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A386916 Nonprimes k such that sopfr(k) = rad(k), where sopfr(k) is sum of the prime factors of k (counting multiplicity), and rad(k) is the product of its distinct prime factors.

%I A386916 #31 Aug 14 2025 15:14:47
%S A386916 18750,22500,24000,27000,28800,30720,32400,34560,36450,38880,43740,
%T A386916 201684,345744,388962,526848,592704,666792,903168,1016064,1143072,
%U A386916 1285956,1376256,1548288,1741824,1959552,2204496,2480058,7730448,8696754,33732864,35644128,37949472
%N A386916 Nonprimes k such that sopfr(k) = rad(k), where sopfr(k) is sum of the prime factors of k (counting multiplicity), and rad(k) is the product of its distinct prime factors.
%C A386916 Nonprimes k such that A001414(k) = A007947(k).
%C A386916 The nonprimes k in this sequence share the property sopfr(k) = rad(k) with the primes.
%C A386916 From _Felix Huber_ and _David A. Corneth_, Aug 13 2025: (Start)
%C A386916 Terms have at least three distinct prime factors.
%C A386916 Proof: If a term had exactly 0 distinct prime factors then it is 1 but 1 is not a term.
%C A386916 If a term had exactly one distinct prime divisor then it is of the form p^m where p is prime. If m = 1 then it is excluded as it is prime. If m > 1 then sopfr(p^m) = m*p > p = rad(p^m) contradicting equality between sopfr(p^m) and rad(p^m).
%C A386916 If a term had exactly two distinct prime factors, p and q, then there would have to be positive integers x and y satisfying p*q = x*p + y*q, or equivalently, p*(q - x) = y*q. Since q divides neither p nor q - x, this is impossible; therefore, no term with exactly two distinct prime factors exists.
%C A386916 Terms with three and more distinct prime factors do exist completing the proof. (End)
%C A386916 Proper subset of A126706. - _Michael De Vlieger_, Aug 13 2025
%H A386916 David A. Corneth, <a href="/A386916/b386916.txt">Table of n, a(n) for n = 1..10344</a> (terms <= 10^35)
%e A386916 18750 = 2*3*5^5 is a term because 2 + 3 + 5 + 5 + 5 + 5 + 5 = 2*3*5 = 30.
%e A386916 1285956 = 2^2*3^8*7^2 is a term because 2 + 2 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 7 + 7 = 2*3*7 = 42.
%e A386916 18537438215625 = 3*5^5*7^11 is a term because 3 + 5*5 + 11*7 = 3*5*7 = 105.
%p A386916 A386916:=proc(n)
%p A386916     option remember;
%p A386916     local k,i;
%p A386916     if n=1 then
%p A386916         18750
%p A386916     else
%p A386916         for k from procname(n-1)+1 do
%p A386916             if not isprime(k) and NumberTheory:-Radical(k)=add(i[1]*i[2],i in ifactors(k)[2]) then
%p A386916                 return k
%p A386916             fi
%p A386916         od
%p A386916     fi;
%p A386916 end proc;
%p A386916 seq(A386916(n),n=1..10);
%t A386916 q[k_] := !PrimeQ[k] && Module[{f = FactorInteger[k]},Plus @@ Times @@@f == Times @@ f[[;;, 1]]]; Select[Range[2, 10^6], q] (* _Amiram Eldar_, Aug 13 2025 *)
%o A386916 (PARI) is(n) = {my(f = factor(n)); if(#f~ < 3, return(0)); prod(i = 1, #f~, f[i, 1]) == sum(i = 1, #f~, f[i,1]*f[i,2])} \\ _David A. Corneth_, Aug 13 2025
%Y A386916 Subsequence of A018252.
%Y A386916 Cf. A000040, A001414, A007947, A085718.
%K A386916 nonn
%O A386916 1,1
%A A386916 _Felix Huber_, Aug 13 2025