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A359235 a(n) is the smallest centered square number with exactly n prime factors (counted with multiplicity).

%I A359235 #19 Feb 16 2025 08:34:04
%S A359235 1,5,25,925,1625,47125,2115625,4330625,83760625,1049140625,6098828125,
%T A359235 224991015625,3735483578125,329495166015625,8193863401953125,
%U A359235 7604781494140625,216431299462890625,148146624615478515625,25926420587158203125,11071085186929931640625
%N A359235 a(n) is the smallest centered square number with exactly n prime factors (counted with multiplicity).
%C A359235 a(14) <= 33811910869140625, a(15) <= 7604781494140625, a(16) <= 216431299462890625. - _Robert Israel_, Dec 22 2022
%H A359235 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>
%H A359235 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>
%e A359235 a(4) = 1625, because 1625 is a centered square number with 4 prime factors (counted with multiplicity) {5, 5, 5, 13} and this is the smallest such number.
%p A359235 cs:= n -> 2*n*(n+1)+1:
%p A359235 V:= Vector(12): count:= 0:
%p A359235 for n from 1 while count < 12 do
%p A359235   v:= cs(n);
%p A359235 w:= numtheory:-bigomega(v);
%p A359235 if V[w] = 0 then V[w]:= v; count:= count+1 fi
%p A359235 od:
%p A359235 convert(V,list); # _Robert Israel_, Dec 22 2022
%o A359235 (PARI)
%o A359235 bigomega_centered_square_numbers(A, B, n) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, if(q%4==1, my(t=m*q); if(issquare(2*t-1), listput(list, t)))), forprime(q=p, sqrtnint(B\m, n), if(q%4==1, my(t=m*q); if(ceil(A/t) <= B\t, list=concat(list, f(t, q, n-1)))))); list); vecsort(Vec(f(1, 2, n)));
%o A359235 a(n) = if(n==0, return(1)); my(x=2^n, y=2*x); while(1, my(v=bigomega_centered_square_numbers(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ _Daniel Suteu_, Dec 29 2022
%Y A359235 Cf. A001222, A001844, A358926, A358929, A359234.
%K A359235 nonn
%O A359235 0,2
%A A359235 _Ilya Gutkovskiy_, Dec 22 2022
%E A359235 a(11)-a(13) from _Robert Israel_, Dec 22 2022
%E A359235 a(14)-a(19) from _Daniel Suteu_, Dec 29 2022