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

1,1

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

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).

Formula

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

From Elmo R. Oliveira, Jul 01 2026: (Start)

a(n) = A335774(n) / 7.

a(n) = a(n-1) + a(n-14) - a(n-15).

G.f.: x * (1 + x^2) * (1 + x + x^2) * (2 - x - 2*x^2 + 3*x^3 + 3*x^4 - 5*x^5 + 3*x^6 + 3*x^7 - 2*x^8 - x^9 + 2*x^10) / ((1 - x) * (1 - x^14)). (End)

Example

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.

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]

Prog

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

Crossrefs

Cf. A047228, A051037, A306044, A335774.

Sequence in context: A329563 A163077 A204810 * A240080 A194714 A054168

Adjacent sequences: A374274 A374275 A374276 * A374278 A374279 A374280

Keyword

nonn,easy,changed

Author

Michael De Vlieger, Jul 26 2024

Status

approved