A025428 Number of partitions of n into 4 nonzero squares.

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

Offset

0,29

Comments

Records occur at n= 4, 28, 52, 82, 90, 130, 162, 198, 202, 210,.... - R. J. Mathar, Sep 15 2015

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Index entries for sequences related to sums of squares

Formula

For n>0, a(n) = ( A063730(n) + 6*A213024(n) + 3*A063725(n/2) + 8*A092573(n) + 6*A010052(n/4) ) / 24. - Max Alekseyev, Sep 30 2012

a(n) = ( A000118(n) - 4*A005875(n) - 6*A004018(n) - 12*A000122(n) - 15*A000007(n) + 12*A014455(n) - 24*A033715(n) - 12*A000122(n/2) + 12*A004018(n/2) + 32*A033716(n) - 32*A000122(n/3) + 48*A000122(n/4) ) / 384. - Max Alekseyev, Sep 30 2012

a(n) = [x^n y^4] Product_{k>=1} 1/(1 - y*x^(k^2)). - Ilya Gutkovskiy, Apr 19 2019

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} A010052(i) * A010052(j) * A010052(k) * A010052(n-i-j-k). - Wesley Ivan Hurt, Apr 19 2019

Maple

A025428 := proc(n)
    local a, i, j, k, lsq ;
    a := 0 ;
    for i from 1 do
        if 4*i^2 > n then
            return a;
        end if;
        for j from i do
            if i^2+3*j^2 > n then
                break;
            end if;
            for k from j do
                if i^2+j^2+2*k^2 > n then
                    break;
                end if;
                lsq := n-i^2-j^2-k^2 ;
                if lsq >= k^2 and issqr(lsq) then
                    a := a+1 ;
                end if;
            end do:
        end do:
    end do:
end proc:
seq(A025428(n), n=1..40) ; # R. J. Mathar, Jun 15 2018
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),
     `if`(i<1 or t<1, 0, b(n, i-1, t)+`if`(i^2>n, 0, b(n-i^2, i, t-1))))
    end:
a:= n-> b(n, isqrt(n), 4):
seq(a(n), n=0..100);  # Alois P. Heinz, Apr 14 2019

Mathematica

nn = 100; lim = Sqrt[nn]; t = Table[0, {nn}]; Do[n = a^2 + b^2 + c^2 + d^2; If[n <= nn, t[[n]]++], {a, lim}, {b, a, lim}, {c, b, lim}, {d, c, lim}]; t (* T. D. Noe, Sep 28 2012 *)
f[n_] := Length@ IntegerPartitions[n, {4}, Range[ Floor[ Sqrt[n - 1]]]^2]; Array[f, 105] (* Robert G. Wilson v, Sep 28 2012 *)

Prog

(PARI) A025428(n)=sum(a=1, n, sum(b=1, a, sum(c=1, b, sum(d=1, c, a^2+b^2+c^2+d^2==n))))
(PARI) A025428(n)=sum(a=1, sqrtint(max(n-3, 0)), sum(b=1, min(sqrtint(n-a^2-2), a), sum(c=1, min(sqrtint(n-a^2-b^2-1), b), issquare(n-a^2-b^2-c^2, &d) & d <= c )))
(PARI) A025428(n)=sum(a=sqrtint(max(n, 4)\4), sqrtint(max(n-3, 0)), sum(b=sqrtint((n-a^2)\3-1)+1, min(sqrtint(n-a^2-2), a), sum(c=sqrtint((t=n-a^2-b^2)\2-1)+1, min(sqrtint(t-1), b), issquare(t-c^2) ))) \\ - M. F. Hasler, Sep 17 2012
for(n=1, 100, print1(A025428(n), ", "))
(PARI) T(n)={a=matrix(n, 4, i, j, 0); for(d=1, sqrtint(n), forstep(i=n, d*d+1, -1, for(j=2, 4, a[i, j]+=sum(k=1, j, if(k<j&&i-k*d*d>0, a[i-k*d*d, j-k], if(k==j&&i-k*d*d==0, 1))))); a[d*d, 1]=1); for(i=1, n, print(i" "a[i, 4]))} /* Robert Gerbicz, Sep 28 2012 */

Crossrefs

Cf. A000414, A000534, A025357-A025375, A216374, A025416 (greedy inverse).

Column k=4 of A243148.

Keyword

nonn,easy

Author

David W. Wilson

Extensions

Values of a(0..10^4) double-checked by M. F. Hasler, Sep 17 2012

Status

approved

A068227 The "genity" sequence of the primes, i.e., a(n) = g(p) = ((p mod 4) + (p mod 6))/2, where p is the n-th prime.

2, 3, 3, 2, 4, 1, 3, 2, 4, 3, 2, 1, 3, 2, 4, 3, 4, 1, 2, 4, 1, 2, 4, 3, 1, 3, 2, 4, 1, 3, 2, 4, 3, 2, 3, 2, 1, 2, 4, 3, 4, 1, 4, 1, 3, 2, 2, 2, 4, 1, 3, 4, 1, 4, 3, 4, 3, 2, 1, 3, 2, 3, 2, 4, 1, 3, 2, 1, 4, 1, 3, 4, 2, 1, 2, 4, 3, 1, 3, 1, 4, 1, 4, 1, 2, 4, 3, 1, 3, 2, 4, 4, 2, 4, 2, 4, 3, 3, 2, 1, 2, 3, 4

Offset

1,1

Comments

The name "genity" was derived from "genes" and "parity", since the fourfold values of g(p) in a sequence corresponding to prime arguments resemble the genetic sequences of the nucleotides in the DNA. Parity is also related, since it originally means a (mod 2) feature, while here we categorize the primes (mod 4) and (mod 6), simultaneously.

The arithmetic function g(p) = ((p mod 4) + (p mod 6))/2 provides integer values for prime arguments, such that 1 <= g(p) <= 4 and is determined by the congruence class of p (mod 12). Specifically, g(p)=1 if p==1 (mod 12), g(p)=2 if p=2 or p==7 (mod 12), g(p)=3 if p=3 or p==5 (mod 12) and g(p)=4 if p==11 (mod 12).

Dickson's conjecture implies that every finite sequence of numbers from 1 to 4 occurs infinitely often in this sequence.

Links

Table of n, a(n) for n=1..103.

The Prime Glossary, Dickson's conjecture

Mathematica

Table[(Mod[Prime[n], 4] + Mod[Prime[n], 6])/2, {n, 1, 100}]

Prog

(PARI) for(i=1, 120, print((prime(i)%4+prime(i)%6)/2))

Crossrefs

Cf. A068228, A068229, A040117, A068231, A068232, A068233, A068234, A068235.

Keyword

easy,nonn

Author

Ferenc Adorjan (fadorjan(AT)freemail.hu), Feb 22 2002

Extensions

Edited by Dean Hickerson and Robert G. Wilson v, Mar 06 2002

Status

approved