A380542 -id:A380542 - cp's OEIS frontend

A337878 a(n) is the smallest m > 0 such that the n-th prime divides Jacobsthal(m).

3, 4, 6, 5, 12, 8, 9, 22, 28, 10, 36, 20, 7, 46, 52, 29, 60, 33, 70, 18, 78, 41, 22, 48, 100, 102, 53, 36, 28, 14, 65, 68, 69, 148, 30, 52, 81, 166, 172, 89, 180, 190, 96, 196, 198, 105, 74, 113, 76, 58, 238, 24, 25, 16, 262, 268, 270, 92, 35, 47, 292, 51

Offset: 2

A.H.M. Smeets, Sep 27 2020

nonn

All positive Jacobsthal numbers are odd, so the index starts at n = 2.
The set of primitive prime factors of J_k is given by {A000040(j) | a(j) = k}.
By definition, a(n) is the multiplicative order of -2 modulo the n-th prime for n > 2. - Jianing Song, Jun 20 2025

			The 4th prime number is 7, and 7 divides 21 which is Jacobsthal(6), so a(4) = 6. The second prime number, 3, divides Jacobsthal(6) as well, but it divides also the smaller Jacobsthal(3), i.e., a(2) = 3.

Jianing Song, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Primitive Prime Factor

Cf. A000040 (primes), A001045 (Jacobsthal numbers), A001602 (similar for Fibonacci numbers), A105874 (primes having primitive root -2), A129738.
Cf. multiplicative orders of 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. multiplicative orders of -2..-10: this sequence (if first term 1), A380482, A380531, A380532, A380533, A380540, A380541, A380542, A385222.

m = 300; j = LinearRecurrence[{1, 2}, {3, 5}, m]; s = {}; p = 3; While[(ind = Select[Range[m], Divisible[j[[#]], p] &, 1]) != {}, AppendTo[s, ind[[1]] + 2]; p = NextPrime[p]]; s (* Amiram Eldar, Sep 28 2020 *)

J(n) = (2^n - (-1)^n)/3; \\ A001045
a(n) = {my(k=1, p=prime(n)); while (J(k) % p, k++); k;} \\ Michel Marcus, Sep 29 2020

n = 1
while n < 63:
    n, J0, J1, a = n+1, 3, 1, 3
    p = A000040(n)
    J0 = J0%p
    while J0 != 0:
        J0, J1, a = (J0+2*J1)%p, J0, a+1
    print(n,a)

A000040(n) == 1 (mod a(n)) for n > 2.

A380482 a(n) is the multiplicative order of -3 modulo prime(n); a(2) = 0 for completion.

Original entry on oeis.org

1, 0, 4, 3, 10, 6, 16, 9, 22, 28, 15, 9, 8, 21, 46, 52, 58, 5, 11, 70, 12, 39, 82, 88, 48, 100, 17, 106, 54, 112, 63, 130, 136, 69, 148, 25, 39, 81, 166, 172, 178, 90, 190, 16, 196, 99, 105, 111, 226, 114, 232, 238, 120, 250, 256, 262, 268, 15, 138, 280

Offset: 1

Jianing Song, Jun 27 2025

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A105875 (primes having primitive root -3).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), this sequence, A380531, A380532, A380533, A380540, A380541, A380542, A385222.

A380482[n_] := If[n == 2, 0, MultiplicativeOrder[-3, Prime[n]]];
Array[A380482, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-3}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380531 a(n) is the multiplicative order of -4 modulo prime(n); a(1) = 0 for completion.

Original entry on oeis.org

0, 2, 1, 6, 10, 3, 4, 18, 22, 7, 10, 9, 5, 14, 46, 13, 58, 15, 66, 70, 18, 78, 82, 22, 24, 25, 102, 106, 9, 7, 14, 130, 17, 138, 37, 30, 13, 162, 166, 43, 178, 45, 190, 48, 49, 198, 210, 74, 226, 19, 58, 238, 12, 50, 8, 262, 67, 270, 23, 70

Offset: 1

Jianing Song, Jun 27 2025

a(n) divides (p-1)/4 if p = prime(n) == 1 (mod 4), since (-4)^((p-1)/4) == (+-1+-i)^(p-1) == 1 (mod p), where i^2 == -1 (mod p).

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A105876 (primes having primitive root -4).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, this sequence, A380532, A380533, A380540, A380541, A380542, A385222.

A380531[n_] := If[n == 1, 0, MultiplicativeOrder[-4, Prime[n]]];
Array[A380531, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-4}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380532 a(n) is the multiplicative order of -5 modulo prime(n); a(3) = 0 for completion.

Original entry on oeis.org

1, 1, 0, 3, 10, 4, 16, 18, 11, 7, 6, 36, 20, 21, 23, 52, 58, 15, 11, 10, 72, 78, 41, 44, 96, 50, 51, 53, 54, 112, 21, 130, 136, 138, 74, 150, 156, 27, 83, 172, 178, 30, 38, 192, 196, 66, 70, 111, 113, 57, 232, 238, 40, 50, 256, 131, 134, 54, 276, 140

Offset: 1

Jianing Song, Jun 27 2025

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A105877 (primes having primitive root -5).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, this sequence, A380533, A380540, A380541, A380542, A385222.

A380532[n_] := If[n == 3, 0, MultiplicativeOrder[-5, Prime[n]]];
Array[A380532, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-5}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380533 a(n) is the multiplicative order of -6 modulo prime(n); a(1) = a(2) = 0 for completion.

Original entry on oeis.org

0, 0, 2, 1, 5, 12, 16, 18, 22, 7, 3, 4, 40, 6, 46, 13, 29, 60, 66, 70, 36, 39, 41, 88, 12, 5, 51, 53, 108, 112, 63, 65, 136, 46, 74, 75, 156, 54, 166, 86, 89, 60, 38, 96, 7, 99, 210, 111, 113, 228, 232, 34, 20, 125, 256, 262, 67, 135, 276, 56

Offset: 1

Jianing Song, Jun 27 2025

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A105878 (primes having primitive root -6).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, this sequence, A380540, A380541, A380542, A385222.

A380533[n_] := If[n < 3, 0, MultiplicativeOrder[-6, Prime[n]]];
Array[A380533, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-6}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380540 a(n) is the multiplicative order of -7 modulo prime(n); a(4) = 0 for completion.

Original entry on oeis.org

1, 2, 4, 0, 5, 12, 16, 6, 11, 14, 30, 18, 40, 3, 46, 13, 58, 60, 33, 35, 24, 39, 82, 88, 96, 100, 102, 53, 54, 7, 63, 130, 68, 138, 37, 75, 52, 81, 166, 172, 89, 12, 5, 24, 49, 198, 105, 74, 226, 228, 116, 119, 240, 250, 256, 131, 268, 270, 69, 20

Offset: 1

Jianing Song, Jun 27 2025

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A105879 (primes having primitive root -7).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, A380533, this sequence, A380541, A380542, A385222.

A380540[n_] := If[n == 4, 0, MultiplicativeOrder[-7, Prime[n]]];
Array[A380540, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-7}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

A380541 a(n) is the multiplicative order of -8 modulo prime(n); a(1) = 0 for completion.

Original entry on oeis.org

0, 1, 4, 2, 5, 4, 8, 3, 22, 28, 10, 12, 20, 7, 46, 52, 29, 20, 11, 70, 6, 26, 41, 22, 16, 100, 34, 53, 12, 28, 14, 65, 68, 23, 148, 10, 52, 27, 166, 172, 89, 60, 190, 32, 196, 66, 35, 74, 113, 76, 58, 238, 8, 25, 16, 262, 268, 90, 92, 35

Offset: 1

Jianing Song, Jun 27 2025

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A105880 (primes having primitive root -8).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, A380533, A380540, this sequence, A380542, A385222.

A380541[n_] := If[n == 1, 0, MultiplicativeOrder[-8, Prime[n]]];
Array[A380541, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-8}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

a(n) = ord(-2,p)/gcd(ord(-2,p),3) for p != 2, where p = prime(n), and ord(a,m) is the multiplicative order of a modulo m. Note that ord(-2,p) = A337878(n) for n > 2.

A385222 a(n) is the multiplicative order of -10 modulo prime(n); a(1) = a(3) = 0 for completion.

Original entry on oeis.org

0, 2, 0, 3, 1, 3, 16, 9, 11, 28, 30, 6, 10, 42, 23, 26, 29, 60, 66, 70, 8, 26, 82, 44, 96, 4, 17, 106, 108, 112, 21, 65, 8, 23, 148, 150, 39, 162, 83, 86, 89, 180, 190, 192, 49, 198, 15, 111, 226, 228, 232, 14, 15, 25, 256, 131, 268, 10, 138, 28

Offset: 1

Jianing Song, Jun 27 2025

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A007348 (primes having primitive root -10).
Cf. bases 2..10: A014664, A062117, A082654, A211241, A211242, A211243, A211244, A211245, A002371.
Cf. bases -2..-10: A337878 (if first term 1), A380482, A380531, A380532, A380533, A380540, A380541, A380542, this sequence.

A385222[n_] := If[n == 1 || n == 3, 0, MultiplicativeOrder[-10, Prime[n]]];
Array[A385222, 100] (* Paolo Xausa, Jun 29 2025 *)

a(n,{k=-10}) = my(p = prime(n)); if(k%p==0, 0, znorder(Mod(k,p)))

cp's OEIS Frontend

A337878 a(n) is the smallest m > 0 such that the n-th prime divides Jacobsthal(m).

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A380482 a(n) is the multiplicative order of -3 modulo prime(n); a(2) = 0 for completion.

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A380531 a(n) is the multiplicative order of -4 modulo prime(n); a(1) = 0 for completion.

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A380532 a(n) is the multiplicative order of -5 modulo prime(n); a(3) = 0 for completion.

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A380533 a(n) is the multiplicative order of -6 modulo prime(n); a(1) = a(2) = 0 for completion.

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A380540 a(n) is the multiplicative order of -7 modulo prime(n); a(4) = 0 for completion.

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A380541 a(n) is the multiplicative order of -8 modulo prime(n); a(1) = 0 for completion.

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A385222 a(n) is the multiplicative order of -10 modulo prime(n); a(1) = a(3) = 0 for completion.

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