A379844 Squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 154, 157, 163, 165, 167, 173, 179, 181, 190, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset

1,1

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. The sum and product of prime indices are A056239 and A003963 respectively.

Links

Table of n, a(n) for n=1..58.

Formula

Satisfies A056239(a(n))|A003963(a(n)).

Mathematica

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[2, 100], SquareFreeQ[#]&&Divisible[Times@@prix[#], Plus@@prix[#]]&]

Crossrefs

Squarefree case of A326149.

For nonprime instead of squarefree we have A326150.

The non-prime case is A326158.

Partitions of this type are counted by A379733, see A379735.

The even case is A379845, counted by A380221.

A003963 multiplies together prime indices.

A005117 lists the squarefree numbers.

A056239 adds up prime indices.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149

- greater than: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less than: A114324, ranks A325037, see A318029, A379720

- less or equal: A319005, ranks A379721, see A025147

- different: A379736, ranks A379722, see A111133

Cf. A000720, A001222, A036844, A112798, A324850, A324851, A324925, A326151, A326153/A326154, A326156, A326157.

Sequence in context: A374595 A365829 A030059 * A201879 A327783 A319333

Adjacent sequences: A379841 A379842 A379843 * A379845 A379846 A379847

Keyword

nonn,new

Author

Gus Wiseman, Jan 19 2025

Status

approved