A059769 -id:A059769 - cp's OEIS frontend

A069758 Frobenius number of the numerical semigroup generated by three consecutive hexagonal numbers.

65, 377, 395, 797, 1589, 6029, 3347, 4571, 6035, 10997, 10979, 12212, 19409, 47246, 24023, 29003, 35357, 52112, 50603, 50411, 73049, 158207, 78155, 90203, 102005, 144443, 138467, 131474, 183077

Offset: 2

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 08 2002

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since three consecutive hexagonal numbers are relatively prime, they generate a numerical semigroup with a Frobenius number.

			a(2)=65 because 65 is not a nonnegative linear combination of 6, 15 and 28, but all integers greater than 65 are.

R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).

Cf. A000384, A037165, A059769, A069755-A069764.

FrobeniusNumber/@Partition[Table[n(2n-1),{n,2,35}],3,1] (* Harvey P. Dale, Jul 25 2011 *)

A069760 Frobenius number of the numerical semigroup generated by consecutive centered square numbers.

Original entry on oeis.org

47, 287, 959, 2399, 5039, 9407, 16127, 25919, 39599, 58079, 82367, 113567, 152879, 201599, 261119, 332927, 418607, 519839, 638399, 776159, 935087, 1117247, 1324799, 1559999, 1825199, 2122847

Offset: 1

Victoria A Sapko (vsapko(AT)canes.gsw.edu), Apr 09 2002

The Frobenius number of a numerical semigroup generated by relatively prime integers a_1,...,a_n is the largest positive integer that is not a nonnegative linear combination of a_1,...,a_n. Since consecutive centered squares are relatively prime, they generate a numerical semigroup with a Frobenius number. The Frobenius number of a 2-generator semigroup is ab-a-b.

			a(1)=47 because 47 is not a nonnegative linear combination of 5 and 13, but all integers greater than 47 are.

R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup Forum, 35 (1987), 63-83 (for definition of Frobenius number).
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001844, A037165, A059769, A069755-A069764.

Table[4n^4+16n^3+20n^2+8n-1,{n,30}] (* or *) LinearRecurrence[ {5,-10,10,-5,1},{47,287,959,2399,5039},30] (* Harvey P. Dale, Apr 25 2011 *)

a(n) = 4*n^4+16*n^3+20*n^2+8*n-1.
a(n) = 5*a(n-1)-10*a(n-2) +10*a(n-3)-5*a(n-4)+a(n-5). - Harvey P. Dale, Apr 25 2011
G.f.: x*(47+52*x-6*x^2+4*x^3-x^4)/(1-x)^5. - Colin Barker, Feb 14 2012

A305415 Numbers k such that F(k)*F(k+1) - F(k+2) is prime, where F = A000045.

Original entry on oeis.org

4, 6, 7, 8, 10, 11, 14, 25, 34, 40, 44, 54, 62, 63, 66, 108, 190, 266, 299, 306, 310, 343, 350, 638, 726, 984, 1626, 2223, 2591, 2843, 3291, 3694, 4198, 4473, 4494, 5128, 7934, 10595, 12515, 17433, 17883, 19979, 23887, 28847, 30071, 64168, 79073, 81971

Offset: 1

Vincenzo Librandi, Jun 09 2018

nonn

Primes in A059769: 7, 83, 239, 659, 4751, 12583, 228983, 9107313407, 52623175261103, 16944503546101559, 796030992711071707, 12041560801669230246323, etc.

Cf. A000045, A059769.

[n: n in [1..800] | IsPrime(Fibonacci(n)*Fibonacci(n+1)-Fibonacci(n+2))];

with(combinat,fibonacci): select(n->isprime(fibonacci(n)*fibonacci(n+1)-fibonacci(n+2)),[$1..8000]); # Muniru A Asiru, Jun 12 2018

Select[Range[3000], PrimeQ[(Fibonacci[#] Fibonacci[# + 1] - Fibonacci[# + 2])]&]

isok(k) = isprime(fibonacci(k)*fibonacci(k+1) - fibonacci(k+2)); \\ Michel Marcus, Jun 13 2018

a(38)-a(46) from Giovanni Resta, Jun 15 2018

a(47)-a(48) from Robert Price, Jun 18 2018

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A069758 Frobenius number of the numerical semigroup generated by three consecutive hexagonal numbers.

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A069760 Frobenius number of the numerical semigroup generated by consecutive centered square numbers.

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A305415 Numbers k such that F(k)*F(k+1) - F(k+2) is prime, where F = A000045.

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