A280892 -id:A280892 - cp's OEIS frontend

A349231 Numbers k such that k and k+4 are consecutive squarefree numbers.

47, 97, 123, 341, 349, 422, 474, 547, 602, 723, 773, 1023, 1249, 1273, 1322, 1374, 1419, 1447, 1518, 1663, 1673, 1847, 1861, 1923, 2006, 2022, 2055, 2105, 2149, 2222, 2274, 2347, 2365, 2522, 2526, 2573, 2643, 2823, 2870, 3049, 3122, 3183, 3210, 3247, 3282, 3427

Offset: 1

Amiram Eldar, Nov 11 2021

nonn

The asymptotic density of this sequence is 0.0149788175410999... (Mossinghoff et al., 2021).

			47 is a term since 47 and 47 + 4 = 51 = 3*17 are squarefree, and 47 + 1 = 48 = 2^4*3, 47 + 2 = 49 = 7^2 and 47 + 3 = 50 = 2*5^2 are not.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael J. Mossinghoff, Tomás Oliveira e Silva, and Tim Trudgian, The distribution of k-free numbers, Mathematics of Computation, Vol. 90, No. 328 (2021), pp. 907-929; arXiv preprint, arXiv:1912.04972 [math.NT], 2019-2020.

Cf. A005117, A007674, A020754, A280892, A349230.

Select[Range[3500], Boole[SquareFreeQ /@ (# + Range[0, 4])] == {1, 0, 0, 0, 1} &]

A381731 a(n) is the least number k with squarefree neighbors such that the number of non-unitary divisors of k (A048105) is equal to n, or 0 if no such k exists.

Original entry on oeis.org

2, 4, 12, 16, 32, 36, 112, 256, 72, 0, 180, 144, 216, 16384, 768, 65536, 432, 1600, 3072, 900, 864, 1296, 720, 12544, 1080, 67108864, 2592, 268435456, 1440, 9216, 196608, 5184, 2160, 17179869184, 2880, 36864, 10368, 3600, 6300

Offset: 0

Juri-Stepan Gerasimov, Mar 05 2025

nonn

From Amiram Eldar, Mar 06 2025: (Start)
For odd k a(k) is a square. a(9) = 0 because for a square m we have tau(m) >= 3^omega(m). Since A048105(m) = tau(m) - 2^omega(m) = 9, we have 2^omega(m) + 9 >= 3^omega(m) so omega(m) = 1.
Because m^2-1 is squarefree, m must be even, so with omega(m) = 1, we have m = 2^k and with tau(2^k) = 2^1 + 9 = 11 we get k = 10, m = 1024. But 1025 is not squarefree. Therefore a(9) = 0. (End)

Cf. A048105, A280892, A309181.

a(25), a(27) and more terms from Amiram Eldar, Mar 06 2025

cp's OEIS Frontend

A349231 Numbers k such that k and k+4 are consecutive squarefree numbers.

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