecture implies that there are infinitely many positive integers m with prime(m) + m (or prime(m) - m) prime.

			a(7) = 1 since 11 and prime(7*(11+1)/2) + 7*(11+1)/2 = prime(42) + 42 = 181 + 42 = 223 are both prime.
a(23) = 1 since 7 and prime(23*(7+1)/2) + 23*(7+1)/2 = prime(92) + 92 = 479 + 92 = 571 are both prime.

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28--Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169--187. (See Conjecture 3.21(i) and note that the typo 2k+1 there should be 2k-1.)

Zhi-Wei Sun, Table of n, a(n) for n = 1..7000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A064269, A064402, A238573, A238576, A238878.

PQ[n_]:=PrimeQ[Prime[n]+n]
p[k_,n_]:=PQ[(Prime[k]+1)/2*n]
a[n_]:=Sum[If[p[k,n],1,0],{k,2,PrimePi[2n-1]}]
Table[a[n],{n,1,80}]

a(n) = {my(nb = 0); forprime(p=3, 2*n, if (isprime(prime(n*(p+1)/2) + n*(p+1)/2), nb++);); nb;} \\ Michel Marcus, Sep 21 2015

A105962 Numbers k such that prime(k^2) - k is prime.

Original entry on oeis.org

2, 30, 38, 44, 74, 82, 88, 96, 106, 114, 132, 138, 140, 160, 162, 184, 230, 276, 278, 280, 298, 304, 316, 318, 332, 342, 414, 420, 428, 450, 470, 480, 540, 580, 584, 600, 638, 668, 672, 678, 680, 684, 728, 750, 754, 766, 768, 788, 798, 848, 858, 860, 902, 930

Offset: 1

Tanya Khovanova, Dec 26 2006

nonn

Jinyuan Wang, Table of n, a(n) for n = 1..500

Cf. A064269 (prime(k) - k is prime), A130135 (prime(k^3) - k is prime).

Select[Range[1000], PrimeQ[Prime[ #^2] - # ] &]

is(k) = isprime(prime(k^2)-k); \\ Jinyuan Wang, Apr 10 2020

A130135 Numbers k such that prime(k^3) - k is prime.

Original entry on oeis.org

2, 4, 12, 30, 40, 42, 62, 66, 70, 72, 84, 88, 98, 112, 130, 150, 180, 248, 264, 298, 336, 342, 354, 408, 438, 450, 468, 482, 486, 546, 550, 558, 602, 606, 682, 686, 700, 710, 720, 732, 744, 770, 774, 804, 840, 848, 854, 948, 968, 976, 994, 996, 1048, 1056

Offset: 1

Jani Melik, Aug 01 2007

			a(1)=2 because prime(2^3) - 2 = 19 - 2 = 17 (a prime),
a(2)=4 because prime(4^3) - 4 = 311 - 4 = 307 (a prime),
a(3)=12 because prime(12^3) - 12 = 14741 (a prime), ...

Charles R Greathouse IV, Table of n, a(n) for n = 1..5000

Cf. A064269 (prime(k) - k is prime), A105962 (prime(k^2) - k is prime).

ts_pra_3:=proc(n) local i,ans; ans := [ ]: for i from 1 to n do if (isprime(ithprime(i^(3))-i)) then ans := [ op(ans), i ]: fi: od; RETURN(ans) end: ts_pra_3(200);

Select[Range[1100],PrimeQ[Prime[#^3]-#]&] (* Harvey P. Dale, Mar 24 2023 *)

is(k) = isprime(prime(k^3)-k); \\ Jinyuan Wang, Apr 10 2020

a(18)-a(32) from Jinyuan Wang, Apr 10 2020
a(33)-a(44) from Tyler NeSmith, Apr 15 2022
a(45)-a(54) from Jon E. Schoenfield, Apr 15 2022

A233539 a(n) = |{0 < k < n-2: m - 1, m + 1, prime(m) - m and prime(m) + m are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 2, 4, 2, 4, 4, 2, 4, 3, 5, 1, 4, 2, 3, 1, 2, 2, 2, 1, 1, 0, 0, 1, 4, 0, 1, 2, 0, 5, 2, 4, 4, 1, 3, 3, 3, 2, 3, 8, 2, 2, 3, 5, 5, 4, 3, 5, 3, 4, 3, 1, 3, 8, 4, 5, 4, 2, 6, 0, 12, 2, 4, 1, 5, 0, 4, 1, 4, 3, 3, 2, 5, 4, 7, 5, 3, 11, 1, 5, 4, 3, 4, 6, 2, 2, 5, 5, 6, 4, 4

Offset: 1

Zhi-Wei Sun, Jan 13 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 794.
(ii) For any integer n > 59, there is a positive integer k < n such that m = phi(k) + phi(n-k)/4 is an integer with prime(m) - m and prime(m) + m both prime.
Clearly, part (i) of the conjecture implies that there are infinitely many positive integers m with m - 1, m + 1, prime(m) - m and prime(m) + m all prime.

			a(21) = 1 since phi(6) + phi(15)/2 = 6 with 6 - 1 = 5, 6 + 1 = 7, prime(6) - 6 = 7 and prime(6) + 6 = 19 all prime.
a(25) = 1 since phi(17) + phi(8)/2 = 18 with 18 - 1 = 17, 18 + 1 = 19, prime(18) - 18 = 43 and prime(18) + 18 = 79 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A014574, A064269, A064402, A232861, A235682.

q[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]&&PrimeQ[Prime[n]-n]&&PrimeQ[Prime[n]+n]
f[n_,k_]:=EulerPhi[k]+EulerPhi[n-k]/2
a[n_]:=Sum[If[q[f[n,k]],1,0],{k,1,n-3}]
Table[a[n],{n,1,100}]

A270821 Numbers n such that F(n) - n is a prime, where F(n) denotes the n-th Fibonacci number.

Original entry on oeis.org

6, 8, 16, 26, 28, 76, 148, 159, 808, 848, 916, 1143, 4036, 4959, 43239, 73432, 98716, 144039, 146132

Offset: 1

Paolo P. Lava, Mar 30 2016

			fibonacci(6) - 6 = 8 - 6 = 2 that is a prime;
fibonacci(8) - 8 = 21 - 8 = 13 that is a prime.

Cf. A000045, A065220, A064269, A069109, A175404.

with(combinat): P:=proc(q) local n;
for n from 0 to q do
if isprime(fibonacci(n)-n) then print(n); fi; od; end: P(10^5);

Select[Range[150000],PrimeQ[Fibonacci[#]-#]&] (* Harvey P. Dale, May 03 2018 *)

lista(nn) = {for(n=1, nn, if(ispseudoprime(fibonacci(n)-n), print1(n, ", ")));} \\ Altug Alkan, Mar 30 2016

a(15)-a(19) from Giovanni Resta, Apr 14 2016

A107295 Numbers k such that prime(k^2) - k^2 is prime.

Original entry on oeis.org

2, 4, 12, 14, 20, 42, 48, 56, 60, 88, 104, 120, 126, 134, 138, 160, 164, 182, 186, 204, 226, 254, 270, 276, 312, 316, 330, 336, 350, 382, 408, 410, 414, 422, 438, 448, 450, 480, 492, 494, 502, 522, 546, 568, 580, 596, 620, 624, 640, 650, 690, 696

Offset: 1

Zak Seidov, May 20 2005

nonn

			88 is in the sequence because prime(88^2) - 88^2 = prime(7744) - 7744 = 78977 - 7744 = 71233 = prime(7051).

Cf. A064269 (prime(n)-n is prime), A141129 (prime(n^2)-n^2).

Select[Range[700],PrimeQ[Prime[#^2]-#^2]&] (* Harvey P. Dale, Jun 20 2015 *)

isok(n) = isprime(prime(n^2) - n^2); \\ Michel Marcus, Oct 09 2013

More terms from Michel Marcus, Oct 09 2013

cp's OEIS Frontend

A234694 a(n) = |{0 < k < n: p = k + prime(n-k) and prime(p) - p + 1 are both prime}|.

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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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Magma

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A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, k*prime(k) - 1, k*prime(k) + 1 all prime.

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A064270 Primes of the form prime(k) - k; or primes arising in A014689.

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A238881 Number of odd primes p < 2*n with prime(n*(p+1)/2) + n*(p+1)/2 prime.

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A105962 Numbers k such that prime(k^2) - k is prime.

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A130135 Numbers k such that prime(k^3) - k is prime.

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Maple

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A233539 a(n) = |{0 < k < n-2: m - 1, m + 1, prime(m) - m and prime(m) + m are all prime with m = phi(k) + phi(n-k)/2}|, where phi(.) is Euler's totient function.

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A232861 Numbers k with k - 1, k + 1, prime(k) - k, prime(k) + k, kprime(k) - 1, kprime(k) + 1 all prime.

A238881 Number of odd primes p < 2n with prime(n(p+1)/2) + n*(p+1)/2 prime.