A111907 Numbers k such that the same number of primes, among primes <= the largest prime dividing k, divide k as do not.

1, 3, 9, 14, 21, 27, 28, 35, 56, 63, 78, 81, 98, 112, 130, 147, 156, 175, 182, 189, 195, 196, 224, 234, 243, 245, 260, 273, 286, 312, 364, 392, 429, 441, 448, 455, 468, 520, 567, 570, 572, 585, 624, 650, 686, 702, 715, 728, 729, 784, 798, 819, 875, 896, 936

Offset

1,2

Comments

Also numbers whose greatest prime index (A061395) is twice their number of distinct prime factors (A001221). - Gus Wiseman, Mar 19 2023

Links

John Tyler Rascoe, Table of n, a(n) for n = 1..1000

Example

28 is included because 7 is the largest prime dividing 28. And of the primes <= 7 (2,3,5,7), 2 and 7 (2 primes) divide 28 and 3 and 5 (also 2 primes) do not divide 28.
From Gus Wiseman, Mar 19 2023: (Start)
The terms together with their prime indices begin:
    1: {}
    3: {2}
    9: {2,2}
   14: {1,4}
   21: {2,4}
   27: {2,2,2}
   28: {1,1,4}
   35: {3,4}
   56: {1,1,1,4}
   63: {2,2,4}
   78: {1,2,6}
   81: {2,2,2,2}
   98: {1,4,4}
  112: {1,1,1,1,4}
  130: {1,3,6}
  147: {2,4,4}
  156: {1,1,2,6}
For example, 156 is included because it has prime indices {1,1,2,6}, with distinct parts {1,2,6} and distinct non-parts {3,4,5}, both of length 3. Alternatively, 156 has greatest prime index 6 and omega 3, and 6 = 2*3.
(End)

Mathematica

Select[Range[100], 2*PrimeNu[#]==PrimePi[FactorInteger[#][[-1, 1]]]&] (* Gus Wiseman, Mar 19 2023 *)

Prog

(PARI) {m=950; v=vector(m); for(n=1, m, f=factor(n)[, 1]~; c=0; pc=0; forprime(p=2, vecmax(f), j=1; s=length(f); while(j<=s&&p!=f[j], j++); if(j<=s, c++); pc++); v[n]=sign(pc-2*c)); for(n=1, m, if(v[n]==0, print1(n, ", ")))} \\ Klaus Brockhaus, Aug 21 2005
(Python)
from itertools import count, islice
from sympy import sieve, factorint
def a_gen():
    yield 1
    for k in count(3):
        f = [sieve.search(i)[0] for i in factorint(k)]
        if 2*len(f) == f[-1]:
            yield k
A111907_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jun 20 2024

Crossrefs

For length instead of maximum we have A067801.

These partitions are counted by A239959.

A001222 (bigomega) counts prime factors, distinct A001221 (omega).

A061395 gives greatest prime index.

A112798 lists prime indices, sum A056239.

Comparing twice the number of distinct parts to greatest part:

less: A360254, ranks A111906

equal: A239959, ranks A111907

greater: A237365, ranks A111905

less or equal: A237363, ranks A361204

greater or equal: A361394, ranks A361395

Cf. A046660, A067340, A324517, A324521, A324562, A361205, A361393.

Sequence in context: A103813 A001968 A305373 * A309149 A294480 A195972

Adjacent sequences: A111904 A111905 A111906 * A111908 A111909 A111910

Keyword

nonn

Author

Leroy Quet, Aug 19 2005

Extensions

More terms from Klaus Brockhaus, Aug 21 2005

Status

approved