A082605 -id:A082605 - cp's OEIS frontend

A082606 n-th powers arising in A082605.

Original entry on oeis.org

3, 9, 125, 810000, 12181395632886429300000

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 28 2003

nonn

The sequence is infinite.

Cf. A083203, A083204, A083205.

a(n) = A083205(n)*A083205(n+1)+1. - David Wasserman, Sep 21 2004

More terms from David Wasserman, Sep 21 2004

A107448 Irregular triangle T(n, k) = b(n) + k^2 + k + 1, where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1, read by rows.

Original entry on oeis.org

5, 7, 11, 17, 13, 17, 23, 31, 41, 53, 67, 83, 101, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033

Offset: 1

Roger L. Bagula, May 26 2005

Former title: Triangular form sequence made from a version of A082605 Euler extension.

			The irregular triangle begins as:
   5;
   7, 11, 17;
  13, 17, 23, 31, 41, 53, 67, 83, 101;
  19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257;

Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Page 155

G. C. Greubel, Rows n = 1..10 of the irregular triangle, flattened

Cf. A056486, A082605.

b:= func< n | n eq 1 select 2 else 2^(n-3)*(9-(-1)^n) >;
A107448:= func< n,k | b(n) +k^2 +k +1 >;
[A107448(n,k): k in [1..b(n)-1], n in [1..8]]; // G. C. Greubel, Mar 23 2024

(* First program *)
a[1] = 3; a[2] = 5; a[3] = 11; a[n_]:= a[n]= Abs[1-4*a[n-2]] -2;
euler= Table[a[n], {n,10}];
Table[k^2 + k + euler[[n]], {n,7}, {k,euler[[i]] -2}]//Flatten
(* Second program *)
b[n_]:= 2^(n-3)*(9-(-1)^n) - Boole[n==1]/2;
T[n_, k_]:= b[n] +k^2+k+1;
Table[T[n,k], {n,8}, {k,b[n]-1}]//Flatten (* G. C. Greubel, Mar 23 2024 *)

def b(n): return 2^(n-3)*(9-(-1)^n) - int(n==1)/2
def A107448(n,k): return b(n) + k^2+k+1;
flatten([[A107448(n,k) for k in range(1,b(n))] for n in range(1,8)]) # G. C. Greubel, Mar 23 2024

T(n, k) = b(n) + k^2 + k + 1, where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1. - G. C. Greubel, Mar 23 2024

Edited by G. C. Greubel, Mar 23 2024

A107449 Irregular triangle T(n, k) = 10 - ( (b(n) + k^2 + k + 1) mod 10 ), where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1, read by rows.

Original entry on oeis.org

5, 3, 9, 3, 7, 3, 7, 9, 9, 7, 3, 7, 9, 1, 7, 1, 3, 3, 1, 7, 1, 3, 3, 1, 7, 1, 3, 3, 7, 3, 7, 9, 9, 7, 3, 7, 9, 9, 7, 3, 7, 9, 9, 7, 3, 7, 9, 9, 7, 3, 7, 9, 9, 7, 3, 7, 9, 9, 7, 3, 7, 9, 9, 7, 3, 7, 9, 3, 9, 3, 5, 5, 3, 9, 3, 5, 5, 3, 9, 3, 5, 5, 3, 9, 3, 5, 5, 3, 9, 3, 5, 5, 3, 9, 3, 5, 5, 3, 9, 3, 5, 5, 3, 9, 3

Offset: 1

Roger L. Bagula, May 26 2005

			The irregular triangle begins as:
  5;
  3, 9, 3;
  7, 3, 7, 9, 9, 7, 3, 7, 9;
  1, 7, 1, 3, 3, 1, 7, 1, 3, 3, 1, 7, 1, 3, 3;

G. C. Greubel, Table of n, a(n) for n = 1..2206

Cf. A056486, A082605.

b:= func< n | n eq 1 select 2 else 2^(n-3)*(9-(-1)^n) >;
A107448:= func< n, k | 10 - ((b(n) +k^2 +k +1) mod 10) >;
[5,3,9,3] cat [A107448(n, k): k in [1..b(n)-1], n in [3..8]]; // G. C. Greubel, Mar 24 2024

b[n_]:= 2^(n-3)*(9-(-1)^n) -Boole[n==1]/2;
T[n_, k_]:= 10  -Mod[k^2+k+1+b[n], 10];
Table[T[n, k], {n,8}, {k,b[n]-1}]//Flatten (* G. C. Greubel, Mar 24 2024 *)

def b(n): return 2^(n-3)*(9-(-1)^n) - int(n==1)/2
def A107449(n, k): return 10 - ((b(n) + k^2+k+1)%10);
flatten([[A107449(n, k) for k in range(1, b(n))] for n in range(1, 8)]) # G. C. Greubel, Mar 24 2024

T(n, k) = 10 - (b(n) + k^2 + k + 1) mod 10, where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1. - G. C. Greubel, Mar 24 2024

Edited by G. C. Greubel, Mar 24 2024

A085613 a(n) = 2^(n-1) + (2 + (-1)^n)^((n-2)/2).

Original entry on oeis.org

2, 3, 5, 11, 17, 41, 65, 155, 257, 593, 1025, 2291, 4097, 8921, 16385, 34955, 65537, 137633, 262145, 543971, 1048577, 2156201, 4194305, 8565755, 16777217, 34085873, 67108865, 135812051, 268435457, 541653881, 1073741825, 2161832555, 4294967297, 8632981313

Offset: 1

Ben de la Rosa & Johan Meyer (MeyerJH.sci(AT)mail.uovs.ac.za), Jul 09 2003

Extends Euler's 6-term sequence.
In Euler's 6-term sequence we have noticed that e(1) = 2 = 2^0 + 1; e(2) = 3 = 2^1 + 3^0; e(3) = 5 = 2^2 + 1; e(4) = 11 = 2^3 + 3^1; e(5) = 17 = 2^4 + 1; e(6) = 41 = 2^5 + 3^2, which of course immediately leads to our formula above. Note: For m>0, we take m^(1/2) to be the unique positive square root of m.
a(2^n + 1), n=0,1,2,... are the Fermat numbers.

Index entries for linear recurrences with constant coefficients, signature (2,4,-8,-3,6).

Cf. A082605.

a(n) = 2^(n-1) + (2 + (-1)^n)^((n-2)/2) \\ Charles R Greathouse IV, Jan 29 2013

More terms from Neven Juric, Apr 10 2008

A117012 Primes of the form n^2+5*n+c (n>=0), where c=3 for even n and c=-3 for odd n.

Original entry on oeis.org

3, 17, 47, 107, 173, 269, 503, 641, 809, 983, 1187, 1637, 2441, 2753, 4157, 4547, 4967, 5393, 5849, 6311, 6803, 7829, 8363, 9497, 11981, 12653, 13331, 14753, 15497, 17027, 22943, 26723, 29753, 31859, 32933, 38609, 39791, 42221, 47297, 49943, 58313

Offset: 1

Roger L. Bagula, Apr 16 2006

Harvey Cohn, Advanced Number Theory,Dover, New York, 1962, page 155.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A014209, A082605.

select(isprime, [seq(n^2 + 5*n + (-1)^n * 3, n=1..1000)]); # Robert Israel, Aug 25 2025

f[n_] := If[Mod[n, 2] == 1, n^2 + 5*n - 3, n^2 + 5*n + 3] b = Flatten[Table[If[PrimeQ[f[n]] == True, f[n], {}], {n, 1, 100}]]

for(n=1, 250, k=n^2+5*n+3-6*(n%2); if(isprime(k), print1(k,", ")))

Edited and extended by N. J. A. Sloane, Apr 17 2006

cp's OEIS Frontend

A082606 n-th powers arising in A082605.

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A107448 Irregular triangle T(n, k) = b(n) + k^2 + k + 1, where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1, read by rows.

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A107449 Irregular triangle T(n, k) = 10 - ( (b(n) + k^2 + k + 1) mod 10 ), where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1, read by rows.

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Programs

Magma

Mathematica

SageMath

Formula

Extensions

A085613 a(n) = 2^(n-1) + (2 + (-1)^n)^((n-2)/2).

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Author

Keywords

Comments

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Crossrefs

Programs

PARI

Extensions

A117012 Primes of the form n^2+5*n+c (n>=0), where c=3 for even n and c=-3 for odd n.

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References

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Programs

Maple

Mathematica

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Extensions