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A359234 a(n) is the smallest centered square number with exactly n distinct prime factors.

%I A359234 #14 Feb 16 2025 08:34:04
%S A359234 1,5,85,1105,99905,2339285,294346585,29215971265,4274253515545,
%T A359234 135890190846085,14289540733429585,10285257499051999685,
%U A359234 659442750659021626765,386961420250791449193065,10019680253112694448155885,7190322949201929673798425205,944550762877225960238953138865
%N A359234 a(n) is the smallest centered square number with exactly n distinct prime factors.
%H A359234 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>
%H A359234 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DistinctPrimeFactors.html">Distinct Prime Factors</a>
%e A359234 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.
%o A359234 (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
%o A359234 (PARI)
%o A359234 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)));
%o A359234 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
%Y A359234 Cf. A001221, A001844, A358894, A358928, A359235.
%K A359234 nonn
%O A359234 0,2
%A A359234 _Ilya Gutkovskiy_, Dec 22 2022
%E A359234 a(8) from _Jon E. Schoenfield_, Dec 23 2022
%E A359234 a(9)-a(16) from _Daniel Suteu_, Dec 29 2022