A366319 Numbers k such that the sum of prime indices of k is not twice the maximum prime index of k, meaning A056239(k) != 2 * A061395(k).

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76

Offset: 1

Author

Gus Wiseman, Oct 10 2023

Keywords

• nonn

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Also Heinz numbers of integer partitions containing n/2, where n is the sum of all parts.

Examples

			The prime indices of 90 are {1,2,2,3}, with sum 8 and twice maximum 6, so 90 is in the sequence.
		

Crossrefs

Partitions of this type are counted by A086543.

For length instead of maximum we have the complement of A340387.

The complement is A344415, counted by A035363.

A001221 counts distinct prime factors, A001222 with multiplicity.

A056239 adds up prime indices, row sums of A112798.

A334201 adds up all prime indices except the greatest.

A344291 lists numbers m with A001222(m) <= A056239(m)/2, counted by A110618.

A344296 lists numbers m with A001222(m) >= A056239(m)/2, counted by A025065.

Cf. A001414, A100959, A238628, A300061, A301987, A316413, A320924, A344414, A344416, A366318.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max[prix[#]]!=Total[prix[#]]/2&]