A379845 Even squarefree numbers x such that the product of prime indices of x is a multiple of the sum of prime indices of x.

2, 30, 154, 190, 390, 442, 506, 658, 714, 874, 1110, 1118, 1254, 1330, 1430, 1786, 1794, 1798, 1958, 2310, 2414, 2442, 2470, 2730, 2958, 3034, 3066, 3266, 3390, 3534, 3710, 3770, 3874, 3914, 4042, 4466, 4526, 4758, 4930, 5106, 5434, 5474, 5642, 6090, 6106

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..45.

Example

The terms together with their prime indices begin:
     2: {1}
    30: {1,2,3}
   154: {1,4,5}
   190: {1,3,8}
   390: {1,2,3,6}
   442: {1,6,7}
   506: {1,5,9}
   658: {1,4,15}
   714: {1,2,4,7}
   874: {1,8,9}
  1110: {1,2,3,12}

Mathematica

Select[Range[2, 1000], EvenQ[#]&&SquareFreeQ[#]&&Divisible[Times@@prix[#], Plus@@prix[#]]&]

Crossrefs

Even squarefree case of A326149.

For nonprime instead of even we have A326158.

Squarefree case of A379319.

Even case of A379844.

Partitions of this type are counted by A380221, see A379733, A379735.

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

- multiple: 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, A112798, A175508, A324850, A324851, A326150, A326151, A326153/A326154, A326156, A326157.

Sequence in context: A285991 A078838 A267851 * A089288 A232602 A154413

Adjacent sequences: A379842 A379843 A379844 * A379846 A379847 A379848

Keyword

nonn,new

Author

Gus Wiseman, Jan 20 2025

Status

approved