A359235 - OEIS

%I #18 Jan 01 2023 02:56:46

%S 1,5,25,925,1625,47125,2115625,4330625,83760625,1049140625,6098828125,

%T 224991015625,3735483578125,329495166015625,8193863401953125,

%U 7604781494140625,216431299462890625,148146624615478515625,25926420587158203125,11071085186929931640625

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

%C a(14) <= 33811910869140625, a(15) <= 7604781494140625, a(16) <= 216431299462890625. - _Robert Israel_, Dec 22 2022

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredSquareNumber.html">Centered Square Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>

%e 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 cs:= n -> 2*n*(n+1)+1:

%p V:= Vector(12): count:= 0:

%p for n from 1 while count < 12 do

%p   v:= cs(n);

%p w:= numtheory:-bigomega(v);

%p if V[w] = 0 then V[w]:= v; count:= count+1 fi

%p od:

%p convert(V,list); # _Robert Israel_, Dec 22 2022

%o (PARI)

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

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Dec 22 2022

%E a(11)-a(13) from _Robert Israel_, Dec 22 2022

%E a(14)-a(19) from _Daniel Suteu_, Dec 29 2022