A262311 -id:A262311 - cp's OEIS frontend

A262785 Number of ordered ways to write n as x^2 + y^2 + p*(p+d)/2, where 0 <= x <= y, d is 1 or -1, and p is prime.

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

Offset: 1

Zhi-Wei Sun, Oct 01 2015

nonn

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 9, 34, 45, 49, 72, 97, 241, 337, 538.
(ii) Any integer n > 9 can be written as x^2 + y^2 + z*(z+1), where x,y,z are nonnegative integers with z-1 or z+1 prime.
In 2015, the author refined a result of Euler by proving that any positive integer can be written as the sum of two squares and a positive triangular number.

			a(1) = 1 since 1 = 0^2 + 0^2 + 2*(2-1)/2 with 2 prime.
a(2) = 1 since 2 = 0^2 + 1^2 + 2*(2-1)/2 with 2 prime.
a(3) = 3 since 3 = 0^2 + 0^2 + 2*(2+1)/2 = 0^2 + 0^2 + 3*(3-1)/2 = 1^2 + 1^2 + 2*(2-1)/2 with 2 and 3 both prime.
a(9) = 1 since 9 = 2^2 + 2^2 + 2*(2-1)/2 with 2 prime.
a(34) = 1 since 34 = 2^2 + 3^2 + 7*(7-1)/2 with 7 prime.
a(45) = 1 since 45 = 1^2 + 4^2 + 7*(7+1)/2 with 7 prime.
a(49) = 1 since 49 = 3^2 + 5^2 + 5*(5+1)/2 with 5 prime.
a(72) = 1 since 72 = 1^2 + 4^2 + 11*(11-1)/2 with 11 prime.
a(97) = 1 since 97 = 1^2 + 9^2 + 5(5+1)/2 with 5 prime.
a(241) = 1 since 241 = 1^2 + 15^2 + 5*(5+1)/2 with 5 prime.
a(337) = 1 since 337 = 5^2 + 6^2 + 23*(23+1)/2 with 23 prime.
a(538) = 1 since 538 = 3^2 + 8^2 + 31*(31-1)/2 with 31 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
Zhi-Wei Sun, On universal sums ax^2+by^2+f(z), aT_x+bT_y+f(z) and aT_x+by^2+f(z), arXiv:1502.03056 [math.NT], 2015.

Cf. A000040, A000290, A000217, A001481, A254885, A262311, A262781.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[d_,n_]:=Prime[n](Prime[n]+(-1)^d)/2
Do[r=0;Do[If[SQ[n-f[d,k]-x^2],r=r+1],{d,0,1},{k,1,PrimePi[(Sqrt[8n+1]-(-1)^d)/2]},{x,0,Sqrt[(n-f[d,k])/2]}];Print[n," ",r];Continue,{n,1,100}]

A262747 Number of ordered ways to write n as x^2 + y^2 + phi(z^2) with 0 <= x <= y, z > 0, 2 | xyz, and phi(k^2) < n for all 0 < k < z, where phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Sep 30 2015

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 3, 4, 5, 8, 13, 16, 23, 32, 35, 39, 68, 75, 96, 215, 219, 243, 363, 471, 723, 759, 923, 1443, 1551, 1839, 2739, 2883.
It is easy to see that all the numbers phi(n^2) = n*phi(n) (n = 1,2,3,...) are pairwise distinct. We have verified that a(n) > 0 for all n = 1,...,10^7.
See also A262311 for a related conjecture.

			a(5) = 1 since 5 = 0^2 + 2^2 + phi(1^2).
a(8) = 1 since 8 = 0^2 + 0^2 + phi(4^2) with 0*0*4 even, and phi(1^2) = 1, phi(2^2) = 2, phi(3^2) = 6 all smaller than 8.
a(13) = 1 since 13 = 1^2 + 2^2 + phi(4^2).
a(32) = 1 since 32 = 2^2 + 4^2 + phi(6^2).
a(68) = 1 since 68 = 0^2 + 6^2 + phi(8^2).
a(96) = 1 since 96 = 0^2 + 8^2 + phi(8^2).
a(363) = 1 since 363 = 0^2 + 19^2 + phi(2^2).
a(471) = 1 since 471 = 0^2 + 19^2 + phi(11^2).
a(723) = 1 since 723 = 17^2 + 18^2 + phi(11^2).
a(759) = 1 since 759 = 9^2 + 26^2 + phi(2^2).
a(923) = 1 since 923 = 16^2 + 25^2 + phi(7^2).
a(1443) = 1 since 1443 = 19^2 + 24^2 +phi(23^2).
a(1551) = 1 since 1551 = 18^2 + 35^2 + phi(2^2).
a(1839) = 1 since 1839 = 3^2 + 30^2 + phi(31^2).
a(2739) = 1 since 2739 = 1^2 + 24^2 + phi(47^2).
a(2883) = 1 since 2883 = 21^2 + 44^2 + phi(23^2).

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

Cf. A000010, A001481, A002618, A262311, A262746, A262750, A262766.

f[n_]:=EulerPhi[n^2]
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[f[x]>n,Goto[aa]];Do[If[SQ[n-f[x]-y^2]&&(Mod[x*y,2]==0||Mod[Sqrt[n-f[x]-y^2],2]==0),r=r+1],{y,0,Sqrt[(n-f[x])/2]}];Continue,{x,1,n}];Label[aa];Print[n," ",r];Continue,{n,1,100}]

A262781 Number of ordered ways to write n as x^2 + phi(y^2) + phi(z^2) (x >= 0 and 0 < y <= z) with y or z prime, where phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

0, 0, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 2, 3, 2, 2, 1, 3, 3, 1, 2, 3, 4, 1, 1, 3, 2, 3, 2, 4, 1, 3, 2, 2, 3, 1, 3, 3, 4, 2, 2, 3, 5, 5, 1, 4, 4, 4, 2, 6, 4, 4, 4, 6, 3, 4, 5, 4, 5, 4, 4, 3, 6, 4, 2, 3, 3, 5, 4, 4, 4, 3, 1, 4, 5, 4, 3, 6, 3, 1, 2, 3, 4, 4, 5, 5, 3, 3, 2, 8, 5, 3, 4, 2, 4, 4, 2, 3, 7, 2

Offset: 1

Zhi-Wei Sun, Oct 01 2015

nonn

Conjecture: (i) a(n) > 0 for all n > 6, and a(n) = 1 only for n = 3, 5, 9, 10, 17, 20, 24, 25, 31, 36, 45, 73, 80, 101, 136, 145, 388, 649.
(ii) For any integer n > 4, we can write 2*n as phi(p^2) + phi(x^2) + phi(y^2) with p prime and p <= x <= y.
See also A262311 for a similar conjecture.

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

Cf. A000010, A000040, A000290, A002618, A262311, A262746, A262747.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=EulerPhi[n^2]
Do[r=0;Do[If[f[z]>n,Goto[aa]];Do[If[SQ[n-f[z]-f[y]]&&(PrimeQ[y]||PrimeQ[z]),r=r+1],{y,1,z}];Label[aa];Continue,{z,1,n}];Print[n," ",r];Continue,{n,1,100}]

a(3) = 1 since 3 = 0^2 + phi(1^2) + phi(2^2) with 2 prime.
a(5) = 1 since 5 = 1^2 + phi(2^2) + phi(2^2) with 2 prime.
a(9) = 1 since 9 = 1^2 + phi(2^2) + phi(3^2) with 2 and 3 both prime.
a(10) = 1 since 10 = 0^2 + phi(2^2) + phi(4^2) with 2 prime.
a(17) = 1 since 17 = 3^2 + phi(2^2) + phi(3^2) with 2 and 3 both prime.
a(20) = 1 since 20 = 4^2 + phi(2^2) + phi(2^2) with 2 prime.
a(24) = 1 since 24 = 4^2 + phi(2^2) + phi(3^2) with 2 and 3 both prime.
a(25) = 1 since 25 = 2^2 + phi(1^2) + phi(5^2) with 5 prime.
a(31) = 1 since 31 = 3^2 + phi(2^2) + phi(5^2) with 2 and 5 both prime.
a(36) = 1 since 36 = 2^2 + phi(5^2) + phi(6^2) with 5 prime.
a(45) = 1 since 45 = 1^2 + phi(2^2) + phi(7^2) with 2 and 7 both prime.
a(73) = 1 since 73 = 5^2 + phi(3^2) + phi(7^2) with 3 and 7 both prime.
a(80) = 1 since 80 = 6^2 + phi(2^2) + phi(7^2) with 2 and 7 both prime.
a(101) = 1 since 101 = 7^2 + phi(5^2) + phi(8^2) with 5 prime.
a(136) = 1 since 136 = 5^2 + phi(1^2) + phi(11^2) with 11 prime.
a(145) = 1 since 145 = 7^2 + phi(7^2) + phi(9^2) with 7 prime.
a(388) = 1 since 388 = 2^2 + phi(7^2) + phi(19^2) with 7 and 19 both prime.
a(649) = 1 since 649 = 11^2 + phi(7^2) + phi(27^2) with 7 prime.

A263998 Number of ordered ways to write n as x^2 + 2y^2 + p(p+d)/2, where x and y are nonnegative integers, d is 1 or -1, and p is prime.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 31 2015

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 2, 8.

This is similar to the conjecture in A262785.

			 a(1) = 1 since 1 = 0^2 + 2*0^2 + 2*(2-1)/2 with 2 prime.
a(2) = 1 since 2 = 1^2 + 2*0^2 + 2*(2-1)/2 with 2 prime.
a(8) = 1 since 8 = 0^2 + 2*1^2 + 3*(3+1)/2 with 3 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Some mysterious representations of integers, a message to Number Theory Mailing List, Oct. 25, 2015.

Cf. A000040, A000217, A000290, A262311, A262785, A263992.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
f[d_, n_]:=f[d,n]=Prime[n](Prime[n]+(-1)^d)/2
Do[r=0; Do[If[SQ[n-f[d, k]-2x^2], r=r+1], {d, 0, 1}, {k, 1, PrimePi[(Sqrt[8n+1]-(-1)^d)/2]}, {x, 0, Sqrt[(n-f[d, k])/2]}]; Print[n, " ", r]; Continue, {n, 1, 100}]

A262887 Number of ordered ways to write n as x^3 + y^2 + pi(z^2) (x >= 0, y >= 0 and z > 0) with z-1 or z+1 prime, where pi(m) denotes the number of primes not exceeding m.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 04 2015

nonn

Conjectures:
(i) a(n) > 0 for all n > 0. Also, each natural number can be written as x^2 + y^2 + pi(z^2) (0 <= x <= y and z > 0) with z-1 or z+1 prime.
(ii) Any integer n > 1 can be written as x^3 + y^2 + pi(z^2) with x >= 0, y >= 0 and z > 0 such that y or z is prime.
(iii) Any integer n > 1 can be written as x^3 + pi(y^2) + pi(z^2) (x >= 0, y > 0 and z > 0) with y or z prime. Also, each integer n > 1 can be written as x^2 + pi(p^2) + pi(q^2) (x >= 0 and p >= q > 0) with p prime.
Compare these conjectures with the conjectures in A262746.

			a(22) = 2 since 22 = 0^3 + 4^2 + pi(4^2) = 0^3 + 2^2 + pi(8^2) with 4+1 = 5 and 8-1 = 7 both prime.
a(24) = 1 since 24 = 2^3 + 4^2 + pi(1^2) with 1+1 = 2 prime.
a(40) = 2 since 40 = 0^3 + 6^2 + pi(3^2) = 3^3 + 3^2 + pi(3^2) with 3-1 = 2 prime.
a(57) = 1 since 57 = 2^3 + 7^2 + pi(1^2) with 1+1 = 2 prime.
a(73) = 1 since 73 = 4^3 + 3^2 + pi(1^2) with 1+1 = 2 prime.

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

Cf. A000040, A000290, A000578, A000720, A262311, A262746, A262785, A262813.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n^2]
Do[r=0;Do[If[f[k]>n,Goto[aa]];If[PrimeQ[k-1]==False&&PrimeQ[k+1]==False,Goto[bb]];Do[If[SQ[n-f[k]-x^3],r=r+1],{x,0,(n-f[k])^(1/3)}];Label[bb];Continue,{k,1,n}];Label[aa];Print[n," ",r];Continue, {n,1,100}]

A262982 Number of ordered ways to write n as x^4 + phi(y^2) + z*(z+1)/2 with x >= 0, y > 0 and z > 0, where phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

0, 1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 4, 3, 2, 2, 3, 3, 4, 3, 1, 3, 4, 7, 4, 2, 1, 5, 4, 3, 5, 3, 2, 3, 5, 3, 3, 4, 5, 5, 1, 3, 5, 6, 3, 4, 5, 4, 5, 6, 3, 5, 4, 4, 5, 3, 5, 8, 7, 3, 3, 5, 4, 5, 7, 3, 2, 4, 6, 7, 4, 3, 3, 5, 2, 3, 6, 5, 3, 6, 3, 2, 1, 4, 6, 7, 6, 5, 6, 1, 6, 5, 5, 6, 6, 4, 3, 4, 6, 7, 5

Offset: 1

Zhi-Wei Sun, Oct 06 2015

nonn

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 6, 20, 26, 40, 82, 89, 105, 305, 416, 470, 725, 6135, 25430, 90285.

Compare this with the conjectures in A262311, A262785 and A262813.

			a(2) = 1 since 1 = 0^4 + phi(1^2) + 1*2/2.
a(6) = 1 since 6 = 1^4 + phi(2^2) + 2*3/2.
a(20) = 1 since 20 = 2^4 + phi(1^2) + 2*3/2.
a(26) = 1 since 26 = 0^4 + phi(5^2) + 3*4/2.
a(40) = 1 since 40 = 0^4 + phi(6^2) + 7*8/2.
a(82) = 1 since 82 = 0^4 + phi(9^2) + 7*8/2.
a(89) = 1 since 89 = 3^4 + phi(2^2) + 3*4/2.
a(105) = 1 since 105 = 0^4 + phi(14^2) + 6*7/2.
a(305) = 1 since 305 = 4^4 + phi(12^2) + 1*2/2.
a(416) = 1 since 416 = 4^4 + phi(10^2) + 15*16/2.
a(470) = 1 since 470 = 2^4 + phi(12^2) + 28*29/2.
a(725) = 1 since 725 = 2^4 + phi(3^2) + 37*38/2.
a(6135) = 1 since 6135 = 6^4 + phi(81^2) + 30*31/2.
a(25430) = 1 since 25430 = 5^4 + phi(152^2) + 166*167/2.
a(90285) = 1 since 90285 = 16^4 + phi(212^2) + 73*74/2.

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

Cf. A000010, A000217, A000290, A000583, A002618, A262311, A262781, A262785, A262813, A262941, A262979.

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]-y^4],r=r+1],{y,0,(n-f[x])^(1/4)}];Label[aa];Continue,{x,1,n}];Print[n," ",r];Continue,{n,1,100}]

A262979 Number of ordered ways to write n as x^4 + phi(y^2) + z(3z-1)/2 with x >= 0 and y > 0, where phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 06 2015

nonn

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

(ii) Any positive integer n can be written as x^4 + phi(y^2) + pi(z^2) (or x^4 + pi(y^2) + pi(z^2)) with y > 0 and z > 0, where pi(m) denotes the number of primes not exceeding m.

			a(5) = 1 since 5 = 1^4 + phi(2^2) + (-1)*(3*(-1)-1)/2.
a(6) = 2 since 6 = 0^4 + phi(1^2) + 2*(3*2-1)/2 = 0^4 + phi(3^2) + 0*(3*0-1)/2.
a(16) = 2 since 16 = 0^4 + phi(1^2) + (-3)*(3*(-3)-1)/2
= 1^4 + phi(4^2) + (-2)*(3*(-2)-1)/2.

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

Cf. A000010, A000290, A000583, A000720, A001318, A002618, A262311, A262746, A262781, A262887, A262941, A262976.

f[n_]:=EulerPhi[n^2]
PenQ[n_]:=IntegerQ[Sqrt[24n+1]]
Do[r=0;Do[If[f[x]>n,Goto[aa]];Do[If[PenQ[n-f[x]-y^4],r=r+1],{y,0,(n-f[x])^(1/4)}];Label[aa];Continue,{x,1,n}];Print[n," ",r];Continue,{n,1,100}]

A263992 Number of ordered ways to write n as x^2 + 2*y^2 + phi(z^2) (x >= 0, y >= 0 and z > 0) such that y or z has the form p-1 with p prime, where phi(.) is Euler's totient function.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Oct 31 2015

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 6, 7, 22, 3447.

This is similar to the conjecture in A262311, and we have verified it for n up to 10^6.

			a(1) = 1 since 1 = 0^2 + 2*0^2 + phi(1^2) with 1 + 1 = 2 prime.
a(6) = 1 since 6 = 2^2 + 2*0^2 + phi(2^2) with 2 + 1 = 3 prime.
a(7) = 1 since 7 = 2^2 + 2*1^2 + phi(1^2) with 1 + 1 = 2 prime.
a(22) = 1 since 22 = 0^2 + 2*1^2 + phi(5^2) with 1 + 1 = 2 prime.
a(3447) = 1 since 3447 = 42^2 + 2*29^2 + phi(1^2) with 1 + 1 = 2 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Some mysterious representations of integers, a message to Number Theory Mailing List, Oct. 25, 2015.

Cf. A000010, A000290, A006093, A262311, A263998.

phi[n_]:=phi[n]=EulerPhi[n]
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[n-z*phi[z]<0,Goto[aa]];Do[If[SQ[n-z*phi[z]-2y^2]&&(PrimeQ[y+1]||PrimeQ[z+1]),r=r+1],{y,0,Sqrt[(n-z*phi[z])/2]}];Label[aa];Continue,{z,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

A262785 Number of ordered ways to write n as x^2 + y^2 + p*(p+d)/2, where 0 <= x <= y, d is 1 or -1, and p is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A262747 Number of ordered ways to write n as x^2 + y^2 + phi(z^2) with 0 <= x <= y, z > 0, 2 | x*y*z, and phi(k^2) < n for all 0 < k < z, where phi(.) is Euler's totient function given by A000010.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A262781 Number of ordered ways to write n as x^2 + phi(y^2) + phi(z^2) (x >= 0 and 0 < y <= z) with y or z prime, where phi(.) is Euler's totient function given by A000010.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A263998 Number of ordered ways to write n as x^2 + 2*y^2 + p*(p+d)/2, where x and y are nonnegative integers, d is 1 or -1, and p is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A262887 Number of ordered ways to write n as x^3 + y^2 + pi(z^2) (x >= 0, y >= 0 and z > 0) with z-1 or z+1 prime, where pi(m) denotes the number of primes not exceeding m.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A262982 Number of ordered ways to write n as x^4 + phi(y^2) + z*(z+1)/2 with x >= 0, y > 0 and z > 0, where phi(.) is Euler's totient function given by A000010.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A262979 Number of ordered ways to write n as x^4 + phi(y^2) + z*(3*z-1)/2 with x >= 0 and y > 0, where phi(.) is Euler's totient function given by A000010.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A263992 Number of ordered ways to write n as x^2 + 2*y^2 + phi(z^2) (x >= 0, y >= 0 and z > 0) such that y or z has the form p-1 with p prime, where phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

A262747 Number of ordered ways to write n as x^2 + y^2 + phi(z^2) with 0 <= x <= y, z > 0, 2 | xyz, and phi(k^2) < n for all 0 < k < z, where phi(.) is Euler's totient function given by A000010.

A263998 Number of ordered ways to write n as x^2 + 2y^2 + p(p+d)/2, where x and y are nonnegative integers, d is 1 or -1, and p is prime.

A262979 Number of ordered ways to write n as x^4 + phi(y^2) + z(3z-1)/2 with x >= 0 and y > 0, where phi(.) is Euler's totient function given by A000010.