A262976 -id:A262976 - cp's OEIS frontend

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.

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}]

A262980 Number of ordered ways to write n as p + 2^k + pi(2^m), where p is prime, and k and m are nonnegative integers, and pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

0, 0, 1, 3, 4, 5, 6, 7, 7, 8, 8, 7, 9, 7, 12, 7, 10, 7, 12, 9, 14, 11, 12, 10, 15, 8, 13, 6, 12, 7, 12, 9, 13, 9, 14, 11, 15, 11, 18, 9, 14, 8, 14, 10, 18, 13, 11, 9, 18, 13, 17, 10, 13, 7, 15, 12, 14, 10, 10, 10, 15, 12, 19, 11, 15, 12, 16, 10, 20, 12, 13, 12, 20, 12, 23

Offset: 1

Zhi-Wei Sun, Oct 06 2015

nonn

Conjecture: a(n) > 0 for all n > 2.
We have verified this for n up to 2*10^8.
In contrast with this conjecture, in 1971 R. Crocker proved that there are infinitely many positive odd numbers not of the form p + 2^k + 2^m, where p is prime, and k and m are positive integers.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Roger Crocker, On the sum of a prime and of two powers of two, Pacific J. Math. 36 (1971), 103-107.

Cf. A000040, A000079, A000720, A007053, A156695, A232398, A262976.

a(3) = 1 since 3 = 2 + 2^0 + pi(2^0) with 2 prime.
a(4) = 3 since 4 = 2 + 2^0 + pi(2) = 2 + 2 + pi(2^0) = 3 + 2^0 + pi(2^0) with 2 and 3 both prime.

f[n_]:=PrimePi[2^n]
Do[r=0;Do[If[f[x]>=n,Goto[aa]];Do[If[PrimeQ[n-f[x]-2^y],r=r+1],{y,0,Log[2,n-f[x]]}];Continue,{x,0,n}];Label[aa];Print[n," ",r];Continue,{n,1,100}]

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.

Original entry on oeis.org

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}]

cp's OEIS Frontend

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.

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A262980 Number of ordered ways to write n as p + 2^k + pi(2^m), where p is prime, and k and m are nonnegative integers, and pi(x) denotes the number of primes not exceeding x.

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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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cp's OEIS Frontend

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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A262980 Number of ordered ways to write n as p + 2^k + pi(2^m), where p is prime, and k and m are nonnegative integers, and pi(x) denotes the number of primes not exceeding x.

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Author

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Comments

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Maple

Mathematica

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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Mathematica

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.