A120320 RF(5): refactorable numbers with smallest prime factor 5.

625, 1500625, 9150625, 17850625, 37515625, 52200625, 73530625, 81450625, 174900625, 442050625, 577200625, 1171350625, 1766100625, 1838265625, 2136750625, 3049800625, 4931550625, 7573350625, 8653650625, 12594450625, 15882300625, 17748900625, 21970650625, 24343800625

Offset: 1

Author

Walter Kehowski, Jun 20 2006

Keywords

• nonn

Comments

Numbers that are odd squares, 5 is their smallest prime factor, and are refactorable.

See A033950 for references. For any prime p, p^(p-1) is the smallest element of RF(p), the refactorable numbers whose smallest prime factor is p. Thus 5^(5-1) = 625 is the first element. Other elements would also be 5^4*17^4 or 5^16*17^4.

All the terms are of the form 5^2 * A084967(k)^2 = 5^4 * A007310(k)^2. - Amiram Eldar, Aug 01 2024

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A033950 and A084967.

Cf. A007310, A036896, A036897.

Programs

Maple

with(numtheory); RF5:=[]: p:=5: for w to 1 do for j from 1 to 12^5 do k:=2*j+1; if k mod 3 <> 0 and k mod p = 0 then n:=k^2; t:=tau(n); if (n mod t = 0) then RF5:=[op(RF5),n]; print(ifactor(n)); fi fi; od od;

PARI

lista(kmax) = {my(m); for(k = 1, kmax, m = 25*(k\2*6-(-1)^k)^2; if(!(m % numdiv(m)), print1(m, ", ")));} \\ Amiram Eldar, Aug 01 2024

Extensions

a(37)-a(40) from Amiram Eldar, Aug 01 2024