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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. 7

%I

%S 139,181,619,2341,3331,4189,4801,5911,6319,8251,9751,11311,12739,

%T 13051,15889,20641,21349,22741,23659,24079,32191,33631,39961,42871,

%U 45769,56131,57511,65341,71839,80149,90919,95989,99181,105271,119131,130651,157261,167359

%N 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.

%D M. A. Stern, Sur un assertion de Goldbach relative aux nombres impairs, Nouvelles Annales Math., 15 (1856) pp. 23-24.

%H Donovan Johnson, <a href="/A016067/b016067.txt">Table of n, a(n) for n = 1..245</a> (terms <= 2*10^9)

%H G. H. Hardy and J. E. Littlewood, <a href="https://doi.org/10.1007/BF02403921">Some problems of 'Partitio numerorum'; III: On the expression of a number as a sum of primes</a>, Acta Math., Vol. 44, No. 1 (1923), pp. 1-70.

%H L. Hodges, <a href="https://www.jstor.org/stable/2690477">A lesser-known Goldbach conjecture</a>, Math. Mag., 66 (1993), 45-47.

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%F Max{A046920(k): k <= a(n)} + 1 < A046920(a(n)). - _Reinhard Zumkeller_, Aug 26 2013, Apr 03 2013

%o (Haskell)

%o import Data.List (findIndices)

%o a016067 n = a016067_list !! (n-1)

%o a016067_list = (map (+ 1) $ findIndices (> 1) $

%o zipWith (-) (tail rs) rs where rs = scanl max 0 a046920_list

%o -- _Reinhard Zumkeller_, Aug 26 2013, Apr 03 2013

%o (PARI) /* 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 */

%Y Cf. A007697, A046920.

%K nonn,changed

%O 1,1

%A _Robert G. Wilson v_

%E Better description and more terms from _Jud McCranie_, Jun 16 2000

%E Invalid first term removed by _Donovan Johnson_, Aug 24 2013

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Last modified July 24 18:34 EDT 2021. Contains 346273 sequences. (Running on oeis4.)