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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<q for n>= 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: A104855 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 November 19 06:03 EST 2019. Contains 329310 sequences. (Running on oeis4.)