A379721 Numbers whose prime indices have sum <= product.

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 102, 103

Offset

1,2

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.

Partitions of this type are counted by A319005.

The complement is A325038.

Links

Table of n, a(n) for n=1..65.

Formula

Number k such that A056239(k) <= A003963(k).

Example

The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}

Mathematica

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

Crossrefs

The case of equality is A301987, inequality A325037.

Nonpositive positions in A325036.

A000040 lists the primes, differences A001223.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798, counted by A001222.

A379681 gives sum plus product of prime indices, firsts A379682.

Counting and ranking multisets by comparing sum and product:

- same: A001055 (strict A045778), ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A326156, A326172, A379733

- greater: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less: A114324, ranks A325037, see A318029

- less or equal: A319005, ranks A379721 (this)

- different: A379736, ranks A379722, see A111133

Cf. A000720, A003963, A046022, A075254, A075255, A178503, A175508, A319000, A325034, A325035, A379720.

Sequence in context: A355330 A338923 A360126 * A004274 A004280 A053224

Adjacent sequences: A379717 A379718 A379719 * A379723 A379724 A379725

Keyword

nonn,new

Author

Gus Wiseman, Jan 05 2025

Status

approved