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Search: id:A133388
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| A133388 |
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Largest integer m such that n-m^2 is a square, or 0 if no such m exists. |
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+0
4
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| 1, 1, 0, 2, 2, 0, 0, 2, 3, 3, 0, 0, 3, 0, 0, 4, 4, 3, 0, 4, 0, 0, 0, 0, 5, 5, 0, 0, 5, 0, 0, 4, 0, 5, 0, 6, 6, 0, 0, 6, 5, 0, 0, 0, 6, 0, 0, 0, 7, 7, 0, 6, 7, 0, 0, 0, 0, 7, 0, 0, 6, 0, 0, 8, 8, 0, 0, 8, 0, 0, 0, 6, 8, 7, 0, 0, 0, 0, 0, 8, 9, 9, 0, 0, 9, 0, 0, 0, 8, 9, 0, 0, 0, 0, 0, 0, 9, 7, 0, 10
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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One could have chosen a(n)=-1 instead of 0 if n is not sum of two squares and then include unambiguously a(0)=0. At present, a(n)=0 <=> A000161(n)=0.
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FORMULA
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a(n) = max( sup { max(a,b) | a^2+b^2 = n ; a,b in Z }, 0 )
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EXAMPLE
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a(3) = 0 since 3 cannot be written as sum of 2 perfect squares;
a(5) = 2 since 5 = 2^2 + 1^2.
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PROGRAM
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(PARI) sum2sqr(n)={ local( t, f, p = 1, s=[] ); if( n<=1, return( if( n<0, [], [[0, n]] ))); if( isprime(n), return( if( n=sum2sqr_prime(n), [n], []))); for( i=1, matsize(f = factor(n))[1], if( f[i, 1] % 4 == 1, s = concat( s, [[sum2sqr_prime( f[i, 1] )*[1, I]~, f[i, 2]]] ), if( f[i, 1] == 2, p = (1+I)^f[i, 2], if( f[i, 2] % 2 == 1, return([]), p *= f[i, 1]^(f[i, 2]>>1)) ) ) ); s = vector( #s, i, f=s[i][1]; vector( t=s[i][2]+1, j, f^(t-j)*conj(f)^(j-1) )); t=[]; forvec( T = vector(#s, i, [1, #s[i]]), f = prod( j=1, #T, s[j][T[j]], p); t = setunion( t, [ vecsort( abs([ real(f), imag(f) ]))] ) ); vecsort( eval( t ), 1 ) } /*helper function: treats the case of a prime*/ sum2sqr_prime(p)={ local(c, d); if( p==2, [1, 1], if( p%4 != 3, if( p%8==5, c=2, if( p%3==2, c=3, c=5; while( Mod(p, c)^(c\2)+1, c=nextprime( c+1 )))); c=lift(Mod(c, p)^(p\4)); d=sqrtint(p); while(c > d, c=(p+0)%(p=c)); [c, p%c] ))} /* now this sequence: take the maximum */ vector(30, i, if(i=sum2sqr(i), vecmax(Mat(i~))))
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CROSSREFS
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Cf. A000161, A001481.
Adjacent sequences: A133385 A133386 A133387 this_sequence A133389 A133390 A133391
Sequence in context: A031124 A063695 A081417 this_sequence A109042 A128540 A051775
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KEYWORD
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nonn
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AUTHOR
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M. F. Hasler (Maximilian.Hasler(AT)gmail.com), Nov 23 2007
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