A362252 -id:A362252 - cp's OEIS frontend

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

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.

A231830 a(0) = 1; for n > 0, a(n) = 1 + 4*Product_{i=1..n-1} a(i)^2.

Original entry on oeis.org

1, 5, 101, 1020101, 1061522231810040101, 1196154511175776540960913502483611007728163340227060101

Offset: 0

Michel Marcus, Nov 14 2013

nonn

Sequence designed to show that there are an infinity of primes congruent to 1 modulo 4 (A002144). Terms are not necessarily prime. Their smallest prime factors from A002144 are: 5, 101, 1020101, 53, 686743037.
Next term is too large to include.
From Max Alekseyev, Apr 21 2023: (Start)
Similarly to Sylvester's sequence (A000058), it is unknown if all terms are squarefree.
Primes dividing terms of this sequence are listed in A362252. Since terms are pairwise coprime, for each n prime A362252(n) divides exactly one term, whose index is A362253(n). That is, A362252(n) divides a(A362253(n)). (End)

S. A. Shirali, A family portrait of primes-a case study in discrimination, Math. Mag. Vol. 70, No. 4 (Oct., 1997), pp. 263-272.

Cf. A000058, A002144, A007018, A231831, A362252, A362253.

lista(nn) = {a = vector(nn); a[1] = 5; 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 25 2023

a(0)=1 prepended by Max Alekseyev, Mar 25 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

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

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A231830 a(0) = 1; for n > 0, a(n) = 1 + 4*Product_{i=1..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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