A231830 -id:A231830 - cp's OEIS frontend

A362252 Primes dividing terms of A231830.

5, 53, 89, 101, 373, 877, 1109, 1181, 1597, 1613, 2029, 2069, 2153, 2213, 2381, 2741, 3617, 4273, 6529, 6737, 7417, 7717, 11321, 12653, 13009, 13309, 16829, 17729, 23581, 23993, 25373, 32569, 33353, 33857, 34841, 35053, 36097, 37201, 38609, 41513, 42461, 48661, 55829, 58369, 59093, 63281

Offset: 1

Max Alekseyev, Apr 21 2023

nonn

Since the terms of A231830 are pairwise coprime, each prime divides at most one term of A231830. Indices of the corresponding terms are listed in A362253, and so a(n) divides A231830(A362253(n)).

Max Alekseyev, Table of n, a(n) for n = 1..8503

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362251, A362253.

A362253 a(n) is the unique index such that prime A362252(n) divides A231830(a(n)).

Original entry on oeis.org

1, 4, 7, 2, 19, 25, 30, 38, 45, 4, 26, 33, 27, 46, 10, 59, 102, 38, 84, 37, 22, 77, 80, 37, 240, 57, 45, 240, 173, 38, 41, 100, 88, 44, 114, 39, 63, 24, 14, 121, 177, 12, 155, 270, 65, 109, 44, 391, 54, 22, 96, 320, 194, 347, 182, 226, 143, 290, 105, 135, 29, 198, 113, 302, 572, 53, 692, 168, 366

Offset: 1

Max Alekseyev, Apr 21 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..8503

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362251, A362252.

A231831 a(0) = 1; for n > 0, a(n) = -1 + 4*Product_{i=0..n-1} a(i)^2.

Original entry on oeis.org

1, 3, 35, 44099, 85762231424099, 630794963141019085083178800095033630804099

Offset: 0

Michel Marcus, Nov 14 2013

nonn

Sequence designed to show that there are an infinity of primes congruent to 3 modulo 4 (A002145). Terms are not necessarily prime. Their smallest prime factor from A002145 are: 3, 7, 11, 23, 4111, 2809343.
Next term is too large to include.
Similarly to Sylvester's sequence (A000058), it is unknown if all terms are squarefree (see also MathOverflow link). - Max Alekseyev, Mar 26 2023
Primes dividing terms of this sequence are listed in A362250. Since terms are pairwise coprime, for each n prime A362250(n) divides exactly one term, whose index is A362251(n). That is, A362250(n) divides a(A362251(n)). - Max Alekseyev, Apr 16 2023

S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.
fredrickmnelson et al., Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, 2023.

Cf. A000058, A002145, A007018, A231830, A362250, A362251.

lista(nn) = {a = vector(nn); a[1] = 3; for (n=2, nn, a[n] = 4*prod(i=1, n-1, a[i]^2) - 1;); a;}

For n > 1, a(n) = (a(n-1) + 1) * a(n-1)^2 - 1. - Max Alekseyev, Mar 26 2023

a(0) = 1 prepended by Max Alekseyev, Mar 26 2023

A362250 Primes dividing terms of A231831.

Original entry on oeis.org

3, 5, 7, 11, 19, 23, 89, 101, 137, 157, 211, 373, 659, 877, 881, 1399, 1597, 1627, 1663, 1811, 2029, 2069, 2087, 2153, 2381, 2677, 2939, 3433, 3491, 3511, 3617, 3673, 4111, 4127, 4547, 4721, 5059, 5483, 6529, 6793, 6827, 7757, 8209, 8297, 8677, 9203, 9463, 9811, 10139, 10159, 11321

Offset: 1

Max Alekseyev, Apr 13 2023

nonn

Since the terms of A231831 are pairwise coprime, each prime divides at most one term of A231831. Indices of the corresponding terms are listed in A362251, and so a(n) divides A231831(A362251(n)).

Max Alekseyev, Table of n, a(n) for n = 1..16944
fredrickmnelson et al., Does a(0)=6, a(n+1)=a(n)^3-a(n), define a square-free sequence?, MathOverflow, 2023.

Cf. A000058, A007996, A180871, A231830, A231831, A362251, A362252, A362253.

A362251 a(n) is the unique index such that prime A362250(n) divides A231831(a(n)).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 9, 4, 4, 8, 3, 31, 12, 7, 7, 9, 44, 8, 22, 29, 36, 37, 8, 21, 5, 26, 4, 20, 24, 12, 76, 30, 5, 47, 5, 13, 9, 25, 6, 41, 51, 9, 53, 6, 27, 39, 5, 12, 78, 64, 10, 185, 113, 205, 91, 85, 43, 195, 32, 117, 20, 133, 142, 119, 64, 70, 199, 41, 125, 79, 243, 70, 35, 105, 67, 156

Offset: 1

Max Alekseyev, Apr 13 2023

nonn

Max Alekseyev, Table of n, a(n) for n = 1..16944

Cf. A000058, A007996, A180871, A231830, A231831, A362250, A362252, A362253.

cp's OEIS Frontend

A362252 Primes dividing terms of A231830.

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A362253 a(n) is the unique index such that prime A362252(n) divides A231830(a(n)).

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A231831 a(0) = 1; for n > 0, a(n) = -1 + 4*Product_{i=0..n-1} a(i)^2.

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A362250 Primes dividing terms of A231831.

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A362251 a(n) is the unique index such that prime A362250(n) divides A231831(a(n)).

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