A046920 -id:A046920 - cp's OEIS frontend

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

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

David W. Wilson

nonn

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

T. D. Noe, Table of n, a(n) for n = 0..10000
L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
Index entries for sequences related to Goldbach conjecture

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

Cf. A007697.

a046921 = a046920 . a005408  -- Reinhard Zumkeller, Apr 03 2013

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

A016067 Consider all ways of writing a number as p+2m^2 where p is 1 or a prime and m >= 0; sequence gives numbers that are expressible in at least 2 more ways than any smaller number.

Original entry on oeis.org

139, 181, 619, 2341, 3331, 4189, 4801, 5911, 6319, 8251, 9751, 11311, 12739, 13051, 15889, 20641, 21349, 22741, 23659, 24079, 32191, 33631, 39961, 42871, 45769, 56131, 57511, 65341, 71839, 80149, 90919, 95989, 99181, 105271, 119131, 130651, 157261, 167359

Offset: 1

Robert G. Wilson v

nonn

Donovan Johnson, Table of n, a(n) for n = 1..245 (terms <= 2*10^9)
G. H. Hardy and J. E. Littlewood, Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.
L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
M. Stern, Sur une assertion de Goldbach relative aux nombres impairs, Nouvelles Annales Math., 15 (1856) pp. 23-24.
Index entries for sequences related to Goldbach conjecture

Cf. A007697, A046920.

import Data.List (findIndices)
a016067 n = a016067_list !! (n-1)
a016067_list = (map (+ 1) $ findIndices (> 1) $
   zipWith (-) (tail rs) rs where rs = scanl max 0 a046920_list
-- Reinhard Zumkeller, Aug 26 2013, Apr 03 2013

/* finds first 80 terms */ mx=6023671; v=vector(mx); p=vector(414391); p[1]=1; pr=1; for(j=2, 414391, pr=nextprime(pr+1); p[j]=pr); for(m=0, 1735, m2=2*m^2; for(j=1, 414391, s=m2+p[j]; if(s<=mx, v[s]++, next(2)))); c=0; n=0; for(j=1, mx, if(v[j]>c, if(v[j]>=c+2, n++; write("b016067.txt", n " " j)); c=v[j])) /* Donovan Johnson, Aug 24 2013 */

Max{A046920(k): k <= a(n)} + 1 < A046920(a(n)). - Reinhard Zumkeller, Aug 26 2013, Apr 03 2013

Better description and more terms from Jud McCranie, Jun 16 2000

Invalid first term removed by Donovan Johnson, Aug 24 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

David W. Wilson

nonn

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

T. D. Noe, Table of n, a(n) for n = 0..10000
L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
Index entries for sequences related to Goldbach conjecture

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

a046923 = a046922 . a005408  -- Reinhard Zumkeller, Apr 03 2013

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

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

David W. Wilson

nonn

T. D. Noe, Table of n, a(n) for n=1..10000
L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
Index entries for sequences related to Goldbach conjecture

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

Cf. A010051, A001105.

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

A346368 Odd numbers that can be written in a single way as 2*k^2+p, k>0, p prime.

Original entry on oeis.org

5, 7, 9, 11, 23, 27, 29, 33, 41, 47, 53, 57, 59, 65, 71, 83, 95, 107, 113, 123, 143, 149, 161, 197, 233, 239, 257, 281, 287, 317, 323, 347, 383, 407, 413, 443, 449, 569, 743, 773, 785, 863, 1227, 1367, 1415, 1703, 1787, 2123, 2507, 2933, 3317, 3515, 3713, 4673, 5987, 6797

Offset: 1

Bernard Pitie, Jul 14 2021

nonn

The next element, if it exists, is greater than 10^8.

Cf. A060003, A046920, A046921, A046923,

isok(m) = (m>3) && (m % 2) && (sum(i=1, sqrtint((m-3)/2), isprime(m-2*i^2)) == 1); \\ Michel Marcus, Jul 22 2021

cp's OEIS Frontend

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

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A016067 Consider all ways of writing a number as p+2m^2 where p is 1 or a prime and m >= 0; sequence gives numbers that are expressible in at least 2 more ways than any smaller number.

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

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Author

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Comments

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Crossrefs

Programs

Haskell

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

A346368 Odd numbers that can be written in a single way as 2*k^2+p, k>0, p prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI