cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

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

Views

Author

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

Keywords

Comments

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]

Examples

			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.

Crossrefs

Cf. A145354, A157884.

Programs

  • Maple
    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

Extensions

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

References

  • 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

Crossrefs

Cf. A145354, A159296, A157884, A264904.

Programs

  • PARI
    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

Extensions

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

References

  • 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

Crossrefs

Cf. A145354, A157884, A014085.

Programs

  • Maple
    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

Extensions

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

References

  • 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

Crossrefs

Cf. A145354, A157884, A014085.

Programs

  • Maple
    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

Extensions

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

Views

Author

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

Keywords

Comments

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

Examples

			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.

References

  • 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

Crossrefs

Cf. A145354, A157884, A014085.

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

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

Keywords

Comments

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]

Examples

			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.

References

  • 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

Crossrefs

Cf. A014085, A089610, A145354, A157884.

Programs

  • PARI
    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

Extensions

More terms from Michel Marcus, May 18 2020
Showing 1-6 of 6 results.

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