A060003 -id:A060003 - 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

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

A042978 Stern primes: primes not of the form p + 2b^2 for p prime and b > 0.

Original entry on oeis.org

2, 3, 17, 137, 227, 977, 1187, 1493

Offset: 1

Jud McCranie

No others < 1299709. Are there any others? Related to a conjecture of Goldbach.
The next element of the sequence, if it exists, is larger than 10^9 ; see A060003. - M. F. Hasler, Nov 16 2007
The next element, if it exists, is larger than 2*10^13. - Benjamin Chaffin, Mar 28 2008
Does not equal A000040(k) + A001105(j) for all k & j >0. - Robert G. Wilson v, Sep 07 2012

Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 226.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 137, p. 46, Ellipses, Paris 2008.
L. E. Dickson, History of the theory of Numbers, vol. 1, page 424.

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
Mark VandeWettering, Toying with a lesser known Goldbach Conjecture
Index entries for sequences related to Goldbach conjecture

Apart from the first term, a subsequence of A060003.

N:= 10^6: # to check primes up to N
P:= select(isprime, {2,seq(i,i=3..N,2)}):
S:= {seq(2*b^2,b=1..floor(sqrt(N/2)))}:
P minus {seq(seq(p+s,p=P),s=S)}; # Robert Israel, Jan 19 2016

fQ[n_] := Block[{k = Floor[ Sqrt[ n/2]]}, While[k > 0 && !PrimeQ[n - 2*k^2], k--]; k == 0]; Select[ Prime[Range[238]], fQ] (* Robert G. Wilson v, Sep 07 2012 *)

forprime( n=1,default(primelimit), for(s=1,sqrtint(n\2), if(isprime(n-2*s^2),next(2)));print(n)) \\ M. F. Hasler, Nov 16 2007

forprime(p=2,4e9,forstep(k=sqrt(p\2),1,-1,if(isprime(p-2*k^2),next(2)));print1(p", ")) \\ Charles R Greathouse IV, Aug 04 2011

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

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, A046921, A046922, A046923, A060003, A143539.

Cf. A016067, A008578, A010051, A001105.

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

A143539 Number of ways to express 2n-1 as p+2a^2; p prime, a > 0.

Original entry on oeis.org

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

Offset: 1

T. D. Noe, Aug 23 2008

nonn

Similar to A046921 and A046923. Sequence A060003 lists the odd numbers having no representations.

			a(11)=3 because 21 = 19+2*1^2 = 13+2*2^2 = 3+2*3^2.

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.

Table[cnt=0; Do[If[PrimeQ[n-2*k^2], cnt++ ], {k,Floor[Sqrt[n/2]]}]; cnt, {n,1,20000,2}]

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

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A042978 Stern primes: primes not of the form p + 2b^2 for p prime and b > 0.

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

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A143539 Number of ways to express 2n-1 as p+2a^2; p prime, a > 0.

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

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

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