A062326 -id:A062326 - cp's OEIS frontend

A238576 Number of odd primes p < 2n with prime(n(p-1)/2)^2 - 2 prime.

0, 1, 2, 3, 2, 3, 1, 4, 3, 2, 2, 4, 2, 2, 2, 3, 2, 2, 4, 5, 2, 2, 1, 8, 2, 2, 3, 3, 2, 2, 4, 4, 5, 6, 2, 5, 4, 3, 3, 7, 2, 2, 8, 8, 5, 4, 6, 3, 3, 7, 6, 5, 3, 3, 9, 4, 8, 3, 5, 3, 1, 5, 6, 4, 6, 7, 7, 8, 6, 6, 2, 7, 1, 5, 9, 7, 5, 6, 5, 7

Offset: 1

Zhi-Wei Sun, Mar 01 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 7, 23, 61, 73.
(ii) For any integer n > 1, there is an odd prime p < 2*n with prime(n*(p+1)/2)^2 - 2 prime.
Clearly, either part of the conjecture implies that there are infinitely many primes of the form p^2 - 2 with p prime.

			a(2) = 1 since 2 and prime(2*(3-1)/2)^2 - 2 = 3^2 - 2 = 7 are both prime.
a(7) = 1 since 5 and prime(7*(5-1)/2)^2 - 2 = 43^2 - 2 = 1847 are both prime.
a(23) = 1 since 29 and prime(23*(29-1)/2)^2 - 2 = 2137^2 - 2 = 4566767 are both prime.
a(61) = 1 since 43 and prime(61*(43-1)/2)^2 - 2 = 10463^2 - 2 = 109474367 are both prime.
a(73) = 1 since 7 and prime(73*(7-1)/2)^2 - 2 = 1367^2 - 2 = 1868687 are both prime.

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

Cf. A000040, A049002, A062326, A237413, A237578, A238573.

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

A247011 Numbers n for which A242719(n) = (prime(n) + 2)^2 + 1.

Original entry on oeis.org

5, 7, 13, 17, 26, 33, 64, 81, 98, 140, 171, 176, 190, 201, 215, 225, 318, 332, 336, 444, 469, 475, 495, 551, 558, 563, 577, 601, 636, 671, 828, 849, 862, 870, 948, 1004, 1064, 1074, 1189, 1198, 1230, 1238, 1305, 1328, 1445, 1449, 1528, 1618, 1634, 1642, 1679

Offset: 1

Vladimir Shevelev, Sep 09 2014

nonn

(prime(n) + 2)^2 + 1 is the second minimal possible value of A242719(n) after prime(n)^2 + 1. Indeed, by the definition lpf(A242719(n) - 3) > lpf(A242719(n) - 1) >= prime(n), thus after prime(n)^2 + 1 we should consider prime(n)*(prime(n) + 2) + 1. Then prime(n) should be lesser number of twin primes, but then prime(n) + 1 == 0 (mod 3). So, prime(n)*(prime(n) + 2) - 2 == 0 (mod 3). Analogously one can prove that prime(n)*(prime(n) + 4) - 2 == 0 (mod 3).

Note that for the sequence prime(n+1) is in intersection of A006512 and A062326, but prime(n) is not in A062326.

François Marques, Table of n, a(n) for n = 1..109

Cf. A242719, A246748.

If prime(n) is not in A062326, then A242719(n) >= (prime(n)+2)^2 + 1.

Intersection of A247011 and A246824 forms sequence 81, 215, 828, 1189, 1634, ... For these values of n we have A242719(n) - A242720(n) = 2*(prime(n) + 1).

More terms from Peter J. C. Moses, Sep 09 2014

A118939 Primes p such that (p^2+3)/4 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 29, 31, 41, 43, 67, 83, 101, 109, 139, 151, 157, 179, 181, 199, 211, 223, 239, 263, 277, 283, 307, 311, 337, 347, 353, 379, 389, 419, 431, 463, 491, 557, 577, 587, 619, 659, 673, 739, 757, 797, 809, 811, 829, 853, 907, 911, 953, 991, 1051

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 4 divides q^2+3. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118940, A118941 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2+3)/4]&]

A137270 Primes p such that p^2 - 6 is also prime.

Original entry on oeis.org

3, 5, 7, 13, 17, 23, 47, 53, 67, 73, 83, 97, 107, 113, 167, 193, 197, 263, 293, 317, 367, 373, 383, 457, 463, 467, 487, 503, 557, 593, 607, 643, 647, 673, 677, 683, 773, 787, 797, 823, 827, 857, 877, 887, 947, 1033, 1063, 1087, 1103, 1187, 1193, 1223, 1303

Offset: 1

Ben de la Rosa and Johan Meyer (meyerjh.sci(AT)ufa.ac.za), Mar 13 2008

Each of the primes p = 2,3,5,7,13 has the property that the quadratic polynomial phi(x) = x^2 + x - p^2 takes on only prime values for x = 1,2,...,2p-2; each case giving exactly one repetition, in phi(p-1) = -p and phi(p) = p.

The only common term in A062718 and A137270 is 5. - Zak Seidov, Jun 16 2015

			The (2 x 7 - 2) -1 = 11 primes given by the polynomial x^2 + x - 7^2 for x = 1, 2, ..., 2 x 7 - 2 are -47, -43, -37, -29, -19, -7, 7, 23, 41, 61, 83, 107.

F. G. Frobenius, Uber quadratische Formen, die viele Primzahlen darstellen, Sitzungsber. d. Konigl. Acad. d. Wiss. zu Berlin, 1912, 966 - 980.

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

Cf. A062326, A062718.

[p: p in PrimesUpTo(1350) | IsPrime(p^2-6)]; // Vincenzo Librandi, Apr 14 2013

isA028879 := proc(n) isprime(n^2-6) ; end: isA137270 := proc(n) isprime(n) and isA028879(n) ; end: for i from 1 to 300 do if isA137270(ithprime(i)) then printf("%d, ",ithprime(i)) ; fi ; od: # R. J. Mathar, Mar 16 2008

Select[Prime[Range[2,300]],PrimeQ[#^2-6]&] (* Harvey P. Dale, Jul 24 2012 *)

A000040 INTERSECT A028879. - R. J. Mathar, Mar 16 2008

Corrected and extended by R. J. Mathar, Mar 16 2008

A238701 Number of primes p < n with q = floor((n-p)/4) and q^2 - 2 both prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Mar 03 2014

nonn

Conjecture: Let m > 0 and n > 2*m + 1 be integers. If m = 1 and 2 | n, or m = 3 and n is not congruent to 1 modulo 6, or m = 2, 4, 5, ..., then there is a prime p < n with q = floor((n-p)/m) and q^2 - 2 both prime.

In the case m = 1, this is a refinement of Goldbach's conjecture. In the case m = 2, this is stronger than Lemoine's conjecture (cf. A046927). The conjecture for m > 2 seems completely new. We view the conjecture as a natural extension of Goldbach's conjecture.

			a(11) = 2 since 2, floor((11-2)/4)= 2 and 2^2 - 2 are all prime, and 3, floor((11-3)/4) = 2 and 2^2 - 2 are all prime.

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

Cf. A000040, A045917, A046927, A049002, A062326, A230502, A237413, A237705.

PQ[n_]:=PrimeQ[n]&&PrimeQ[n^2-2]
p[n_,k_]:=PQ[Floor[(n-Prime[k])/4]]
a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,80}]

A245363 Numbers n for which A242719(n) = prime(n)*(prime(n)+6) + 1.

Original entry on oeis.org

19, 37, 49, 69, 73, 102, 165, 236, 253, 365, 465, 542, 595, 694, 713, 723, 762, 920, 962, 979, 1119, 1162, 1259, 1334, 1387, 1441, 1706, 1797, 1843, 1906

Offset: 1

Vladimir Shevelev, Sep 09 2014

nonn

These are such numbers n for which prime(n), prime(n)+6, prime(n)*(prime(n)+6)-2 are primes, but prime(n) is not in A062326; besides, if prime(n) is in A001359, then also prime(n+1) is not in A062326.

Prime(n)*(prime(n)+6) + 1 is the third minimal possible value of A242719(n) after prime(n)^2 + 1 and (prime(n)+2)^2 + 1 (cf. A247011 and comment there).

Cf. A242719, A246748, A247011.

prime(n) == 7 (mod 10), A242719(n) == 92 (mod 100).

More terms from Peter J. C. Moses, Sep 09 2014

A260080 Least positive integer k such that prime(kn)^2 - 2 = prime(in)prime(jn) for some integers 0 < i < j.

Original entry on oeis.org

5, 18, 18, 9, 115, 208, 69, 373, 68, 430, 8, 214, 57, 1887, 1255, 295, 880, 542, 5612, 767, 1562, 40, 853, 884, 753, 4332, 4750, 6077, 799, 1394, 639, 5442, 4785, 440, 7417, 1290, 15830, 27745, 3927, 5701, 1891, 22008, 8243, 6031, 9172, 5949, 43286, 20778, 9876, 12472

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

Conjecture: a(n) exists for any n > 0.

			a(1) = 5 since prime(5*1)^2-2 = 11^2-2 = 119 = 7*17 = prime(4*1)*prime(7*1).
a(66) = 149073 since prime(149073*66)^2-2 = 176365951^2-2 = 31104948672134399 = 3160879*9840600881 = prime(3448*66)*prime(9840600881*66).

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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..105
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A062326, A253257, A257926, A257938, A260078.

Dv[n_]:=Divisors[Prime[n]^2-2]
L[n_]:=Length[Dv[n]]
P[k_,n_]:=L[k*n]==4&&PrimeQ[Part[Dv[k*n],2]]&&Mod[PrimePi[Part[Dv[k*n],2]],n]==0&&PrimeQ[Part[Dv[k*n],3]]&&Mod[PrimePi[Part[Dv[k*n],3]],n]==0
Do[k=0;Label[bb];k=k+1;If[P[k,n],Goto[aa]];Goto[bb];Label[aa];Print[n," ", k];Continue,{n,1,50}]

A261354 Primes p such that prime(p)^2 - 2 = prime(q) for some prime q.

Original entry on oeis.org

31, 191, 541, 809, 1153, 1301, 2221, 3037, 3847, 4049, 4159, 5441, 8243, 10177, 12277, 13681, 14783, 15619, 17903, 19463, 20897, 22697, 24517, 25163, 25847, 25849, 26633, 26647, 27329, 27407, 28051, 32653, 35059, 35747, 36341, 36527, 37369, 37811, 38609, 40949, 42737, 46679, 51061, 51607, 54443, 54679, 56113, 57637, 60887, 61493

Offset: 1

Zhi-Wei Sun, Aug 15 2015

nonn

Conjecture: The sequence has infinitely many terms. In general, for any integers a,b,c with a>0 and gcd(a,b,c)=1, if b^2-4*a*c is not a square, a+b+c is odd, and gcd(b,a+c) is not divisible by 3, then there are infinitely many prime pairs {p,q} such that a*prime(p)^2+b*prime(p)+c = prime(q).

			a(1) = 31 since 31 is a prime, and prime(31)^2-2 = 127^2-2 = 16127 = prime(1877) with 1877 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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..100
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A049002, A062326, A237413, A260120, A261281, A261352, A261361.

PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
f[k_]:=Prime[Prime[k]]^2-2
n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,6200}]

A118941 Primes p such that (p^2-5)/4 is prime.

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 31, 41, 43, 53, 61, 71, 79, 83, 89, 97, 101, 107, 109, 113, 131, 137, 167, 173, 179, 193, 229, 241, 251, 263, 269, 277, 281, 283, 307, 311, 317, 349, 353, 373, 383, 419, 431, 439, 461, 463, 467, 563, 571, 577, 593, 607, 613, 619, 647

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>2, we have q=4k+-1 for some k, which makes it easy to show that 4 divides q^2-5. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118940 and A118942.

Select[Prime[Range[200]],PrimeQ[(#^2-5)/4]&]

A118942 Primes p such that (p^2-13)/12 is prime.

Original entry on oeis.org

7, 13, 17, 19, 23, 31, 37, 41, 53, 67, 71, 73, 89, 103, 107, 113, 131, 139, 157, 163, 181, 199, 211, 233, 239, 257, 269, 283, 307, 311, 337, 359, 373, 379, 401, 419, 463, 487, 491, 499, 509, 521, 577, 593, 607, 617, 631, 647, 653, 683, 701, 733, 761, 769, 787

Offset: 1

T. D. Noe, May 06 2006

nonn

For all primes q>3, we have q=6k+-1 for some k, which makes it easy to show that 12 divides q^2-13. Similar sequences, with p and (p^2+a)/b both prime, are A048161, A062324, A062326, A062718, A109953, A110589, A118915, A118918, A118939, A118940 and A118941.

Select[Prime[Range[200]],PrimeQ[(#^2-13)/12]&]

cp's OEIS Frontend

A238576 Number of odd primes p < 2*n with prime(n*(p-1)/2)^2 - 2 prime.

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A247011 Numbers n for which A242719(n) = (prime(n) + 2)^2 + 1.

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A118939 Primes p such that (p^2+3)/4 is prime.

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A137270 Primes p such that p^2 - 6 is also prime.

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A238701 Number of primes p < n with q = floor((n-p)/4) and q^2 - 2 both prime.

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A245363 Numbers n for which A242719(n) = prime(n)*(prime(n)+6) + 1.

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A260080 Least positive integer k such that prime(k*n)^2 - 2 = prime(i*n)*prime(j*n) for some integers 0 < i < j.

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A261354 Primes p such that prime(p)^2 - 2 = prime(q) for some prime q.

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A238576 Number of odd primes p < 2n with prime(n(p-1)/2)^2 - 2 prime.

A260080 Least positive integer k such that prime(kn)^2 - 2 = prime(in)prime(jn) for some integers 0 < i < j.