A016067 -id:A016067 - 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

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

A046903 Largest odd number that can be represented in no more than n ways as p + 2*i^2 where p is 1 or a prime and i >= 0.

Original entry on oeis.org

5993, 6797, 59117, 59117, 87677, 148397, 148397, 268157, 285863, 361127, 597473, 597473, 597473, 809057, 809057, 944567, 1281473, 1281473, 1417697, 2148827, 2148827, 2419337, 2550137, 2550137, 2571263, 2571263, 2884823, 2931167, 3383837, 3601067, 3756407

Offset: 0

N. J. A. Sloane

These are just the largest numbers presently known - it has not been proved that they are really the largest.

L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
Index entries for sequences related to Goldbach conjecture

Cf. A055108, A007697, A016067.

a(3) corrected, a(6)-a(10) by Jud McCranie, Jun 12 2000

a(3) corrected and a(11)-a(30) from Donovan Johnson, Mar 21 2012

A055108 Largest odd number that can be represented in exactly n ways as p+2*i^2 where p is 1 or a prime and i >= 0.

Original entry on oeis.org

5993, 6797, 59117, 48143, 87677, 148397, 147347, 268157, 285863, 361127, 597473, 448667, 542627, 809057, 753257, 944567, 1281473, 1237007, 1417697, 2148827, 1612067, 2419337, 2550137, 2490587, 2571263, 2565893, 2884823, 2931167, 3383837, 3601067, 3756407

Offset: 0

N. J. A. Sloane

These are just the largest numbers presently known - it has not been proved that they are really the largest.

Sean A. Irvine, Table of n, a(n) for n = 0..60
L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.
Sean A. Irvine, Java program (github)
Index entries for sequences related to Goldbach conjecture

Cf. A046903, A007697, A016067.

The sequence as given in the Hodge paper is incorrect; corrected and extended by Jud McCranie, Jun 12 2000

a(11)-a(30) from Donovan Johnson, Mar 21 2012

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]))

A055202 Smallest integer that can be expressed as p+2m^2 in more ways than any smaller number, where m >= 0 and p = 1 or prime.

Original entry on oeis.org

3, 13, 19, 55, 61, 139, 181, 391, 439, 559, 619, 829, 859, 1069, 1081, 1459, 1489, 1609, 1741, 1951, 2029, 2341, 3331, 3961, 4189, 4261, 4801, 5911, 6319, 8251, 9751, 11311, 12739, 13051, 15889, 20641, 21349, 22741, 23659, 24079, 32191, 33631

Offset: 1

Jud McCranie, Jun 18 2000

nonn

			13 = 13+2*0^2 = 11+2*1^2 = 5+2*2^2, 3 ways, more than any smaller number, so 13 is in the sequence.

L. Hodges, A lesser-known Goldbach conjecture, Math. Mag., 66 (1993), 45-47.

Cf. A016067, A007697.

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

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A046903 Largest odd number that can be represented in no more than n ways as p + 2*i^2 where p is 1 or a prime and i >= 0.

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A055108 Largest odd number that can be represented in exactly n ways as p+2*i^2 where p is 1 or a prime and i >= 0.

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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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PARI

A055202 Smallest integer that can be expressed as p+2m^2 in more ways than any smaller number, where m >= 0 and p = 1 or prime.

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