A071312 -id:A071312 - cp's OEIS frontend

A384498 Squarefree numbers whose distinct prime factors can be partitioned into two sets with equal sums.

1, 30, 70, 286, 646, 1798, 2145, 2310, 2730, 3135, 3526, 3570, 4641, 4845, 5005, 5610, 6006, 6279, 6630, 7198, 7410, 7854, 8778, 8855, 8970, 9177, 10366, 10374, 10626, 10695, 11305, 11571, 11730, 13110, 13485, 13566, 13585, 15470, 16095, 16302, 16422, 16530

Offset: 1

Alois P. Heinz, May 31 2025

nonn

			2145 = 3*5*11*13 is a term because it is squarefree and 3+13 = 5+11.
16422 = 2*3*7*17*23 is squarefree and 2+7+17 = 3+23.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Intersection of A005117 and A221054.

Cf. A071141, A071142, A071312.

q:= n-> (l-> {l[.., 2][]} minus {1}={} and (s->
        (m-> m::even and coeff(mul(1+x^j, j=s), x, m/2)>0)
        (add(i, i=s)))({l[.., 1][]}))(ifactors(n)[2]):
select(q, [$1..20000])[];

Join[{1},Select[Range[16600],SquareFreeQ[#]&&MemberQ[Total/@Subsets[First/@FactorInteger[#]],Total[First/@FactorInteger[#]]/2]&]] (* James C. McMahon, Jun 02 2025 *)

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.

Original entry on oeis.org

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

Leroy Quet, Dec 01 2005

nonn

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

			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.

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

Cf. A008475, A034699, A298010.

A071312 is a subsequence.

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 *)

Edited by Ray Chandler, Dec 04 2005

cp's OEIS Frontend

A384498 Squarefree numbers whose distinct prime factors can be partitioned into two sets with equal sums.

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