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.

18750, 22500, 24000, 27000, 28800, 30720, 32400, 34560, 36450, 38880, 43740, 201684, 345744, 388962, 526848, 592704, 666792, 903168, 1016064, 1143072, 1285956, 1376256, 1548288, 1741824, 1959552, 2204496, 2480058, 7730448, 8696754, 33732864, 35644128, 37949472

Offset: 1

Author

Felix Huber, Aug 13 2025

Keywords

• nonn

Comments

Nonprimes k such that A001414(k) = A007947(k).

The nonprimes k in this sequence share the property sopfr(k) = rad(k) with the primes.

From Felix Huber and David A. Corneth, Aug 13 2025: (Start)

Terms have at least three distinct prime factors.

Proof: If a term had exactly 0 distinct prime factors then it is 1 but 1 is not a term.

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).

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.

Terms with three and more distinct prime factors do exist completing the proof. (End)

Proper subset of A126706. - Michael De Vlieger, Aug 13 2025

Examples

			18750 = 2*3*5^5 is a term because 2 + 3 + 5 + 5 + 5 + 5 + 5 = 2*3*5 = 30.
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.
18537438215625 = 3*5^5*7^11 is a term because 3 + 5*5 + 11*7 = 3*5*7 = 105.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10344 (terms <= 10^35)

Crossrefs

Subsequence of A018252.

Cf. A000040, A001414, A007947, A085718.

Programs

Maple

A386916:=proc(n)
    option remember;
    local k,i;
    if n=1 then
        18750
    else
        for k from procname(n-1)+1 do
            if not isprime(k) and NumberTheory:-Radical(k)=add(i[1]*i[2],i in ifactors(k)[2]) then
                return k
            fi
        od
    fi;
end proc;
seq(A386916(n),n=1..10);

Mathematica

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 *)

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