A208296 -id:A208296 - cp's OEIS frontend

A024699 a(n) = (prime(n+2)-1)/6 if this is an integer or (prime(n+2)+ 1)/6 otherwise.

1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 16, 17, 17, 18, 18, 19, 21, 22, 23, 23, 25, 25, 26, 27, 28, 29, 30, 30, 32, 32, 33, 33, 35, 37, 38, 38, 39, 40, 40, 42, 43, 44, 45, 45, 46, 47, 47, 49, 51, 52, 52, 53, 55, 56, 58, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70

Offset: 1

Clark Kimberling

nonn

Also number of partitions of n-th prime > 3 into a sum of 2's or 3's (inclusive or).
From Wolfdieter Lang, Mar 13 2012: (Start)
The primes of the form 6*k+1 are given in A002476.
For n >= 1 such that prime(n+2) is from A002476, one has 8*T(prime(n+2)-1) + 1 = r(n)^2, n >= 1, with the triangular numbers T(n) = A000217(n) and r(n) = A208296(n). Therefore, 24*prime(n+2)*a(n) + 1 = r(n)^2. E.g., n=2: prime(4)=7, a(2)=1, 8*21 + 1 = 13^2 = A208296(2)^2 = 24*7*1 + 1.
The primes of the form 6*k-1 are given in A007528.
For n >= 1 such that prime(n+2) is from A007528, one has 8*T(prime(n+2)) + 1 = r(n)^2. For T and r see the preceding comment. Therefore, 24*prime(n+2)*a(n) + 1 = r(n)^2. E.g., n=1, prime(3)=5, a(1)=1, 8*15 + 1 = 11^2 = A208296(1)^2 = 24*5*1 + 1.
(End)

[(NthPrime(n+2)+3) div 6: n in [1..80]]; // Vincenzo Librandi, Sep 06 2016

From R. J. Mathar, May 02 2010: (Start)
A103221 := proc(n) a := 0 ; for t from 0 do if 2*t > n then return a; end if; if n-2*t mod 3 = 0 then a := a+1 ; end if; end do : end proc:
A024699 := proc(n) A103221(ithprime(n+2)) ; end proc: seq(A024699(n),n=1..120) ; (End)

pi6[n_]:=Module[{p=Prime[n+2],c},c=(p-1)/6;If[IntegerQ[c],c,(p+1)/6]]; Array[pi6,80] (* Harvey P. Dale, Aug 19 2013 *)
Table[Floor[(Prime[n + 2] + 3) / 6], {n, 100}] (* Vincenzo Librandi, Sep 06 2016 *)

a(n) = (prime(n+2)+3)\6; \\ Michel Marcus, Sep 06 2016; after Wolfdieter Lang

a(n) = A103221(prime(n+2)). - R. J. Mathar, May 02 2010

a(n) = floor((prime(n+2)+3)/6), n >= 1, prime(n)=A000040(n). Consider the two cases prime(n+2) == 1 (mod 6) and == -1 (mod 6) separately. See the formula above. - Wolfdieter Lang, Mar 15 2012

A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = f(prime(n),prime(k)), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

Original entry on oeis.org

0, 5, 5, 9, 0, 9, 13, 11, 11, 13, 21, 13, 0, 13, 21, 25, 23, 29, 29, 23, 25, 33, 25, 21, 0, 21, 25, 33, 37, 35, 51, 43, 43, 51, 35, 37, 45, 37, 69, 27, 0, 27, 69, 37, 45, 57, 47, 39, 69, 131, 131, 69, 39, 47, 57, 61, 59, 91, 113, 67, 0, 67, 113, 91, 59, 61, 73, 61, 59, 139, 153, 103, 103, 153, 139, 59, 61, 73

Offset: 1

N. J. A. Sloane, Sep 18 2021

If u, v are positive integers with gcd(u,v) = 1, the "reduced inverse" red_inv(u,v) of u mod v is u^(-1) mod v if u^(-1) mod v <= v/2, otherwise it is v - u^(-1) mod v.
That is, we map u to whichever of +-u has a representative mod v in the range 0 to v/2. Stated another way, red_inv(u,v) is a number r in the range 0 to v/2 such that r*u == +-1 mod v.
For example, red_inv(3,11) = 4, since 3^(-1) mod 11 = 4. But red_inv(2,11) = 5 = 11-6, since red_inv(2,11) = 6.
Arises in the study of A344005.

			The array begins:
0, 5, 9, 13, 21, 25, 33, 37, 45, 57, 61, 73,...
5, 0, 11, 13, 23, 25, 35, 37, 47, 59, 61, 73,...
9, 11, 0, 29, 21, 51, 69, 39, 91, 59, 61, 149,...
13, 13, 29, 0, 43, 27, 69, 113, 139, 57, 125, 223,...
21, 23, 21, 43, 0, 131, 67, 153, 45, 175, 309, 221,...
25, 25, 51, 27, 131, 0, 103, 77, 183, 233, 311, 443,...
33, 35, 69, 69, 67, 103, 0, 305, 137, 407, 373, 443,...
37, 37, 39, 113, 153, 77, 305, 0, 229, 115, 495, 75,...
...
The first few antidiagonals are:
[0]
[5, 5]
[9, 0, 9]
[13, 11, 11, 13]
[21, 13, 0, 13, 21]
[25, 23, 29, 29, 23, 25]
[33, 25, 21, 0, 21, 25, 33]
[37, 35, 51, 43, 43, 51, 35, 37]
...

N. J. A. Sloane, Table of n, a(n) for n = 1..5050 [First 100 antidiagonals, flattened]

Cf. A344005, A347681, A347683, A347684.

Rows 1 and 2 are (essentially) A076274 and A208296.

myfun1 := proc(A,B) local Ar,Br;
if igcd(A,B) > 1 then return(0); fi;
  Ar:=(A)^(-1) mod B;
   if 2*Ar > B then Ar:=B-Ar; fi;
  Br:=(B)^(-1) mod A;
   if 2*Br > A then Br:=A-Br; fi;
A*Ar+B*Br;
end;
myfun2:=(i,j)->myfun1(ithprime(i),ithprime(j));
for i from 1 to 30 do lprint([seq(myfun2(i-j+1,j),j=1..i)]); od:

A207336 One half of smallest positive nontrivial even solution of the congruence x^2 == 1 (mod A001748(n+2)), n>=1.

Original entry on oeis.org

2, 4, 5, 7, 8, 10, 11, 14, 16, 19, 20, 22, 23, 26, 29, 31, 34, 35, 37, 40, 41, 44, 49, 50, 52, 53, 55, 56, 64, 65, 68, 70, 74, 76, 79, 82, 83, 86, 89, 91, 95, 97, 98, 100, 106, 112, 113, 115, 116, 119

Offset: 1

Wolfdieter Lang, Mar 14 2012

nonn

See the comments on A208296, which gives the representatives of the odd nontrivial solutions of the congruence x^2 == 1 (mod 3*prime(n+2)), with primes prime(n+2)=A000040(n+2), n>=1.

			The actual solutions are 4, 8, 10, 14, 16, 20, 22, 28, 32, 38, 40, 44, 46, 52, 58, 62, 68, 70, 74, 80, 82, 88, 98, 100, 104, 106, 110, 112, 128, 130, 136, 140, 148, 152, 158, 164, 166, 172, 178, 182, 190, 194, 196, 200, 212, 224, 226, 230, ...
n=4: 2*a(4) = 14 = 3*13 - 25. 14^2 = 196 == 1 (mod 39), 25^2 = 625 == 1 (mod 39). Representatives of the trivial solutions are 1 and 39-1= 38. All-together there are 4 incongruent solutions.

Jon Maiga, Table of n, a(n) for n = 1..1000

Cf. A001748, A208296.

Table[(3*Prime[n+2]-SelectFirst[Solve[x^2==1 && x !=1,x,Modulus->3*Prime[n+2]][[All,1,2]],OddQ])/2, {n, 50}] (* Jon Maiga, Sep 28 2019 *)

a(n) = (3*prime(n+2) - A208296(n))/2, with the primes prime(n+2) = A000040(n+2), n>=1.

cp's OEIS Frontend

A024699 a(n) = (prime(n+2)-1)/6 if this is an integer or (prime(n+2)+ 1)/6 otherwise.

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A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = f(prime(n),prime(k)), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

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Maple

A207336 One half of smallest positive nontrivial even solution of the congruence x^2 == 1 (mod A001748(n+2)), n>=1.

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Formula

cp's OEIS Frontend

A024699 a(n) = (prime(n+2)-1)/6 if this is an integer or (prime(n+2)+ 1)/6 otherwise.

Views

Author

Keywords

Comments

Programs

Magma

Maple

Mathematica

PARI

Formula

A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = f(prime(n),prime(k)), where f(x,y) = x*red_inv(x,y) + y*red_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A207336 One half of smallest positive nontrivial even solution of the congruence x^2 == 1 (mod A001748(n+2)), n>=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A347682 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = f(prime(n),prime(k)), where f(x,y) = xred_inv(x,y) + yred_inv(y,x) if gcd(x,y)=1, or 0 if gcd(x,y)>1, and red_inv is defined in the comments.