A067754 -id:A067754 - cp's OEIS frontend

A067752 Number of unordered solutions of xy + xz + yz = n in nonnegative integers.

1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 3, 4, 2, 3, 4, 4, 3, 3, 3, 5, 4, 2, 4, 6, 3, 4, 5, 4, 4, 4, 4, 6, 4, 3, 6, 7, 2, 4, 6, 6, 5, 4, 3, 7, 6, 3, 6, 8, 4, 5, 6, 5, 4, 6, 6, 9, 4, 2, 7, 8, 4, 5, 8, 7, 6, 6, 3, 8, 6, 4, 8, 9, 3, 6, 8, 7, 6, 4, 6, 11, 7, 3, 7, 10, 4, 6, 8, 6, 7

Offset: 1

Colin Mallows, Jan 31 2002

An upper bound on the number of solutions appears to be 1.5*sqrt(n). - T. D. Noe, Jun 14 2006

a(n) is also the total number of distinct quadratic forms of discriminant -4n. A232551 counts only the primitive quadratic forms of discriminant -4n (those with all coefficients pairwise coprime) and A234287 includes those by which some prime can be represented (those with all coefficients pairwise coprime or coefficient of x^2 is prime or coefficient of y^2 is prime). This sequence includes all quadratic forms like 2x^2 + 2xy + 4y^2 and 2x^2 + 4y^2 which are non-primitive and those like 4x^2 + 2xy + 4y^2 and 4x^2 + 4xy + 4y^2 by which no prime can be represented (those with no restrictions). - V. Raman, Dec 24 2013

			a(12)=4 because of (0,1,12), (0,2,6), (0,3,4), (2,2,2).
a(20)=5 because of (0,1,20), (0,2,10), (0,4,5), (1,2,6), (2,2,4).

T. D. Noe, Table of n, a(n) for n = 1..10000
V. Raman, Examples of these distinct quadratic forms for n = 1..100

Cf. A000003, A000926, A067751, A067753, A067754, A232550, A232551, A234287.

Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z], cnt++ ], {x,0,Sqrt[n/3]}, {y, Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] (* T. D. Noe, Jun 14 2006 *)

Corrected, extended and edited by John W. Layman, Dec 03 2004

A067753 Number of primitive solutions in nonnegative integers of xy+xz+y*z = n.

Original entry on oeis.org

3, 6, 7, 6, 9, 12, 9, 9, 9, 12, 15, 12, 9, 18, 18, 9, 15, 12, 15, 18, 18, 12, 21, 18, 9, 24, 15, 12, 21, 24, 21, 15, 18, 18, 30, 18, 9, 24, 30, 18, 27, 24, 15, 24, 18, 18, 33, 18, 15, 24, 30, 18, 21, 24, 30, 30, 18, 12, 39, 24, 21, 30, 30, 15, 30, 36, 15, 30, 30, 24, 45, 18, 15, 36

Offset: 1

Colin Mallows, Jan 31 2002

An upper bound on the number of solutions appears to be 9*sqrt(n). - T. D. Noe, Jun 14 2006

			a(9)=9 because of permutations of (0,1,9) and (1,1,4) (but not (0,3,3)).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A066751, A067752, A067754.

CntFunc[s_List] := Module[{len=Length[Union[s]]}, If[len==3,6,If[len==2,3,1]]]; Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z] && GCD[x,y,z]==1, cnt=cnt+CntFunc[{x,y,z}]], {x,0,Sqrt[n/3]}, {y, Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] (* T. D. Noe, Jun 14 2006 *)

Corrected and extended by T. D. Noe, Jun 14 2006

A067751 Number of solutions in nonnegative integers of xy+xz+y*z = n.

Original entry on oeis.org

3, 6, 7, 9, 9, 12, 9, 15, 12, 12, 15, 19, 9, 18, 18, 18, 15, 18, 15, 27, 18, 12, 21, 30, 12, 24, 22, 21, 21, 24, 21, 30, 18, 18, 30, 36, 9, 24, 30, 30, 27, 24, 15, 39, 27, 18, 33, 37, 18, 30, 30, 27, 21, 36, 30, 48, 18, 12, 39, 42, 21, 30, 39, 33, 30, 36, 15, 45, 30, 24, 45, 45, 15

Offset: 1

Colin Mallows, Jan 31 2002

An upper bound on the number of solutions appears to be 9*sqrt(n). - T. D. Noe, Jun 14 2006

			a(3)=7 because of (0,1,3),(0,3,1),(1,0,3),(1,3,0),(3,0,1),(3,1,0),(1,1,1).

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A067752, A067753, A067754.

CntFunc[s_List] := Module[{len=Length[Union[s]]}, If[len==3,6,If[len==2,3,1]]]; Table[cnt=0; Do[z=(n-x*y)/(x+y); If[IntegerQ[z], cnt=cnt+CntFunc[{x,y,z}]], {x,0,Sqrt[n/3]}, {y, Max[1,x],Sqrt[x^2+n]-x}]; cnt, {n,100}] (* T. D. Noe, Jun 14 2006 *)

Corrected and extended by T. D. Noe, Jun 14 2006

A328932 Number of primes that are a concatenation of two positive integers whose sum is prime(n).

Original entry on oeis.org

1, 0, 2, 2, 4, 3, 2, 6, 5, 9, 6, 7, 10, 11, 12, 14, 12, 16, 14, 17, 12, 15, 16, 20, 19, 19, 20, 17, 23, 23, 18, 27, 28, 24, 30, 25, 26, 26, 28, 30, 27, 30, 32, 27, 25, 27, 37, 42, 38, 32, 32, 33, 30, 39, 38, 36, 43, 38, 43, 42, 36, 36, 47, 47, 49, 38, 45, 48, 51, 50

Offset: 1

Juri-Stepan Gerasimov, Oct 31 2019

			a(3) = 2 because primes 23, 41 are concatenations of prime(3) = 5 = 2 + 3 = 4 + 1.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Subsequence of A328903.

Cf. A000040, A066686, A067754.

a:= n-> (p-> add(`if`(isprime(parse(cat(i,
        p-i))), 1, 0), i=1..p-1))(ithprime(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Oct 31 2019

a(n) = A328903(A000040(n)).

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A067752 Number of unordered solutions of xy + xz + yz = n in nonnegative integers.

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A067753 Number of primitive solutions in nonnegative integers of x*y+x*z+y*z = n.

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A067751 Number of solutions in nonnegative integers of x*y+x*z+y*z = n.

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A328932 Number of primes that are a concatenation of two positive integers whose sum is prime(n).

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A067753 Number of primitive solutions in nonnegative integers of xy+xz+y*z = n.

A067751 Number of solutions in nonnegative integers of xy+xz+y*z = n.