A390799 - OEIS

A390799

Number of ways to write n as (p-1)^4/16 + (q-1)^2/2 + u^2 + 3*v^2, where each of p and q is 1 or an odd prime, and u and v are in the set {m >= 0: 2*m is practical} with u > 0.

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

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OFFSET

1,4

COMMENTS

Conjecture: a(n) > 0 for all n > 0. This has been verified for n <= 2*10^5.

It seems that a(n) = 1 only for n = 1, 2, 3, 8, 11, 26, 42, 56, 74, 123, 139, 161, 203, 314, 363, 1003.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

EXAMPLE

a(74) = 1 since 74 = (3-1)^4/16 + (13-1)^2/2 + 1^2 + 3*0^2 with 3 and 13 odd primes, and 2*1 practical.

a(1003) = 1 since 1003 = (1-1)^4/16 + (13-1)^2/2 + 16^2 + 3*15^2 with 13 prime, and both 2*16 and 2*15 practical.

MATHEMATICA

f[n_]:=f[n]=FactorInteger[n];

Pow[n_, i_]:=Pow[n, i]=Part[Part[f[n], i], 1]^(Part[Part[f[n], i], 2]);

Con[n_]:=Con[n]=Sum[If[Part[Part[f[n], s+1], 1]<=DivisorSigma[1, Product[Pow[n, i], {i, 1, s}]]+1, 0, 1], {s, 1, Length[f[n]]-1}];

pr[n_]:=pr[n]=n>0&&(n<3||Mod[n, 2]+Con[n]==0);

qq[n_]:=qq[n]=IntegerQ[n]&&(n==0||pr[2n]);

pq[n_]:=pq[n]=(n==0||PrimeQ[2n+1]);

tab={}; Do[m=0; Do[If[pq[x]&&pq[y]&&qq[z]&&qq[Sqrt[n-x^4-2y^2-3z^2]], m=m+1], {x, 0, (n-1)^(1/4)},

{y, 0, Sqrt[(n-1-x^4)/2]}, {z, 0, Sqrt[(n-1-x^4-2y^2)/3]}]; tab=Append[tab, m], {n, 1, 100}]; Print[tab]

CROSSREFS

Cf. A000040, A000290, A000583, A005097, A390762, A390793, A390795.

Sequence in context: A183049 A178086 A353296 * A281977 A240666 A052901

Adjacent sequences: A390796 A390797 A390798 * A390800 A390801 A390802

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Nov 19 2025

STATUS

approved