

A111907


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


29



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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



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.
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 nonparts {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(pc2*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


CROSSREFS

For length instead of maximum we have A067801.
These partitions are counted by A239959.
A061395 gives greatest prime index.
Comparing twice the number of distinct parts to greatest part:


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



