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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. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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.

LINKS

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

FORMULA

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.

MATHEMATICA

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}]

CROSSREFS

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

Sequence in context: A144082 A145579 A167655 * A157218 A004718 A157225

Adjacent sequences:  A262778 A262779 A262780 * A262782 A262783 A262784

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 01 2015

STATUS

approved

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Last modified October 16 03:37 EDT 2019. Contains 328040 sequences. (Running on oeis4.)