A338093 - OEIS

%I #39 Sep 12 2021 11:19:37

%S 16,27,256,540,756,1200,1890,2940,3060,3125,4050,4200,4320,5460,6000,

%T 6048,7920,8232,10080,10164,10368,10530,11232,11286,12960,13104,13524,

%U 13800,14000,14157,14175,15708,15960,17280,18200,18480,19278,19683,19992,20295,23814

%N Composite numbers which are multiples of the sum of the squares of their prime factors (taken with multiplicity).

%C If a(n)=p1*p2*..*pk where p1,p2,..pk primes, then a(n)=m(p1^2+p2^2+..+pk^2) with m a positive integer.

%C For the special case of m=1, a(n) is equal to the sum of the squares of its prime factors.

%C There are only 5 known numbers to have this property:

%C 16, 27 and three more numbers with 123, 163 and 179 digits found by _Giorgos Kalogeropoulos_ (see Rivera links).

%C It is not known if any smaller numbers than those three exist for the case of m=1.

%C From _Robert Israel_, Oct 16 2020: (Start)

%C Suppose n is in the sequence with n = k*A067666(n).  Then n^m is in the sequence if m divides k^m (in particular for m=k).

%C For any prime p, p^(p^j) is in the sequence if j >= 1 (except j>=2 if p=2). (End)

%H Robert Israel, <a href="/A338093/b338093.txt">Table of n, a(n) for n = 1..2000</a>

%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_625.htm">Puzzle 625. Sum of squares of prime divisors</a>, The Prime Puzzles and Problems Connection.

%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_1019.htm">Puzzle 1019. Follow-up to Puzzle 625</a>, The Prime Puzzles and Problems Connection.

%e 16 = 2*2*2*2 = 1*(2^2 + 2^2 + 2^2 + 2^2).

%e 7920 = 2*2*2*2*3*3*5*11 = 44*(2^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2 + 5^2 + 11^2).

%p filter:= proc(n) local t;

%p   if isprime(n) then return false fi;

%p   n mod add(t[1]^2*t[2],t=ifactors(n)[2]) = 0

%p end proc:

%p select(filter, [$4..30000]); # _Robert Israel_, Oct 16 2020

%t Select[Range@20000,Mod[#,Total[Flatten[Table@@@FactorInteger@#]^2]]==0&]

%o (PARI) isok(m) = if (!isprime(m) && (m>1), my(f=factor(m)); (m % sum(k=1, #f~, f[k,1]^2*f[k,2])) == 0); \\ _Michel Marcus_, Oct 11 2020

%Y Cf. A067666.

%K nonn

%O 1,1

%A _Giorgos Kalogeropoulos_, Oct 09 2020