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 A143697 Least square k^2 such that n^2-k^2 = p*q with p and q odd primes and p= 4. 3
 1, 4, 1, 16, 9, 4, 9, 36, 1, 36, 9, 4, 9, 36, 1, 144, 9, 4, 81, 36, 25, 36, 9, 16, 81, 144, 1, 144, 81, 16, 9, 36, 25, 36, 81, 4, 9, 144, 1, 576, 9, 4, 225, 36, 25, 144, 9, 64, 81, 36, 49, 144, 9, 16, 225, 144, 1, 324, 81, 16, 9, 36, 25, 36, 225, 4, 9, 144, 1, 36, 225 (list; graph; refs; listen; history; text; internal format)
 OFFSET 4,2 COMMENTS The product p*q is the sum of p consecutive odd numbers with 2*n-1 the greatest. For n=4 p*q=3*5=15, 15=7+5+3 For n=5 p*q=3*7=21, 21=9+7+5 For n=6 p*q=5*7=35, 35=11+9+7+5+3 For n=7 p*q=3*11=33, 33=13+11+9 k^2 is the sum of the k first consecutive odd numbers p=n-k and q=n+k. Assuming a strong version of the Goldbach conjecture, every term exists and we have a(n)=A082467(n)^2, p(n)=A078587(n) and q(n)=A078496(n). [T. D. Noe, Jan 22 2009] LINKS Pierre CAMI, Table of n, a(n) for n = 4..60000 EXAMPLE 4*4-1=3*5 p=3 q=5 5*5-4=3*7 p=3 q=7 6*6-1=5*7 p=5 q=7 7*7-16=3*11 p=3 q=11 PROG (PARI) a(n) = {for (k=1, n-1, my(x=n^2-k^2); if ((omega(x)==2) && (bigomega(x)==2) && (x%2), return(k^2); ); ); } \\ Michel Marcus, Sep 23 2019 CROSSREFS Cf. A078587, A078496. Sequence in context: A351419 A303054 A143496 * A272088 A271135 A271601 Adjacent sequences: A143694 A143695 A143696 * A143698 A143699 A143700 KEYWORD nonn AUTHOR Pierre CAMI, Aug 29 2008 STATUS approved

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Last modified September 21 02:47 EDT 2023. Contains 365486 sequences. (Running on oeis4.)