A380410 Numbers with greater product of prime indices than sum of prime factors.

1, 45, 49, 63, 75, 77, 81, 91, 99, 105, 117, 119, 121, 125, 126, 133, 135, 143, 147, 150, 153, 161, 162, 165, 169, 171, 175, 182, 187, 189, 195, 198, 203, 207, 209, 210, 217, 221, 225, 231, 234, 238, 242, 243, 245, 247, 250, 253, 255, 259, 261, 266, 270, 273

Offset: 1

Author

Gus Wiseman, Jan 25 2025

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, with product A003963.

Examples

			126 has prime indices {1,2,2,4} and prime factors {2,3,3,7}, and 16 > 15, so 126 is in the sequence.
The terms together with their prime indices begin:
     1: {}
    45: {2,2,3}
    49: {4,4}
    63: {2,2,4}
    75: {2,3,3}
    77: {4,5}
    81: {2,2,2,2}
    91: {4,6}
    99: {2,2,5}
   105: {2,3,4}
   117: {2,2,6}
   119: {4,7}
   121: {5,5}
   125: {3,3,3}
   126: {1,2,2,4}
   133: {4,8}
   135: {2,2,2,3}
		

Crossrefs

For factors instead of indices we have A002808.

The case of prime powers is A244623.

For indices instead of factors we have A325037, see also A325038.

The version for equality is A331384, counted by A331383.

Positions of positive terms in A380344.

Partitions of this type are counted by A380411.

A000040 lists the primes, differences A001223.

A001222 counts prime factors with multiplicity.

A055396 gives least prime index, greatest A061395.

Triangles:

- A027746 = prime factors

- A112798 = prime indices

Statistics:

- A000027 = product of prime factors = row products of A027746

- A001414 = sum of prime factors = row sums of A027746

- A003963 = product of prime indices = row products of A112798

- A056239 = sum of prime indices = row sums of A112798

Combinations:

- A075254 = product of factors + sum of factors = A000027 + A001414

- A075255 = product of factors - sum of factors = A000027 - A001414

- A178503 = product of factors - sum of indices = A000027 - A056239

- A325036 = product of indices - sum of indices = A003963 - A056239

- A379681 = product of indices + sum of indices = A003963 + A056239

- A380344 = product of indices - sum of factors = A003963 - A001414

- A380345 = product of factors + sum of indices = A000027 + A056239

- A380409 = product of indices + sum of factors = A003963 + A001414

Cf. A000720, A175508, A301987, A319000, A325040, A325041, A325044, A379682, A380220.

Programs

Mathematica

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

Formula

A003963(a(n)) > A001414(a(n)).