A359234 a(n) is the smallest centered square number with exactly n distinct prime factors.
1, 5, 85, 1105, 99905, 2339285, 294346585, 29215971265, 4274253515545, 135890190846085, 14289540733429585, 10285257499051999685, 659442750659021626765, 386961420250791449193065, 10019680253112694448155885, 7190322949201929673798425205, 944550762877225960238953138865
Offset: 0
Keywords
Examples
a(4) = 99905, because 99905 is a centered square number with 4 distinct prime factors {5, 13, 29, 53} and this is the smallest such number.
Links
- Eric Weisstein's World of Mathematics, Centered Square Number
- Eric Weisstein's World of Mathematics, Distinct Prime Factors
Programs
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PARI
a(n) = for(k=0, oo, my(t=2*k*k + 2*k + 1); if(omega(t) == n, return(t))); \\ Daniel Suteu, Dec 29 2022
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PARI
omega_centered_square_numbers(A, B, n) = A=max(A, vecprod(primes(n))); (f(m, p, j) = my(list=List()); forprime(q=p, sqrtnint(B\m, j), if(q%4==1, my(v=m*q, r=nextprime(q+1)); while(v <= B, if(j==1, if(v>=A, if (issquare((8*(v-1))/4 + 1) && ((sqrtint((8*(v-1))/4 + 1)-1)%2 == 0), listput(list, v))), if(v*r <= B, list=concat(list, f(v, r, j-1)))); v *= q))); list); vecsort(Vec(f(1, 2, n))); a(n) = if(n==0, return(1)); my(x=vecprod(primes(n)), y=2*x); while(1, my(v=omega_centered_square_numbers(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Dec 29 2022
Extensions
a(8) from Jon E. Schoenfield, Dec 23 2022
a(9)-a(16) from Daniel Suteu, Dec 29 2022