A317682 -id:A317682 - cp's OEIS frontend

A317685 Number of partitions of n into a prime and two positive squares.

0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 2, 1, 2, 3, 0, 4, 2, 1, 2, 3, 3, 4, 3, 3, 3, 4, 1, 4, 4, 3, 3, 6, 3, 4, 4, 2, 6, 6, 1, 8, 3, 3, 6, 6, 4, 6, 4, 5, 7, 6, 3, 6, 6, 5, 6, 9, 5, 8, 6, 3, 7, 8, 2, 12, 6, 4, 7, 7, 6, 10, 7, 7, 9, 7, 5, 9, 9, 7, 9, 10, 4

Offset: 0

R. J. Mathar, Michel Marcus, Aug 04 2018

As in A025426, the two squares do not need to be distinct.

			a(7) = 2 counts 7 = 5 + 1^2 + 1^2 = 2 + 1^2 + 2^2.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A025426, A317682 - A317684.

A317685 := proc(n)
    a := 0 ;
    p := 2;
    while p <= n do
        a := a+A025426(n-p);
        p := nextprime(p) ;
    end do:
    a ;
end proc:

p2sQ[{a_,b_,c_}]:=PrimeQ[a]&&AllTrue[{Sqrt[b],Sqrt[c]},IntegerQ]||PrimeQ[b] && AllTrue[{Sqrt[c],Sqrt[a]},IntegerQ]||PrimeQ[c]&&AllTrue[{Sqrt[b],Sqrt[a]},IntegerQ]; Table[Count[IntegerPartitions[n,{3}],?(p2sQ[#]&)],{n,0,80}] (* _Harvey P. Dale, Mar 10 2023 *)

a(n) = Sum_{primes p} A025426(n-p).

A317683 Number of partitions of n into a prime and two distinct positive squares.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 2, 2, 1, 2, 1, 2, 1, 3, 2, 3, 1, 1, 3, 4, 2, 3, 3, 3, 3, 3, 0, 6, 3, 1, 5, 3, 2, 6, 4, 4, 3, 4, 4, 7, 2, 3, 4, 5, 4, 6, 4, 5, 7, 6, 2, 7, 3, 2, 9, 6, 3, 7, 5, 6, 6, 7, 6, 9, 4, 4, 5, 9, 5, 9, 5, 4

Offset: 0

R. J. Mathar, Michel Marcus, Aug 04 2018

As in A025441, the two squares must be distinct and positive.

			a(12)=2 counts 12 = 7 +1^2 +2^2 = 2 + 1^2 +3^2.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A025441, A317682 - A317685.

A317683 := proc(n)
    a := 0 ;
    p := 2;
    while p <= n do
        a := a+A025441(n-p);
        p := nextprime(p) ;
    end do:
    a ;
end proc:

p2sQ[n_]:=Length[Union[n]]==3&&Count[n,?(IntegerQ[Sqrt[#]]&)]==2&&Count[ n,?(PrimeQ[#]&)]==1; Table[Count[IntegerPartitions[n,{3}],?p2sQ],{n,0,80}] (* _Harvey P. Dale, Sep 21 2019 *)

a(n) = Sum_{primes p} A025441(n-p).

A317684 Number of partitions of n into a prime and two squares.

Original entry on oeis.org

0, 0, 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 4, 4, 2, 5, 3, 3, 4, 5, 5, 6, 4, 6, 4, 4, 2, 7, 6, 5, 5, 7, 6, 6, 4, 4, 7, 7, 5, 10, 4, 6, 8, 8, 6, 8, 5, 9, 9, 7, 4, 8, 8, 8, 9, 10, 8, 10, 6, 6, 9, 9, 6, 14, 6, 6, 10, 10, 10, 12, 8, 10, 12, 9, 6, 12, 10, 11, 11, 12, 7

Offset: 0

R. J. Mathar, Michel Marcus, Aug 04 2018

As in A000161, the squares may be zero and do not need to be distinct.

			a(11) = 4 counts 11 = 11+0^2+0^2 = 7+0^2+2^2 = 2+0^2+3^2 = 3+2^2+2^2.

Robert Israel, Table of n, a(n) for n = 0..10000
C. Hooley, On the representation of a number as the sum of two squares and a prime, Acta Mathem. 97 (1957) 189-210

Cf. A000161, A317682-A317685.

A317684 := proc(n)
    a := 0 ;
    p := 2;
    while p <= n do
        a := a+A000161(n-p);
        p := nextprime(p) ;
    end do:
    a ;
end proc:

a(n) = Sum_{primes p} A000161(n-p).

cp's OEIS Frontend

A317685 Number of partitions of n into a prime and two positive squares.

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A317683 Number of partitions of n into a prime and two distinct positive squares.

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Maple

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A317684 Number of partitions of n into a prime and two squares.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula