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

1, 5, 25, 925, 1625, 47125, 2115625, 4330625, 83760625, 1049140625, 6098828125, 224991015625, 3735483578125, 329495166015625, 8193863401953125, 7604781494140625, 216431299462890625, 148146624615478515625, 25926420587158203125, 11071085186929931640625

Offset: 0

Author

Ilya Gutkovskiy, Dec 22 2022

Keywords

• nonn

Comments

a(14) <= 33811910869140625, a(15) <= 7604781494140625, a(16) <= 216431299462890625. - Robert Israel, Dec 22 2022

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Centered Square Number
• Eric Weisstein's World of Mathematics, Prime Factor

Crossrefs

Cf. A001222, A001844, A358926, A358929, A359234.

Programs

Maple

cs:= n -> 2*n*(n+1)+1:
V:= Vector(12): count:= 0:
for n from 1 while count < 12 do
  v:= cs(n);
w:= numtheory:-bigomega(v);
if V[w] = 0 then V[w]:= v; count:= count+1 fi
od:
convert(V,list); # Robert Israel, Dec 22 2022

PARI

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)));
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

Extensions

a(11)-a(13) from Robert Israel, Dec 22 2022

a(14)-a(19) from Daniel Suteu, Dec 29 2022