A330939 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.

42, 140, 290, 618, 2058, 6747, 131430, 531531, 2098830, 5124615, 14356161, 34797196, 40265322, 67239938, 1164192201, 1220704045, 2191309850, 3486789945, 8789700524, 17700471298, 68719510772, 305419896610, 2261852491428, 4398046548368, 8822667321452

Offset

1,1

Comments

For each term k, indeed, the number of prime factors r must be an odd number > 1.

Also, m = log_(q_1)(k - (q_2 + ... + q_r)) must be an integer.

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

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

10^13 < a(26) <= 17592196382708. - Giovanni Resta, Jan 10 2020

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

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

References

J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 263 pp. 42-187, Ellipses, Paris 2004.

Links

Table of n, a(n) for n=1..25.

Example

2058 = 2 * 3 * 7^3 = 2^11 + 3 + 7.
531531 = 3^2 * 7 * 11 * 13 * 59 = 3^12 + 7 + 11 + 13 + 59.

Mathematica

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 *)

Prog

(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
(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

Crossrefs

Subsequence of A024619.

Sequence in context: A303860 A002759 A346856 * A044374 A044755 A262424

Adjacent sequences: A330936 A330937 A330938 * A330940 A330941 A330942

Keyword

nonn,more

Author

Bernard Schott, Jan 04 2020

Extensions

a(9) and a(15)-a(17) from Amiram Eldar, Jan 04 2020

a(18) from Michel Marcus, Jan 10 2020

a(19)-a(25) from Giovanni Resta, Jan 10 2020

Status

approved