A145354 -id:A145354 - cp's OEIS frontend

A157884 For each positive integer m there exist at least one prime Q=Q(m) and at least one prime P=P(m) such that (2m-1)^2 < Q < (2m)^2 - (2m-1) <= P < (2m)^2. Sequence lists pairs P(m), Q(m) for m >= 1. If more than one prime for P or Q exists, we take the smallest.

2, 3, 11, 13, 29, 31, 53, 59, 83, 97, 127, 137, 173, 191, 227, 241, 293, 307, 367, 383, 443, 463, 541, 557, 631, 653, 733, 757, 853, 877, 967, 997, 1091, 1123, 1229, 1277, 1373, 1409, 1523, 1567, 1693, 1723, 1861, 1901, 2027, 2081, 2213, 2267, 2411, 2459

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 08 2009

nonn

In some intervals there is one prime only: Q(1)=2, P(1)=3, Q(2)=11, P(2)=13, Q(3)=29, P(3)=31, Q(4)=53, P(5)=97.

Second part of numerical results to the problem: There is always a prime p in the interval between two consecutive square numbers: n^2 <= p <= (n+1)^2.

			m=1: 1 < Q < 3 <= P < 4; the only such prime Q and the only such prime P are Q(1)=2 and P(1)=3, so a(1)=2, a(2)=3.
m=2: 9 < Q < 13 <= P < 16; the only such prime Q and the only such prime P are Q(2)=11 and P(2)=13, so a(3)=11, a(4)=13.
m=4: 49 < Q < 57 <= P < 64; the only such prime Q is Q(4)=53, but there are two such primes P (59 and 61), so we take the smaller one, thus P(4)=59, so a(7)=53, a(8)=59.

Dickson, History of the theory of numbers

Cf. A145354.

277 replaced with 241, 347 with 307, 431 with 383, etc. by R. J. Mathar, Nov 01 2010

A159296 a(n) is the smaller number in the pair (L,m) which minimizes the primes of the form L^2 + m^2 under the constraint L + m = 2n + 1.

Original entry on oeis.org

1, 2, 2, 4, 5, 5, 7, 7, 9, 8, 10, 12, 10, 14, 11, 14, 17, 15, 19, 18, 20, 22, 22, 24, 25, 25, 23, 26, 29, 30, 29, 32, 30, 34, 35, 34, 34, 37, 39, 31, 40, 42, 41, 40, 43, 44, 47, 45, 40, 50, 50, 47, 51, 52, 53, 55, 54, 56, 55, 60, 59, 61, 62, 55, 65, 65, 64, 66, 69, 70, 64, 72, 67, 72, 65

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 09 2009

1) It is known that this sequence is infinite.
2) L and m with odd sum L + m are necessarily coprime if L^2 + M^2 is prime.
3) The "singular" case m = L = 1, L + m = 2 (even) with 1^2 + 1^2 = 2 is skipped. It would define a(0)=1.
4) a(n) <= n.
It has not been proved that a(n) exists for all n. See A036468. [T. D. Noe, Apr 22 2009]

			n=1: 1^2 + 2^2 = 5; a(1)=1.
n=2: 2^2 + 3^2 = 13 < 1^2 + 4^2 = 17; a(2)=2.
n=3: 2^2 + 5^2 = 29 < 1^2 + 6^2 = 37. 3^2 + 4^2 = 5^2 not prime; a(3)=2.
n=27: 23^2 + 32^2 = 1553 < 1597, 1657, 1693, 1733, 1777, 1877, 1933, 1993, 2273, 2437, 2617, 2713, 2917, a(27)=23.

Cf. A145354, A157884.

A159296 := proc(n) local a,pmin,l,m ; a := 0 ; pmin := 2*(2*n+1)^2 ; for l from 1 to n do m := 2*n+1-l ; if isprime(m^2+l^2) then if m^2+l^2 < pmin then pmin := m^2+l^2 ; a := l ; fi; fi; od: RETURN(a) ; end: seq(A159296(n),n=1..80) ; # R. J. Mathar, Apr 18 2009

Edited and extended by R. J. Mathar, Apr 18 2009

A159351 Smallest prime of the form a^2 + b^2 with 0 < a < b such that a + b = 2n+1.

Original entry on oeis.org

5, 13, 29, 41, 61, 89, 113, 149, 181, 233, 269, 313, 389, 421, 521, 557, 613, 709, 761, 853, 929, 1013, 1109, 1201, 1301, 1409, 1553, 1637, 1741, 1861, 1997, 2113, 2269, 2381, 2521, 2677, 2837, 2969, 3121, 3461, 3449, 3613, 3797, 4001, 4153, 4337, 4513, 4729, 5081

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 11 2009

nonn

Conjecture: there is always such a prime number.

Primes of the form x^2 + y^2 with 0 < x < y such that there are no primes of the form (x+z)^2 + (y-z)^2 for 0 < z < (y-x)/2. Note: a(40) = 3461 > a(41) = 3449, so the order is not maintained. - Thomas Ordowski, Jan 21 2017

			1^2 + 2^2 = 5 = a(1) = 1.
2^2 + 3^2 = 13 = a(2) < 1^2 + 4^2 = 17.
2^2 + 5^2 = 29 = a(3) < 1^2 + 6^2 = 37.
23^2 + 32^2 = 1553 = a(27) < 1597, 1657, 1693, 1733, 1777, 1877, 1933, 1993, 2273, 2437, 2617, 2713, 2917, 14 prime representations as sum of two squares.

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
David Wells, Prime Numbers: The Most Mysterious Figures in Math. John Wiley and Sons. 2005

Cf. A145354, A159296, A157884, A264904.

isok(p, n) = for (i=1, 2*n, if (i^2 + (2*n+1-i)^2 == p, return (1));); 0;
a(n) = {my(p = 2); while (! isok(p, n), p = nextprime(p+1)); p;} \\ Michel Marcus, Jan 29 2017

A-number in definition and cross-reference corrected, and more terms from R. J. Mathar, Apr 24 2009

Edited by Thomas Ordowski, Jan 25 2017

A159803 Number of primes p with (2m+1)^2 - 2m <= p < (2m+1)^2.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 2, 3, 4, 4, 3, 5, 3, 5, 4, 4, 5, 2, 6, 4, 4, 7, 3, 8, 5, 7, 6, 5, 7, 8, 10, 5, 8, 7, 10, 8, 7, 10, 9, 7, 10, 9, 13, 10, 11, 11, 11, 11, 11, 12, 9, 9, 11, 14, 12, 11, 12, 12, 11, 15, 12, 11, 14, 12, 12, 14, 15, 12, 15, 14, 17, 18, 20, 18, 17, 14, 18, 12, 15, 15, 15, 14, 21

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009

nonn

1) Immediate connection to unsolved problem, is there always a prime between n^2 and (n+1)^2 ("full" interval of two consecutive squares).
2) See sequence A145354 and A157884 for more details to this new improved conjecture.
3) Second ("right") half interval: number of primes p with (2m+1)^2-2m <= p < (2m+1)^2.
4) It is conjectured that a(m) >= 1.
5) No a(m) with m>5 is known, where a(m)=1.
This is a bisection of A094189 and hence related to a conjecture of Oppermann. - T. D. Noe, Apr 22 2009

			1) m=1: 7 <= p < 9 => prime 7: a(1)=1.
2) m=2: 21 <= p < 25 => prime 23: a(2)=1.
3) m=3: 43 <= p < 49 => primes 43, 47: a(3)=2.
4) m=30: 3661 <= p < 3721 => primes 3671,3673,3677,3691,3697,3701,3709,3719: a(30)=8.

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
P. Ribenboim, The New Book of Prime Number Records. Springer. 1996

Cf. A145354, A157884, A014085.

A159803 := proc(n) local a,p; a := 0 ; for p from 4*n^2+2*n+1 to 4*n^2+4*n do if isprime(p) then a := a+1 ; fi; od: a ; end: seq(A159803(n),n=1..120) ; # R. J. Mathar, Apr 22 2009

More terms from R. J. Mathar, Apr 22 2009

A159805 Number of primes p with (2m)^2-(2m-1) <= p < (2m)^2.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 4, 3, 3, 4, 4, 4, 4, 5, 4, 5, 6, 5, 6, 5, 7, 7, 4, 10, 5, 5, 10, 8, 6, 7, 5, 7, 5, 7, 10, 7, 10, 12, 11, 10, 7, 11, 10, 10, 10, 12, 8, 9, 11, 9, 9, 8, 9, 15, 15, 9, 14, 11, 14, 17, 11, 11, 10, 17, 14, 15, 13, 17, 17, 13, 12, 16, 13, 20, 17, 11, 14, 14, 17, 16, 17, 16

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009

nonn

1) Immediate connection to unsolved problem, is there always a prime between n^2 and (n+1)^2 ("full" interval of two consecutive squares).
2) See sequence A145354 and A157884 for more details to this new improved conjecture.
3) Second ("right") half-interval: number of primes p with (2m)^2-(2m-1) <= p < (2m)^2.
4) It is conjectured that a(m) >= 1.
5) No a(m) with m > 5 is known, where a(m)=1.
Except for a(1), this is a bisection of A094189 and hence related to a conjecture of Oppermann. - T. D. Noe, Apr 22 2009

			m=1: 3 <= p < 4 => prime 3: a(1)=1;
m=4: 57 <= p < 64 => primes 59,61: a(4)=2;
m=5: 91 <= p < 100 => prime 97: a(5)=1;
m=30: 3541 <= p < 3600 => primes 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593: a(30)=8.

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
P. Ribenboim, The New Book of Prime Number Records. Springer, 1996

Cf. A145354, A157884, A014085.

A159805 := proc(n) local a,p; a := 0 ; for p from 4*n^2-2*n+1 to 4*n^2-1 do if isprime(p) then a := a+1 ; fi; od: a ; end: seq(A159805(n),n=1..120) ; # R. J. Mathar, Apr 22 2009

More terms from R. J. Mathar, Apr 22 2009

A159802 Number of primes q with (2m)^2+1 <= q < (2m+1)^2-2m.

Original entry on oeis.org

1, 2, 2, 2, 4, 2, 2, 4, 2, 3, 4, 4, 4, 4, 4, 5, 4, 7, 6, 8, 5, 4, 7, 7, 6, 9, 7, 7, 6, 8, 7, 9, 7, 10, 11, 7, 10, 12, 9, 6, 9, 8, 8, 8, 9, 8, 10, 10, 12, 11, 11, 12, 13, 9, 12, 14, 13, 11, 10, 14, 11, 14, 15, 12, 16, 14, 16, 11, 12, 11, 12, 14, 14, 15, 15, 13, 17, 15, 16, 18, 17, 15, 12, 12

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009

nonn

1) Immediate connection to unsolved problem, is there always a prime between n^2 and (n+1)^2 ("full" interval of two consecutive squares).
2) See sequence A145354 and A157884 for more details to this new improved conjecture.
3) First ("left") half interval, primes q with (2m)^2+1 <= q < (2m+1)^2-2m.
4) It is conjectured that a(m) >= 1.
5) No a(m) with m>1 is known, where a(m)=1.
This is a bisection of A089610 and hence related to a conjecture of Oppermann. - T. D. Noe, Apr 22 2009

			1) m=1: 5 <= q < 7 => prime 5: a(1)=1.
2) m=2: 17 <= q < 21 => primes 17, 19: a(2)=2.
3) m=3: 37 <= q < 43 => primes 37, 41: a(3)=2.
4) m=30: 3601 <= q < 3661 => primes 3607,3613,3617,3623,3631,3637,3643,3659: a(30)=8.

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
P. Ribenboim, The New Book of Prime Number Records. Springer. 1996

Cf. A145354, A157884, A014085.

f[n_] := PrimePi[(2 n + 1)^2 - 2 n - 1] - PrimePi[(2 n)^2]; Table[ f@n, {n, 85}] (* Robert G. Wilson v, May 04 2009 *)

More terms from Robert G. Wilson v, May 04 2009

A159804 Number of primes q with (2n-1)^2+1 <= q < (2n)^2-(2n-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 4, 1, 3, 4, 2, 4, 4, 4, 5, 6, 5, 3, 6, 5, 7, 6, 6, 6, 5, 7, 6, 7, 8, 7, 8, 11, 8, 8, 8, 7, 11, 4, 11, 9, 10, 9, 11, 8, 10, 13, 9, 14, 12, 11, 12, 12, 11, 15, 12, 9, 13, 15, 8, 14, 13, 16, 12, 14, 11, 11, 15, 9, 16, 16, 11, 14, 14, 13, 13, 13

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 22 2009

nonn

Immediate connection to unsolved problem, is there always a prime between n^2 and (n+1)^2 ("full" interval of two consecutive squares).
See sequence A145354 and A157884 for more details to this new improved conjecture.
First ("left") half interval: number of primes q (2m-1)^2+1 <= q < (2m)^2-(2m-1).
It is conjectured that a(n) >= 1 for n >= 1.
No a(m) with m>9 is known, where a(m)=1.
This is a bisection of A089610 and hence related to a conjecture of Oppermann. [T. D. Noe, Apr 22 2009]

			n=1: 2 <= q < 3 => prime 2: a(1)=1;
n=5: 82 <= q < 91 => primes 83,89: a(5)=2;
n=9: 290 <= q < 307 => prime 293: a(9)=1;
n=30: 3482 <= q < 3541 => prime 3491,3499,3511,3517,3527,3529,3533,3539: a(30)=8.

L. E. Dickson, History of the Theory of Numbers, Vol, I: Divisibility and Primality, AMS Chelsea Publ., 1999
R. K. Guy, Unsolved Problems in Number Theory (2nd ed.) New York: Springer-Verlag, 1994
P. Ribenboim, The New Book of Prime Number Records. Springer. 1996

Cf. A014085, A089610, A145354, A157884.

a(n) = if (n==1, 1, primepi((2*n)^2-(2*n-1)-1) - primepi((2*n-1)^2+1)); \\ Michel Marcus, May 18 2020

More terms from Michel Marcus, May 18 2020

cp's OEIS Frontend

A157884 For each positive integer m there exist at least one prime Q=Q(m) and at least one prime P=P(m) such that (2m-1)^2 < Q < (2m)^2 - (2m-1) <= P < (2m)^2. Sequence lists pairs P(m), Q(m) for m >= 1. If more than one prime for P or Q exists, we take the smallest.

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A159296 a(n) is the smaller number in the pair (L,m) which minimizes the primes of the form L^2 + m^2 under the constraint L + m = 2n + 1.

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A159351 Smallest prime of the form a^2 + b^2 with 0 < a < b such that a + b = 2n+1.

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A159803 Number of primes p with (2m+1)^2 - 2m <= p < (2m+1)^2.

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A159805 Number of primes p with (2m)^2-(2m-1) <= p < (2m)^2.

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A159802 Number of primes q with (2m)^2+1 <= q < (2m+1)^2-2m.

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A159804 Number of primes q with (2n-1)^2+1 <= q < (2n)^2-(2n-1).

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PARI

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