A112343 Positive integers m such that the largest prime-power divisor of m equals the sum of the other maximal prime-power divisors (> 1) of m.

1, 30, 70, 84, 120, 126, 180, 198, 264, 286, 308, 468, 520, 624, 646, 880, 884, 912, 1008, 1150, 1224, 1350, 1566, 1672, 1748, 1798, 2484, 2576, 2784, 2900, 3135, 3348, 3400, 3526, 3570, 3600, 4104, 4320, 4606, 4752, 5600, 5704, 5920, 6032, 6068, 6279

Offset: 1

Author

Leroy Quet, Dec 01 2005

Keywords

• nonn

Comments

Sequence consists of those positive integers m where, if m = Product_{p prime, p|m} p^k(p), each k(p) = positive integer, then Sum_{p prime, p|m} p^k(p) = twice the largest prime power dividing m. The inclusion of 1 in the sequence is debatable.

There is substantial overlap between the terms here and in A298010, which has a straightforward cause in the two definitions. Initially (looking at the 46 terms currently in the data section) the majority of the terms that are in A298010 but not here are the oblong (a.k.a. pronic) numbers, A002378; and the terms that are here but not in A298010 are in the subsequence A071312, except for the "debatable" 1. The 2nd term not in A071312 or A298010 is 7980. - Peter Munn, Apr 07 2024

Examples

			84 = 2^2 * 3 * 7. Now 7 = 2^2 + 3, so 84 is in the sequence.
120 = 2^3 * 3 * 5. Now 2^3 = 3 + 5, so 120 is in the sequence.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008475, A034699, A298010.

A071312 is a subsequence.

Programs

Mathematica

f[n_] := Block[{pp}, If[n == 1, Return[True]]; pp = Power @@@ FactorInteger[n]; Return[2Max[pp] == Plus @@ pp]; ]; Select[Range[6500], f] (* Ray Chandler, Dec 04 2005 *)

Extensions

Edited by Ray Chandler, Dec 04 2005