A046921 -id:A046921 - cp's OEIS frontend

A007697 Smallest odd number expressible in at least n ways as p+2*m^2 where p is 1 or a prime and m >= 0.

1, 3, 13, 19, 55, 61, 139, 139, 181, 181, 391, 439, 559, 619, 619, 829, 859, 1069, 1081, 1459, 1489, 1609, 1741, 1951, 2029, 2341, 2341, 3331, 3331, 3331, 3961, 4189, 4189, 4261, 4801, 4801, 5911, 5911, 5911, 6319, 6319, 6319, 8251, 8251, 8251, 8251, 8251

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Donovan Johnson, Table of n, a(n) for n = 1..10000
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. A016067, A046921.

import Data.List (findIndex)
import Data.Maybe (fromJust)
a007697 n = 2 * (fromJust $ findIndex (>= n) a046921_list) + 1
-- Reinhard Zumkeller, Apr 03 2013

max = 9000; sp = Outer[Plus, Prepend[Prime /@ Range[PrimePi[max]], 1], 2*Range[0, Ceiling[Sqrt[max/2]]]^2] // Flatten // Sort // Split;
a[1] = 3; a[n_] := (sel = Select[sp, Length[#] >= n &];
If[sel == {}, {}, sel[[1, 1]]]); a /@ Range[47]
(* Jean-François Alcover, Apr 29 2011 *)

Stern and Hardy-Littlewood references suggested by Ctibor O. Zizka, Apr 14 2008
Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar
a(1) changed to 1 at the suggestion of Donovan Johnson by N. J. A. Sloane, May 10 2011

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

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

A228466 Smallest odd number expressible in exactly n ways as p + 2*m^2 where p is 1 or a prime and m >= 0.

Original entry on oeis.org

5777, 1, 3, 13, 19, 55, 61, 169, 139, 271, 181, 391, 439, 559, 661, 619, 829, 859, 1069, 1081, 1459, 1489, 1609, 1741, 1951, 2029, 2509, 2341, 3631, 3769, 3331, 3961, 4525, 4189, 4261, 5281, 4801, 6229, 6361, 5911, 6439, 7111, 6319, 13081, 9931, 8869, 10321

Offset: 0

Donovan Johnson, Aug 22 2013

nonn

			a(3) = 13 = 5+2*2^2 = 11+2*1^2 = 13+2*0^2. 13 is the smallest odd number expressible in exactly 3 ways.
a(4) = 19 = 1+2*3^2 = 11+2*2^2 = 17+2*1^2 = 19+2*0^2. 19 is the smallest odd number expressible in exactly 4 ways.
a(5) = 55 = 5+2*5^2 = 23+2*4^2 = 37+2*3^2 = 47+2*2^2 = 53+2*1^2. 55 is the smallest odd number expressible in exactly 5 ways.

Donovan Johnson, Table of n, a(n) for n = 0..10000
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., Vol. 66, No. 1 (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, A016067, A046921.

(* finds terms < mx *) upto[mx_] := Block[{r = Floor[1+mx/2], k, t, p, s = {}}, t = 0*Range@r; p = Prime@ Range@ PrimePi@ mx; p[[1]] = 1; t[[# + Range[0, Sqrt[r - #]]^2]]++ & /@ ((1 + p)/2); k = 0; While[(r = Position[t, k, 1, 1]) != {}, k++; AppendTo[s, 2 r[[1,1]] - 1]]; s]; upto[10^5] (* Giovanni Resta, Aug 23 2013 *)

/* finds terms up to a(1000) */ mx=10602619; v=vector(mx); nn=vector(1000); p=vector(701940); p[1]=1; pr=2; for(j=2, 701940, pr=nextprime(pr+1); p[j]=pr); for(m=0, 2302, m2=2*m^2; for(j=1, 701940, s=m2+p[j]; if(s<=mx, v[s]++, next(2)))); forstep(j=1, mx, 2, if(v[j]==0, write("b228466.txt", 0 " " j); j=mx)); forstep(j=1, mx, 2, if(v[j]>0, if(v[j]<=1000, if(nn[v[j]]==0, nn[v[j]]=j)))); for(n=1, 1000, write("b228466.txt", n " " nn[n]))

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

A007697 Smallest odd number expressible in at least n ways as p+2*m^2 where p is 1 or a prime and m >= 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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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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Haskell

A228466 Smallest odd number expressible in exactly n ways as p + 2*m^2 where p is 1 or a prime and m >= 0.

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Mathematica

PARI

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

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PARI