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.

A046921 Number of ways to express 2n+1 as p+2a^2; p = 1 or prime, a >= 0.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 3, 2, 1, 4, 3, 2, 3, 1, 2, 4, 2, 2, 4, 3, 2, 3, 3, 2, 4, 3, 2, 5, 1, 2, 6, 3, 1, 3, 4, 2, 5, 4, 2, 6, 3, 2, 4, 2, 3, 6, 2, 1, 4, 3, 4, 6, 4, 2, 6, 5, 2, 6, 3, 2, 5, 1, 2, 3, 5, 4, 5, 4, 1, 8, 4, 1, 6, 3, 2, 6, 2, 2, 6, 6, 1, 4, 5, 3, 7, 4, 3, 6, 2, 3, 10, 2, 3, 4, 4, 3, 3, 4, 2
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

Goldbach conjectured this sequence is never zero.
The only zero terms appear to be for the odd numbers 5777 and 5993. - T. D. Noe, Aug 23 2008

Links

Crossrefs

Cf. A042978, A046920, A046922, A046923, A060003, A143539.
Cf. A007697.

Programs

Formula

a(n) = A046920(A005408(n)). - Reinhard Zumkeller, Apr 03 2013

A046923 Number of ways to express 2n+1 as p+2a^2; p prime, a >= 0.

Original entry on oeis.org

0, 1, 2, 2, 1, 2, 3, 2, 1, 3, 3, 2, 3, 1, 2, 4, 1, 2, 4, 3, 2, 3, 3, 2, 4, 2, 2, 5, 1, 2, 6, 3, 1, 3, 4, 2, 4, 4, 2, 6, 3, 2, 4, 2, 3, 6, 2, 1, 4, 2, 4, 6, 4, 2, 6, 5, 2, 6, 3, 2, 5, 1, 2, 3, 4, 4, 5, 4, 1, 8, 4, 1, 6, 3, 2, 6, 2, 2, 6, 6, 1, 3, 5, 3, 7, 4, 3, 6, 2, 3, 10, 2, 3, 4, 4, 3, 3, 4, 2
Offset: 0

Views

Author

David W. Wilson

Keywords

Comments

The only zero terms appear to be for the odd numbers 1, 5777 and 5993. - T. D. Noe, Aug 23 2008
a(n) = A046922(A005408(n)). - Reinhard Zumkeller, Apr 03 2013

Links

Crossrefs

Cf. A042978, A046920, A046921, A046922, A060003, A143539

Programs

Extensions

Definition corrected by T. D. Noe, Aug 23 2008

A060003 Odd numbers not of the form p + 2*k^2, k>0, p prime.

Original entry on oeis.org

1, 3, 17, 137, 227, 977, 1187, 1493, 5777, 5993
Offset: 1

Views

Author

Robert G. Wilson v, Mar 14 2001

Keywords

Comments

This sequence is probably finite.
Goldbach conjectured that all odd composites are sum of a prime and twice a square. a(9) = 5777 and a(10) = 5993 are the only known exceptions. Elements a(2) .. a(8) are the odd Stern primes (cf. A042978). The next element of the sequence, if it exists, is larger than 10^9. - M. F. Hasler, Nov 16 2007
The next term, if it exists, is larger than 2 * 10^13. - Benjamin Chaffin, Mar 28 2008
Terms greater than 3 are of the form 6*n-1. - Dan Graham, Mar 03 2015

References

  • David Wells, Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, 1997, page 76.

Links

Crossrefs

Cf. A042978.

Programs

  • Mathematica
    Do[ k = 1; While[ n - 2*k^2 > 1 && !PrimeQ[ n - 2*k^2 ], k++ ]; If[ n - 2*k^2 < 0, Print[n] ], { n, 5, 10^8 } ]
  • PARI
    forstep( n=1,2^30,2, for(s=1,sqrtint(n\2), if(isprime(n-2*s^2),next(2)));print(n)) \\ M. F. Hasler, Nov 16 2007

A046920 Number of ways to express n as p+2a^2; p = 1 or prime, a >= 0.

Original entry on oeis.org

1, 1, 2, 1, 2, 0, 2, 0, 2, 1, 2, 0, 3, 0, 2, 0, 1, 0, 4, 1, 3, 0, 2, 0, 3, 0, 1, 0, 2, 0, 4, 0, 2, 1, 2, 0, 4, 0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 4, 0, 3, 1, 2, 0, 5, 0, 1, 0, 2, 0, 6, 0, 3, 0, 1, 0, 3, 0, 4, 0, 2, 0, 5, 1, 4, 0, 2, 0, 6, 0, 3, 0, 2, 0, 4, 0, 2, 0, 3, 0, 6, 0, 2, 0, 1, 0, 4, 0, 3
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A042978, A046921, A046922, A046923, A060003, A143539.
Cf. A016067, A008578, A010051, A001105.

Programs

  • Haskell
    a046920 n = length $ filter ((\x -> x == 1 || a010051 x == 1) . (n -)) $
                            takeWhile (< n) a001105_list
-- Reinhard Zumkeller, Apr 03 2013

A046922 Number of ways to express n as p+2a^2; p prime, a >= 0.

Original entry on oeis.org

0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 2, 0, 3, 0, 2, 0, 1, 0, 3, 1, 3, 0, 2, 0, 3, 0, 1, 0, 2, 0, 4, 0, 1, 1, 2, 0, 4, 0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 4, 0, 2, 1, 2, 0, 5, 0, 1, 0, 2, 0, 6, 0, 3, 0, 1, 0, 3, 0, 4, 0, 2, 0, 4, 1, 4, 0, 2, 0, 6, 0, 3, 0, 2, 0, 4, 0, 2, 0, 3, 0, 6, 0, 2, 0, 1, 0, 4, 0, 2
Offset: 1

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A042978, A046920, A046921, A046923, A060003, A143539.
Cf. A010051, A001105.

Programs

  • Haskell
    a046922 n = sum $ map (a010051 . (n -)) $ takeWhile (< n) a001105_list
-- Reinhard Zumkeller, Apr 03 2013

A375170 Primes that can be expressed as the sum of a prime and twice a positive square.

Original entry on oeis.org

5, 7, 11, 13, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307
Offset: 1

Views

Author

Stefano Spezia, Aug 16 2024

Keywords

Examples

			23 is a term since it equals 5 + 2*3^2.

References

  • Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 226.

Crossrefs

Complement of A042978 in A000040.
Cf. A001105.

Programs

  • Mathematica
    a={}; For[i=1, i<=63, i++,For[j=1, (d=Prime[i]-2j^2)>0, j++,If[PrimeQ[d]&&!MemberQ[a,Prime[i]],AppendTo[a,Prime[i]]]]]; a
Showing 1-6 of 6 results.

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