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

1,1

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

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;

Prog

(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

Crossrefs

Intersection of A033950 and A084967.

Cf. A007310, A036896, A036897.

Sequence in context: A187456 A223318 A222278 * A013837 A050448 A364073

Adjacent sequences: A120317 A120318 A120319 * A120321 A120322 A120323

Keyword

nonn,changed

Author

Walter Kehowski, Jun 20 2006

Extensions

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

Status

approved