A263317 -id:A263317 - cp's OEIS frontend

A263319 a(n) = pi(n^2)*phi(n)/2, where pi(x) denotes the number of primes not exceeding x, and phi(.) is Euler's totient function given by A000010.

0, 1, 4, 6, 18, 11, 45, 36, 66, 50, 150, 68, 234, 132, 192, 216, 488, 198, 648, 312, 510, 460, 1089, 420, 1140, 732, 1161, 822, 2044, 616, 2430, 1376, 1810, 1528, 2400, 1260, 3942, 2052, 2880, 2008, 5260, 1644, 5943, 2950, 3672, 3509, 7567, 2736, 7497, 3670

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Conjecture: (i) All the terms of this sequence are pairwise distinct.
(ii) All the numbers phi(n)*pi(n*(n-1)) (n = 1,2,3,...) are pairwise distinct.
(iii) All the numbers phi(n^2)*pi(n^2) = n*phi(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct.
We have checked this conjecture via Mathematica. For example, we have verified that a(n) (n = 1..4*10^5) are indeed pairwise distinct.
See also A263325 for a similar conjecture.

			a(1) = 0 since pi(1^2)*phi(1)/2 = 0*1/2 = 0.
a(2) = 1 since pi(2^2)*phi(2)/2 = 2*1/2 = 1.
a(3) = 4 since pi(3^2)*phi(3)/2 = 4*2/2 = 4.

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

Cf. A000010, A000290, A000720, A002618, A038107, A263317, A263321, A263325.

[#PrimesUpTo(n^2)*EulerPhi(n)/2: n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

a[n_]:=a[n]=PrimePi[n^2]*EulerPhi[n]/2
Do[Print[n," ",a[n]],{n,1,50}]

a(n) = primepi(n^2)*eulerphi(n)/2; \\ Michel Marcus, Oct 15 2015

A263325 a(n) = sigma(n)*pi(n^2), where sigma(n) is the sum of all (positive) divisors of n, and pi(x) is the number of primes not exceeding x.

Original entry on oeis.org

0, 6, 16, 42, 54, 132, 120, 270, 286, 450, 360, 952, 546, 1056, 1152, 1674, 1098, 2574, 1440, 3276, 2720, 3312, 2376, 6300, 3534, 5124, 5160, 7672, 4380, 11088, 5184, 10836, 8688, 10314, 9600, 19110, 8322, 13680, 13440, 22590, 11046, 26304, 12452, 24780, 23868, 22968, 15792, 42408, 20349, 34131

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Conjecture: (i) All the terms of this sequence are pairwise distinct.
(ii) All the numbers sigma(n)*pi(n*(n+1)) (n = 1,2,3,...) are pairwise distinct.
(iii) All the numbers n*sigma(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct, and all the numbers sigma(n^2)*pi(n^2) (n = 1,2,3,...) are also pairwise distinct.
(iv) All the numbers n*phi(n)*sigma(n^2) = phi(n^2)*sigma(n^2) (n = 1,2,3,...) are pairwise distinct, where phi(.) is Euler's totient function.
We have verified that the terms a(n) (n = 1..4*10^5) are indeed pairwise distinct.
See also A263319 for a similar conjecture.

			a(1) = 0 since sigma(1)*pi(1^2) = 1*0 = 0.
a(2) = 6 since sigma(2)*pi(2^2) = 3*2 = 6.

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

Cf. A000010, A000203, A000290, A000720, A002618, A038107, A065764, A263317, A263319.

[#PrimesUpTo(n^2)*SumOfDivisors(n): n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

a[n_]:=a[n]=DivisorSigma[1,n]*PrimePi[n^2]
Do[Print[n," ",a[n]],{n,1,50}]

a(n) = sigma(n)*primepi(n^2); \\ Michel Marcus, Oct 15 2015

A263321 Least positive integer m such that the numbers phi(k)*pi(k^2) (k = 1..n) are pairwise incongruent modulo m.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 16, 19, 19, 19, 29, 29, 29, 37, 37, 59, 59, 59, 59, 59, 59, 59, 59, 101, 101, 101, 133, 133, 133, 133, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 175, 245, 269, 269, 269, 269, 379, 379

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Part (i) of the conjecture in A263319 implies that a(n) exists for any n > 0.

Conjecture: a(n) <= n^2 for all n > 0, and the only even term is a(7) = 16.

			a(7) = 16 since the 7 numbers phi(1)*pi(1^2) = 0, phi(2)*pi(2^2) = 2, phi(3)*pi(3^2) = 8, phi(4)*pi(4^2) = 12, phi(5)*pi(5^2) = 36, phi(6)*pi(6^2) = 22 and phi(7)*pi(7^2) = 90 are pairwise incongruent modulo 16, but not so modulo any positive integer smaller than 16.

Zhi-Wei Sun, Table of n, a(n) for n = 1..4000
Zhi-Wei Sun, On functions taking only prime values, J. Number Theory 133(2013), 2794-2812.

Cf. A000010, A000290, A000720, A038107, A263317, A263319.

f[n_]:=f[n]=EulerPhi[n]*PrimePi[n^2]
Le[n_,m_]:=Le[m,n]=Length[Union[Table[Mod[f[k],m],{k,1,n}]]]
Do[n=1;m=1;Label[aa];If[Le[n,m]==n,Goto[bb],m=m+1;Goto[aa]];
Label[bb];Print[n," ",m];If[n<50,n=n+1;Goto[aa]]]

A263326 Denominator of the rational number Sum_{d|n}1/(d+1).

Original entry on oeis.org

2, 6, 4, 30, 3, 84, 8, 90, 20, 11, 12, 5460, 7, 40, 48, 1530, 9, 7980, 20, 1155, 88, 276, 24, 81900, 78, 189, 35, 1160, 15, 38192, 32, 16830, 51, 315, 72, 3838380, 19, 780, 280, 142065, 21, 132440, 44, 828, 5520, 376, 48, 9746100, 200, 14586

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Conjecture: For any positive integers k and s, all the numbers Sum_{d|n}1/(d+k)^s (n = 1,2,3,...) have pairwise distinct fractional parts, and none of them is an integer.
This implies that a(n) > 1 for all n > 0.
See also A001157 for a similar conjecture involving Sum_{d|n}1/d^s.
I have verified that Sum_{d|n}1/(d+1) (n = 1..2*10^5) indeed have pairwise distinct fractional parts and none of them is an integer. For each k = 2,3,4,5,6 I have verified that Sum_{d|n}1/(d+k) (n = 1..10^5) have pairwise distinct fractional parts and none of them is integral. - Zhi-Wei Sun, Oct 20 2015.

			a(1) = 2 since sum_{d|1}1/(d+1) = 1/2.
a(2) = 6 since sum_{d|2}1/(d+1) = 1/2 + 1/3 = 5/6.
a(3) = 4 since sum_{d|3}1/(d+1) = 1/2 + 1/4 = 3/4.

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

Cf. A000203, A001157, A263317, A263319, A263325.

f:= n -> denom(add(1/(d+1),d=numtheory:-divisors(n))):
map(f, [$1..100]); # Robert Israel, Oct 20 2015

Dv[n_]:=Dv[n]=Divisors[n]
a[n_]:=a[n]=Denominator[Sum[1/(Part[Dv[n],i]+1),{i,1,Length[Dv[n]]}]]
Do[Print[n," ",a[n]],{n,1,50}]

a(n) = denominator(sumdiv(n, d, 1/(d+1))); \\ Michel Marcus, Oct 15 2015

cp's OEIS Frontend

A263319 a(n) = pi(n^2)*phi(n)/2, where pi(x) denotes the number of primes not exceeding x, and phi(.) is Euler's totient function given by A000010.

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A263325 a(n) = sigma(n)*pi(n^2), where sigma(n) is the sum of all (positive) divisors of n, and pi(x) is the number of primes not exceeding x.

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A263321 Least positive integer m such that the numbers phi(k)*pi(k^2) (k = 1..n) are pairwise incongruent modulo m.

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A263326 Denominator of the rational number Sum_{d|n}1/(d+1).

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