Jack Braxton - cp's OEIS frontend

A362376 a(n) is the least k such that Fibonacci(n)*Fibonacci(k) + 1 is a prime, and -1 if no such k exists.

1, 1, 1, 3, 3, 3, 9, 3, 4, 9, 3, 4, 3, 27, 4, 24, 24, 4, 3, 6, 3, 3, 444, 3, 12, 9, 3, 63, 6, 8, 36, 6, 36, 12, 12, 4, 21, 60, 4, 3, 24, 73, 51, 3, 11, 51, 12, 4, 504, 12, 3, 33, 21, 6, 9, 6, 4, 384, 21, 7, 54, 3, 4, 51, 24, 63, 30, 24, 11, 45, 72, 6, 39, 9, 22, 42, 12, 16, 60, 30

Offset: 1

Jack Braxton, Apr 17 2023

nonn

The frequencies seem interesting. In the early terms, 5 appears notably rarely, i.e., not until at a(240), whereas several other numbers appear notably frequently, e.g., 24 appears 13 times before a(240). - Peter Munn, May 03 2023

			For n=4, Fibonacci(4)=3 and 3*Fibonacci(k)+1 is not prime until k reaches 3, so a(4)=3.

Chai Wah Wu, Table of n, a(n) for n = 1..448

Cf. A000040, A000045, A034693, A363533.

Table[m = Fibonacci[n]; k = 1; While[! PrimeQ[m*Fibonacci[k] + 1], k++]; k, {n, 120}] (* Michael De Vlieger, May 03 2023 *)

a(n) = my(F=fibonacci(n), k=1); while (!ispseudoprime(F*fibonacci(k) + 1), k++); k; \\ Michel Marcus, Apr 18 2023

from itertools import count
from sympy import fibonacci, isprime
def A362376(n):
    a = b = fibonacci(n)
    for k in count(1):
        if isprime(a+1):
            return k
        a, b = b, a+b # Chai Wah Wu, May 03 2023

a(n) = A363533(A000045(n)). - Pontus von Brömssen, Jun 20 2023

More terms from Michel Marcus, Apr 18 2023

A362035 Numbers that occur three or more times in A362034.

Original entry on oeis.org

2, 17, 29, 43, 59, 61, 83, 103, 107, 137, 167, 173, 211, 251, 257, 307, 359, 379, 419, 479, 499, 541, 613, 631, 673, 691, 761, 769, 853, 887, 937, 991, 1031, 1129, 1151, 1231, 1361, 1399, 1459, 1471, 1493, 1583, 1697, 1823, 1831, 1973, 2111, 2179, 2243, 2251, 2393

Offset: 1

Jack Braxton, Apr 05 2023

nonn

All terms are prime because all terms in A362034 are.

			2 is a term because in A362034, 2 occurs at positions 1, 2, and 3.
17 is a term because in A362034, 17 occurs at positions 13, 23, and 27.

Cf. A000040, A362034.

nn = 300; q = Prime[nn]; c[] = 0; p[] = False; T[n_, 0] := T[n, n] = 2; T[n_, k_] := T[n, k] = NextPrime[T[n - 1, k - 1] + T[n - 1, k] - 1]; TakeWhile[Sort@ Reap[Do[(c[#]++; If[And[c[#] >= 3, ! p[#]], Sow[#]; p[#] = True]) &@ T[n, k], {n, 0, nn}, {k, 0, n}] ][[-1, -1]], # <= q &] (* Michael De Vlieger, Apr 06 2023 *)

A362037 Sums of the rows in A362034.

Original entry on oeis.org

2, 4, 9, 18, 43, 88, 183, 370, 761, 1536, 3081, 6202, 12427, 24908, 49875, 99834, 199769, 399572, 799289, 1598726, 3197557, 6395400, 12791045, 25582310, 51164767, 102329800, 204659929, 409320270, 818640731, 1637281876, 3274564199, 6549128694, 13098257783, 26196515972

Offset: 1

Jack Braxton, Apr 05 2023

nonn

			a(3) = 9, the 3rd row in A362034 is 2 5 2; and 2 + 5 + 2 = 9.
a(4) = 18, the 4th row in A362034 is 2, 7, 7, 2; and 2+7+7+2 = 18.

Cf. A362034.

T[n_, 0] := T[n, n] = 2; T[n_, k_] := T[n, k] = NextPrime[T[n - 1, k - 1] + T[n - 1, k] - 1]; Total /@ Table[T[n, k], {n, 0, 33}, {k, 0, n}] (* Michael De Vlieger, Apr 06 2023 *)

A362036 The prime indices of A362034.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 5, 7, 5, 1, 1, 6, 10, 10, 6, 1, 1, 7, 14, 17, 14, 7, 1, 1, 8, 18, 27, 27, 18, 8, 1, 1, 9, 23, 39, 47, 39, 23, 9, 1, 1, 10, 28, 54, 75, 75, 54, 28, 10, 1, 1, 11, 33, 72, 115, 135, 115, 72, 33, 11, 1, 1, 12, 40, 95, 167, 222, 222, 167, 95, 40, 12, 1

Offset: 0

Jack Braxton, Apr 05 2023

			Triangle begins:
      k=0  1  2  3  4
  n=0:  1;
  n=1:  1, 1;
  n=2:  1, 3, 1;
  n=3:  1, 4, 4, 1;
  n=4:  1, 5, 7, 5, 1;
  n=5:  ...

Cf. A000720, A362034.

T[n_, 0] := T[n, n] = 2; T[n_, k_] := T[n, k] = NextPrime[T[n - 1, k - 1] + T[n - 1, k] - 1]; Table[PrimePi@ T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2023 *)

t(n,k) = if (n==0, 2, if (k==0, 2, if (k==n, 2, nextprime(t(n-1,k-1) + t(n-1,k))))); \\ A362034
T(n,k) = primepi(t(n,k)); \\ Michel Marcus, Apr 07 2023

T(n,k) = A000720(A362034(n,k)).

A362034 Triangle read by rows: T(n,0) = T(n,n) = 2, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).

Original entry on oeis.org

2, 2, 2, 2, 5, 2, 2, 7, 7, 2, 2, 11, 17, 11, 2, 2, 13, 29, 29, 13, 2, 2, 17, 43, 59, 43, 17, 2, 2, 19, 61, 103, 103, 61, 19, 2, 2, 23, 83, 167, 211, 167, 83, 23, 2, 2, 29, 107, 251, 379, 379, 251, 107, 29, 2, 2, 31, 137, 359, 631, 761, 631, 359, 137, 31, 2

Offset: 0

Jack Braxton, Apr 05 2023

In order to get the next number in the row, you add the two numbers above it, and find the next prime.
3 is the only prime number that never shows up.
5 is the only prime number that only shows up once; every prime number above 5 shows up at least twice.

			Triangle begins:
      k=0  1   2   3   4   5   6   7   8  9 10
  n=0:  2
  n=1:  2  2
  n=2:  2  5   2
  n=3:  2  7   7   2
  n=4:  2 11  17  11   2
  n=5:  2 13  29  29  13   2
  n=6:  2 17  43  59  43  17   2
  n=7:  2 19  61 103 103  61  19   2
  n=8:  2 23  83 167 211 167  83  23   2
  n=9:  2 29 107 251 379 379 251 107  29  2
 n=10:  2 31 137 359 631 761 631 359 137 31  2

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0..150, flattened)

Cf. A000040, A007318, A132403, A199333.

for n from 0 to 10 do
  T[n,0]:= 2: T[n,n]:= 2:
  for k from 1 to n-1 do
    T[n,k]:= nextprime(T[n-1,k-1]+T[n-1,k]-1)
  od
od:
for n from 0 to 10 do
  seq(T[n,k],k=0..n)
od; # Robert Israel, Apr 05 2023

T[n_, 0] := T[n, n] = 2; T[n_, k_] := T[n, k] = NextPrime[T[n - 1, k - 1] + T[n - 1, k] - 1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2023, after Maple *)

T(n,k) = if (n==0, 2, if (k==0, 2, if (k==n, 2, nextprime(T(n-1,k-1) + T(n-1,k))))); \\ Michel Marcus, Apr 07 2023

T(n,k) = A007918(T(n-1,k-1) + T(n-1,k)) for 0 < k < n. - Robert Israel, Apr 05 2023

A361977 a(n) is the largest prime p such that 2^p - 1 <= 10^n.

Original entry on oeis.org

3, 5, 7, 13, 13, 19, 23, 23, 29, 31, 31, 37, 43, 43, 47, 53, 53, 59, 61, 61, 67, 73, 73, 79, 83, 83, 89, 89, 89, 97, 101, 103, 109, 109, 113, 113, 113, 113, 127, 131, 131, 139, 139, 139, 149, 151, 151, 157, 157, 163, 167, 167, 173, 179, 181, 181, 181, 191, 193

Offset: 1

Jack Braxton, Apr 02 2023

			a(5) = 13, 2^13 - 1 = 8192 < 10^5.

Cf. A000040, A000043, A000668, A011557, A067497, A151799.

from sympy import prevprime
def A361977(n): return prevprime((10**n-1).bit_length()) # Chai Wah Wu, Apr 02 2023

a(n) = A151799(A067497(n)). - Alois P. Heinz, Apr 02 2023

A361436 Primes of the form k! - Sum_{i=1..k-1} (-1)^(k-i)*i!.

Original entry on oeis.org

3, 7, 29, 139, 821, 5659, 44741, 515616581, 1389068025019, 2390389721955353653838200398484730341485707553165512827613149996957838364422981

Offset: 1

Jack Braxton, Mar 11 2023

Primes of the form k! + A005165(k - 1).

			139 is in the sequence because it is 5! + (4! - 3! + 2! - 1!).

Cf. A005165 (alternating factorials), A071828, A361437 (the k's).

\\ here b(n) is n! + A005165(n-1).
b(n) = {n! - sum(i=1, n-1, (-1)^(n-i)*i!)}
{ for(k=1, 150, if(ispseudoprime(b(k)), print1(b(k), ", "))) } \\ Andrew Howroyd, Mar 12 2023

A361509 a(n) = smallest Fibonacci number F(k) such that F(k) + F(n) is a prime, or -1 if no such F(k) exists.

Original entry on oeis.org

2, 1, 1, 0, 0, 0, 3, 0, 2, 3, 34, 0, 5, 0, 2, 3, 34, 0, 987, 46368, 2584, 3, 2, 0, 13, 144

Offset: 0

Jack Braxton, Mar 26 2023

a(26) is currently unknown.

a(26) > 10^7000 if it is not -1. - Robert Israel, Apr 03 2023

Cf. A000045, A361510, A361902.

with(combinat):
a:=[]; b:=[]; for n from 0 to 25 do
k:=0; t1:=fibonacci(n);
while not isprime( fibonacci(k)+t1) do k:=k+1; od:
a:=[op(a),fibonacci(k)]; b:=[op(b),k];
od:
a; # A361509
b; # A361510

a[n_] := Module[{fn = Fibonacci[n], k = 0}, While[! PrimeQ[fn + Fibonacci[k]], k++]; Fibonacci[k]]; Array[a, 26, 0] (* Amiram Eldar, Mar 30 2023 *)

a(n) = my(k=0, fn=fibonacci(n)); while (!isprime(fn+fibonacci(k)), k++); fibonacci(k); \\ Michel Marcus, Mar 30 2023

a(n) = A000045(A361902(A000045(n))) (unless A361902(A000045(n)) = -1). - Pontus von Brömssen, Mar 30 2023

Edited by N. J. A. Sloane, Mar 30 2023

A361510 a(n) = smallest k >= 0 such that Fibonacci(k) + Fibonacci(n) is a prime, or -1 if no such k exists.

Original entry on oeis.org

3, 1, 1, 0, 0, 0, 4, 0, 3, 4, 9, 0, 5, 0, 3, 4, 9, 0, 16, 24, 18, 4, 3, 0, 7, 12

Offset: 0

Jack Braxton, Mar 26 2023

a(26) is currently unknown.

Cf. A000045, A361509, A361902.

Maple
```
See A361509.
```

a[n_] := Module[{fn = Fibonacci[n], k = 0}, While[! PrimeQ[fn + Fibonacci[k]], k++]; k]; Array[a, 26, 0] (* Amiram Eldar, Mar 30 2023 *)

a(n) = my(k=0, fn=fibonacci(n)); while (!isprime(fn+fibonacci(k)), k++); k; \\ Michel Marcus, Mar 30 2023

a(n) = A361902(A000045(n)). - Pontus von Brömssen, Mar 30 2023

Edited by N. J. A. Sloane, Mar 30 2023

A361437 Numbers k such that k! - Sum_{i=1..k-1} (-1)^(k-i)*i! is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 12, 15, 58, 59, 102, 111, 118, 164, 291, 589, 685, 1671, 1900, 1945, 4905, 9564

Offset: 1

Jack Braxton, Mar 11 2023

Numbers k such that k! + A005165(k - 1) is prime.

a(23) > 7000. - Hugo Pfoertner, Mar 15 2023

			2 is in the sequence because 2! + 1! = 3.
3 is in the sequence because 3! + (2! - 1!) = 7.
4 is in the sequence because 4! + (3! - 2! + 1!) = 29.
5 is in the sequence because 5! + (4! - 3! + 2! - 1!) = 139.

Cf. A361436 (the corresponding primes).

Cf. A001272, A005165 (alternating factorials), A071828.

isok(k) = isprime(k! + sum(i=1, k-1, (-1)^(i+1)*(k-i)!)); \\ Michel Marcus, Mar 12 2023

Missing a(10) inserted and a(12)-a(18) from Andrew Howroyd, Mar 12 2023
a(19)-a(22) from Hugo Pfoertner, Mar 13 2023
a(23) from Michael S. Branicky, Oct 02 2024

cp's OEIS Frontend

User: Jack Braxton

A362376 a(n) is the least k such that Fibonacci(n)*Fibonacci(k) + 1 is a prime, and -1 if no such k exists.

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A362035 Numbers that occur three or more times in A362034.

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A362037 Sums of the rows in A362034.

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A362036 The prime indices of A362034.

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A362034 Triangle read by rows: T(n,0) = T(n,n) = 2, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).

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A361977 a(n) is the largest prime p such that 2^p - 1 <= 10^n.

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A361436 Primes of the form k! - Sum_{i=1..k-1} (-1)^(k-i)*i!.

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A361509 a(n) = smallest Fibonacci number F(k) such that F(k) + F(n) is a prime, or -1 if no such F(k) exists.

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