A262746 -id:A262746 - cp's OEIS frontend

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.

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.

A262976 Number of ordered ways to write n as 2^x + y^2 + pi(z^2) with x >= 0, y >= 0 and z > 0, where pi(m) denotes the number of primes not exceeding m.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 6, 4, 6, 6, 7, 6, 7, 5, 6, 10, 5, 9, 10, 7, 7, 9, 9, 4, 12, 10, 9, 8, 7, 10, 9, 10, 7, 15, 10, 6, 13, 10, 9, 10, 16, 10, 10, 9, 8, 15, 9, 8, 15, 12, 12, 7, 12, 11, 14, 12, 8, 16, 6, 10, 11, 14, 8, 11, 17, 10, 16, 9, 13, 16, 15, 8, 18, 13, 10, 14, 10, 12, 16, 12, 13, 18, 11, 9, 17, 17, 9, 15, 16, 15, 9, 12, 12, 17, 12, 9, 21, 10, 11

Offset: 1

Zhi-Wei Sun, Oct 05 2015

nonn

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

(ii) Each positive integer can be written as 2^x + pi(y^2) + pi(z^2) with x >= 0, y > 0 and z > 0.

			a(1) = 1 since 1 = 2^0 + 0^2 + pi(1^2).
a(2) = 2 since 2 = 2^0 + 1^2 + pi(1^2) = 2 + 0^2 + pi(1^2).
a(3) = 2 since 3 = 2^0 + 0^2 + pi(2^2) = 2 + 1^2 + pi(1^2).

Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000079, A000290, A000720, A038107, A262746, A262887.

SQ[n_]:=IntegerQ[Sqrt[n]]
f[n_]:=PrimePi[n^2]
Do[r=0;Do[If[f[x]>=n,Goto[aa]];Do[If[2^y>n-f[x],Goto[bb]];If[SQ[n-f[x]-2^y],r=r+1],{y,0,Log[2,n-f[x]]}];Label[bb];Continue,{x,1,n}];Label[aa];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}]

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}]

cp's OEIS Frontend

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.

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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.

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A262976 Number of ordered ways to write n as 2^x + y^2 + pi(z^2) with x >= 0, y >= 0 and z > 0, where pi(m) denotes the number of primes not exceeding m.

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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.

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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.

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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.

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.