A093194 Minimal values of m=a^2+b^2=c^2+d^2 for each x=a+b+c+d=6*p (p=any odd prime).

50, 130, 250, 610, 850, 1450, 1810, 2650, 4210, 4810, 6850, 8410, 9250, 11050, 14050

Offset

1,1

Comments

The number s of different m values for each x=6*p (p prime>=7) is s=((p-1)/2)+l+1 with l=floor((p-5)/10). For a pair of twin primes p, p+2 p>=7:s(x=6*(p+2))=s(x=6*p)+1. If we consider now p=k=odd composite, then m=5*(k^2+1)=(k-2, 2*k+1, k+2, 2*k-1) is still an m value of x=6*k but is never the minimal m value for k>=15. Example:If k=15 m=5*(15^2+1)=1130=(13, 31, 17, 29) is an m value of x=6*k=90, but minimal m=(1/2)*(5^2+1)*(9^2+1)=1066 for x=90. (See A092541). This is in fact a (very slow?) algorithm to find primes.

Links

Table of n, a(n) for n=1..15.

Formula

We denote m=a^2+b^2=c^2+d^2 as m=(a, b, c, d) minimal m=5*(p^2+1)=(p-2, 2*p+1, p+2, 2*p-1) for p prime>=3 maximal m=(13/2)*(P^2+(2*l+1)^2) with l=floor((p-5)/10) for p prime>=7

Example

If p=7 minimal m=5*(7^2+1)=250=(5,15,9,13), maximal m=(13/2)*(7^2+1)=325 for x=6*p=42
If p=29 minimal m=5*(29^2+1)=4210=(27,59,31,57), maximal m=(13/2)*(29^2+5^2)=5629 for x=6*p=174

Crossrefs

See A093300 for maximal values. Cf. A092357 A092541.

Sequence in context: A212226 A031397 A045181 * A043474 A044301 A044682

Adjacent sequences: A093191 A093192 A093193 * A093195 A093196 A093197

Keyword

nonn

Author

Robin Garcia, Apr 21 2004

Status

approved