A064270 -id:A064270 - 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.

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

Original entry on oeis.org

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}]

A064269 Numbers k such that prime(k) - k is prime.

Original entry on oeis.org

3, 4, 6, 8, 10, 14, 16, 18, 28, 30, 42, 44, 50, 54, 66, 68, 76, 84, 90, 94, 110, 144, 148, 154, 168, 174, 178, 192, 196, 214, 220, 242, 264, 266, 268, 278, 280, 282, 294, 304, 308, 336, 346, 348, 354, 358, 360, 370, 376, 380, 382, 384, 390, 400, 402, 408, 414

Offset: 1

Jason Earls, Sep 23 2001

			n=54: prime(54) - 54 = 197, a prime.

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

Cf. A014689, A064270.

Select[ Range[ 415 ], PrimeQ[ Prime[ # ] - # ] & ]

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

{ n=0; for (m=1, 10^9, if (isprime(prime(m) - m), write("b064269.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 10 2009

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}]

A172096 Primes having no representation of the form prime(n)-+n.

Original entry on oeis.org

13, 17, 23, 31, 41, 47, 53, 59, 61, 67, 71, 73, 89, 97, 103, 107, 109, 131, 137, 151, 157, 167, 181, 191, 193, 199, 211, 227, 229, 233, 239, 241, 257, 263, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 379, 389

Offset: 1

Juri-Stepan Gerasimov, Nov 21 2010

nonn

			13 is a term because it's prime and prime(n)+n < 13 for n <= 4 and > 13 for n > 4, and prime(n)-n < 13 for n <= 8 and > 13 for n > 8.

Cf. A064270.

nn=100; p=Select[Union[Flatten[Table[Prime[n]+{-n,n}, {n,nn}]]], PrimeQ]; Complement[Prime[Range[PrimePi[p[[-1]]]]], p]

Corrected by T. D. Noe, Nov 21 2010

cp's OEIS Frontend

A128996 Intersection of A061068 and A064270.

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

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A064269 Numbers k such that prime(k) - k is prime.

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PARI

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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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A172096 Primes having no representation of the form prime(n)-+n.

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Extensions

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