A374277 Numbers k divisible by exactly one of the prime factors of 30.

2, 3, 4, 5, 8, 9, 14, 16, 21, 22, 25, 26, 27, 28, 32, 33, 34, 35, 38, 39, 44, 46, 51, 52, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 74, 76, 81, 82, 85, 86, 87, 88, 92, 93, 94, 95, 98, 99, 104, 106, 111, 112, 115, 116, 117, 118, 122, 123, 124, 125, 128, 129, 134

Offset: 1

Author

Michael De Vlieger, Jul 26 2024

Keywords

• nonn
• easy

Comments

Numbers k congruent to r (mod 30), where r is in {2, 3, 4, 5, 8, 9, 14, 16, 21, 22, 25, 26, 27, 28}, residues r = p^m mod 30 and r = (30 - p^m) mod 30.

The asymptotic density of this sequence is 7/15. - Amiram Eldar, Jul 26 2024

Examples

			8 is in this sequence since it is even and a multiple of neither 3 nor 5.
10 is not in this sequence since 10 = 2*5; both 2 and 5 divide 30.
14 is in this sequence since it is even and a multiple of neither 3 nor 5, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A047228, A051037, A306044.

Programs

Mathematica

s = Prime@ Range[3]; k = Times @@ s; r = Union[#, k - #] &@ Flatten@ Map[PowerRange[#, k, #] &, s]; m = Length[r]; Array[k*#1 + r[[1 + #2]] & @@ QuotientRemainder[# - 1, m] &, 60]

PARI

is(k) = isprime(gcd(k, 30)); \\ Amiram Eldar, Jul 26 2024

Formula

Intersection of this sequence and 5-smooth numbers (A051037) is A306044 \ {1}.