A007697 -id:A007697 - 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

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

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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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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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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