A371782 Numbers with non-biquanimous prime signature.

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 102

Offset: 1

Author

Gus Wiseman, Apr 09 2024

Keywords

• nonn

Comments

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 (aerated) and ranked by A357976.

Also numbers n without a unitary divisor d|n having exactly half as many prime factors as n, counting multiplicity.

Examples

			The prime signature of 120 is (3,1,1), which is not biquanimous, so 120 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

A number's prime signature is given by A124010.

The complement for prime indices is A357976, counted by A002219 aerated.

For prime indices we have A371731, counted by A371795, even case A006827.

The complement is A371781, counted by A371839.

Partitions of this type are counted by A371840.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A237258 (aerated) counts biquanimous strict partitions, ranks A357854.

A321142 and A371794 count non-biquanimous strict partitions.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A371792 counts non-biquanimous sets, complement A371791.

Cf. A035470, A064914, A232466, A299729, A367094, A371736, A371783, A371789.

Subsequence of A026424.

Programs

Mathematica

g[n_]:=Select[Divisors[n],GCD[#,n/#]==1&&PrimeOmega[#]==PrimeOmega[n/#]&];
Select[Range[100],g[#]=={}&]
(* second program: *)
q[n_] := Module[{e = FactorInteger[n][[;; , 2]], sum, x}, sum = Plus @@ e; OddQ[sum] || CoefficientList[Product[1 + x^i, {i, e}], x][[1 + sum/2]] == 0]; q[1] = False; Select[Range[120], q] (* Amiram Eldar, Jul 24 2024 *)