A036303 Composite numbers whose prime factors contain no digits other than 1 and 3.

9, 27, 33, 39, 81, 93, 99, 117, 121, 143, 169, 243, 279, 297, 339, 341, 351, 363, 393, 403, 429, 507, 729, 837, 891, 933, 939, 961, 993, 1017, 1023, 1053, 1089, 1179, 1209, 1243, 1287, 1331, 1441, 1469, 1521, 1573, 1703, 1859, 2187, 2197, 2511, 2673, 2799

Offset: 1

Author

Patrick De Geest, Dec 15 1998

Keywords

• nonn
• easy
• base

Comments

All terms are a product of at least two terms of A020451. - David A. Corneth, Oct 09 2020

Examples

			The composite 117 = 3^2 * 13 is in the sequence as the digits of the prime factors are either 1 or 3. - _David A. Corneth_, Oct 17 2020
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Alois P. Heinz)
• Index entries for sequences related to prime factors.

Crossrefs

Cf. A003597, A020451, A036302-A036325.

Programs

Mathematica

Select[Range[3000],CompositeQ[#]&&SubsetQ[{1,3},Union[Flatten[IntegerDigits/@FactorInteger[#][[;;,1]]]]]&] (* Harvey P. Dale, Jan 08 2025 *)

Python

from sympy import factorint
def ok(n):
    f = factorint(n)
    return sum(f.values()) > 1 and all(set(str(p)) <= set("13") for p in f)
print(list(filter(ok, range(2800)))) # Michael S. Branicky, Sep 27 2021

Formula

Sum_{n>=1} 1/a(n) = Product_{p in A020451} (p/(p - 1)) - Sum_{p in A020451} 1/p - 1 = 0.3374936085... . - Amiram Eldar, May 18 2022