A353887 -id:A353887 - cp's OEIS frontend

A353886 Nonnegative numbers k such that k^2 + k + 1 is squarefree.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72

Offset: 1

Rémy Sigrist, May 09 2022

Dimitrov proved that this sequence is infinite.

The number of terms not exceeding X is Product_{p prime} (1 - A000086(p)/p^2) * X + O(X^(4/5+eps)) (Dimitrov, 2023). The coefficient of X, which is the asymptotic density of this sequence, equals Product_{primes p == 1 (mod 3)} (1 - 2/p^2) = 0.93484201367... . - Amiram Eldar, Dec 11 2023

			For k = 4, 4^2 + 4 + 1 = 21 = 3 * 7 is squarefree, so 4 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stoyan Ivanov Dimitrov, Square-free values of n^2+n+1, Georgian Mathematical Journal, Vol. 30, No. 3 (2023), pp. 333-348; arXiv preprint, arXiv:2205.02488 [math.NT], 2022-2023.

Cf. A000086, A002061, A005117, A353887 (corresponding squarefree numbers).

Select[Range[0, 72], SquareFreeQ[#^2 + # + 1] &] (* Amiram Eldar, Dec 11 2023 *)

PARI
```
is(k) = issquarefree(k^2 + k + 1);
```

A368083 Numbers k such that k^2 + k + 1 and k^2 + k + 2 are both squarefree numbers.

Original entry on oeis.org

0, 3, 4, 7, 8, 11, 12, 16, 19, 20, 23, 24, 27, 28, 31, 35, 36, 39, 40, 43, 44, 47, 48, 51, 52, 55, 56, 59, 60, 63, 64, 71, 72, 75, 76, 80, 83, 84, 87, 88, 91, 92, 95, 96, 99, 100, 103, 104, 107, 111, 112, 115, 119, 120, 123, 124, 127, 131, 132, 135, 139, 140, 143

Offset: 1

Amiram Eldar, Dec 11 2023

Dimitrov (2023) proved that this sequence is infinite and gave the formula for its asymptotic density.

			0 is a term since 0^2 + 0 + 1 = 1 and 0^2 + 0 + 2 = 2 are both squarefree numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stoyan Dimitrov, Square-free pairs n^2 + n + 1, n^2 + n + 2, HAL preprint, hal-03735444, 2023; ResearchGate link.

Subsequence of A353886.

Cf. A002061, A005117, A007674, A014206, A335962, A353887, A368084.

Select[Range[0, 150], And @@ SquareFreeQ /@ (#^2 + # + {1, 2}) &]

is(k) = {my(m = k^2 + k + 1); issquarefree(m) && issquarefree(m + 1);}

A368084 Squarefree numbers of the form k^2 + k + 1 such that k^2 + k + 2 is also squarefree.

Original entry on oeis.org

1, 13, 21, 57, 73, 133, 157, 273, 381, 421, 553, 601, 757, 813, 993, 1261, 1333, 1561, 1641, 1893, 1981, 2257, 2353, 2653, 2757, 3081, 3193, 3541, 3661, 4033, 4161, 5113, 5257, 5701, 5853, 6481, 6973, 7141, 7657, 7833, 8373, 8557, 9121, 9313, 9901, 10101, 10713, 10921

Offset: 1

Amiram Eldar, Dec 11 2023

Dimitrov (2023) proved that this sequence is infinite.

			1 is a term since 1 is squarefree, 1 = 0^2 + 0 + 1, and 0^2 + 0 + 2 = 2 is also squarefree.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stoyan Dimitrov, Square-free pairs n^2 + n + 1, n^2 + n + 2, HAL preprint, hal-03735444, 2023; ResearchGate link.

Intersection of A007674 and A353887.

Cf. A002061, A005117, A368083.

Select[Table[n^2 + n + 1, {n, 0, 100}], And @@ SquareFreeQ /@ {#, #+1} &]

lista(kmax) = {my(m); for(k = 0, kmax, m = k^2 + k + 1; if(issquarefree(m) && issquarefree(m + 1), print1(m, ", ")));}

a(n) = A002061(A368083(n) + 1).

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A353886 Nonnegative numbers k such that k^2 + k + 1 is squarefree.

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A368083 Numbers k such that k^2 + k + 1 and k^2 + k + 2 are both squarefree numbers.

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A368084 Squarefree numbers of the form k^2 + k + 1 such that k^2 + k + 2 is also squarefree.

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