%I #76 Sep 08 2022 08:46:24
%S 42,140,290,618,2058,6747,131430,531531,2098830,5124615,14356161,
%T 34797196,40265322,67239938,1164192201,1220704045,2191309850,
%U 3486789945,8789700524,17700471298,68719510772,305419896610,2261852491428,4398046548368,8822667321452
%N Composite numbers k such that k = q_1^b_1 * ... * q_r^b_r with r >= 2, where q_1 < q_2 < ... < q_r are the prime factors of k and there exists some positive integer m that satisfies k = q_1^m + q_2 + q_3 + ... + q_r.
%C For each term k, indeed, the number of prime factors r must be an odd number > 1.
%C Also, m = log_(q_1)(k - (q_2 + ... + q_r)) must be an integer.
%C Without the conditions "composite numbers" and "r >= 2", all primepowers p^t with t >= 1, A246655, would be in the sequence. - _Marius A. Burtea_ and _Michel Marcus_, Jan 09 2020
%C Is there a further term > a(25) with omega(k) not 3 or 5, so with omega(k) = 7, 9, 11 ... ? - _Michel Marcus_, Jan 10 2020
%C 10^13 < a(26) <= 17592196382708. - _Giovanni Resta_, Jan 10 2020
%C One example of a term k with omega(k)=7 is provided by 37778932653963899150610 = 2*3^3*5*11*13*1237*791006737439773. - _Giovanni Resta_, Jan 11 2020
%C While the first 25 terms have q_1<=5, other values of q1 are possible. For example, 7*53*107*151*364318444053146400583044515149 and 11*1877*18393385333 are terms. Other terms with q_1=11 are 45949836663342271, 3740488174520014333270574609829106805891, and 6626407607736641103900310601529495873176214551. - _Giovanni Resta_, Jan 12 2020
%D J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 263 pp. 42-187, Ellipses, Paris 2004.
%e 2058 = 2 * 3 * 7^3 = 2^11 + 3 + 7.
%e 531531 = 3^2 * 7 * 11 * 13 * 59 = 3^12 + 7 + 11 + 13 + 59.
%t seqQ[n_] := Module[{f = FactorInteger[n]}, p = f[[1, 1]]; Length[f[[;; , 1]]] > 1 && IntegerQ[Log[p, n - Total[f[[;; , 1]]] + p]]]; Select[Range[10^4], seqQ] (* _Amiram Eldar_, Jan 04 2020 *)
%o (Magma) [k:k in [2..1000000]|#v ge 3 and Log(v[1], k-&+v+v[1]) eq Floor(Log(v[1], k-&+v+v[1])) where v is PrimeDivisors(k)]; // _Marius A. Burtea_, Jan 04 2020
%o (PARI) isok(k) = {my(f = factor(k), p); !isprimepower(k) && isprimepower(k-sum(i=2, #f~, f[i,1]), &p) && (p==f[1,1]);} \\ _Michel Marcus_, Jan 09 2020
%Y Subsequence of A024619.
%K nonn,more
%O 1,1
%A _Bernard Schott_, Jan 04 2020
%E a(9) and a(15)-a(17) from _Amiram Eldar_, Jan 04 2020
%E a(18) from _Michel Marcus_, Jan 10 2020
%E a(19)-a(25) from _Giovanni Resta_, Jan 10 2020