A262982 -id:A262982 - cp's OEIS frontend

A262985 Number of ordered ways to write n as 2^x + phi(y^2) + z*(z+1)/2 with x, y and z positive integers, where phi(.) is Euler's totient function given by A000010.

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

Offset: 1

Zhi-Wei Sun, Oct 06 2015

nonn

Conjecture: a(n) > 0 for all n > 3.

We have verified this for n up to 1.3*10^8.

			a(4) = 1 since 4 = 2 + phi(1^2) + 1*2/2.
a(5) = 1 since 5 = 2 + phi(2^2) + 1*2/2.
a(8) = 1 since 8 = 2^2 + phi(1^2) + 2*3/2.
a(36) = 2 since 36 = 2 + phi(3^2) + 7*8/2 = 2^5 + phi(1^2) + 2*3/2.

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

Cf. A000010, A000079, A000217, A000290, A002618, A262976, A262980, A262982.

 f[n_]:=EulerPhi[n^2]
TQ[n_]:=n>0&&IntegerQ[Sqrt[8n+1]]
Do[r=0;Do[If[f[x]>=n,Goto[aa]];Do[If[TQ[n-f[x]-2^y],r=r+1], {y,1,Log[2,n-f[x]]}]; Label[aa];Continue,{x,1,n}];Print[n," ",r];Continue,{n,1,100}]

A308342 Number of ways to write 2*n as phi(x^2) + phi(y^2) + phi(z^2), where x,y,z are positive integers with x <= y <= z, and phi(.) is Euler's totient function (A000010).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 1, 2, 3, 4, 4, 3, 4, 5, 4, 6, 5, 5, 5, 5, 5, 5, 4, 5, 4, 4, 2, 5, 5, 3, 6, 6, 3, 7, 6, 6, 6, 5, 6, 6, 4, 5, 5, 5, 5, 6, 4, 5, 8, 7, 5, 9, 6, 7, 8, 8, 7, 6, 6, 8, 5, 7, 7, 6, 5, 6, 8, 8, 8, 10, 6, 10, 13, 10, 10, 9, 6, 11, 9, 7, 3, 9, 6, 6, 9, 7, 5, 12

Offset: 1

Zhi-Wei Sun, May 20 2019

nonn

Conjecture 1: a(n) > 0 for all n > 1. In other words, the set {phi(x^2) + phi(y^2) + phi(z^2): x,y,z = 1,2,3,...} contains all even numbers greater than two.

Conjecture 2: For any integer n > 3, we can write 2*n+1 as phi(x^2) + phi(y^2) + sigma(z^2) with x,y,z positive integers, where the function sigma(.) is given by A000203.

			a(2) = 1 with 2*2 = phi(1^2) + phi(1^2) + phi(2^2).
a(3) = 1 with 2*3 = phi(2^2) + phi(2^2) + phi(2^2).
a(4) = 1 with 2*4 = phi(1^2) + phi(1^2) + phi(3^2).
a(6) = 1 with 2*6 = phi(2^2) + phi(2^2) + phi(4^2).
a(19) = 1 with 2*19 = phi(3^2) + phi(5^2) + phi(6^2).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Does phi(x^2) + phi(y^2) + phi(z^2) represent all even numbers greater than two?, Question 332067 on MathOverflow, May 20, 2019.

Cf. A000010, A000203, A000290, A002618, A262311, A262982, A262985.

f[n_]:=f[n]=n*EulerPhi[n]
T={};Do[If[f[n]<=200,T=Append[T,f[n]]],{n,1,200}];
tab={};Do[r=0;Do[If[f[k]>2n/3,Goto[cc]];Do[If[f[m](2n-f[k])/2,Goto[bb]];If[MemberQ[T,2n-f[k]-f[m]],r=r+1];Label[bb],{m,1,(2n-f[k])/2}];Label[cc],{k,1,2n/3}];tab=Append[tab,r],{n,1,100}];Print[tab]

cp's OEIS Frontend

A262985 Number of ordered ways to write n as 2^x + phi(y^2) + z*(z+1)/2 with x, y and z positive integers, where phi(.) is Euler's totient function given by A000010.

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