A116477 a(n) = Sum_{1<=k<=n, gcd(k,n)=1} floor(n/k).

1, 2, 4, 5, 9, 7, 15, 12, 18, 15, 28, 16, 36, 23, 31, 30, 51, 26, 59, 34, 50, 43, 75, 37, 77, 52, 72, 55, 102, 42, 112, 69, 90, 73, 106, 61, 141, 84, 109, 80, 159, 66, 169, 97, 119, 108, 187, 84, 185, 103, 155, 121, 218, 97, 193, 126, 179, 142, 248, 95, 262, 152, 185

Offset: 1

Author

Leroy Quet, Mar 18 2006

Keywords

• easy
• nonn

Comments

sum{k|n} a(k) = sum{k=1 to n} d(k), where d(k) is the number of positive divisors of k.

Equals A054525 * A006218 (Mobius transform of A006218). - Gary W. Adamson, Aug 07 2008

Examples

			a(6)=7 because the numbers relatively prime to 6 and not exceeding 6 are 1 and 5, yielding floor(6/1) + floor(6/5) = 7.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006218. Row sums of A122191.

Cf. A054525. - Gary W. Adamson, Aug 07 2008

Programs

Maple

a:=proc(n) local s,j: s:=0: for j from 1 to n do if gcd(j,n)=1 then s:=s+floor(n/j) else s:=s: fi od: s: end: seq(a(n),n=1..75);

Mathematica

Table[a := Select[Range[n], GCD[n, # ] == 1 &]; Sum[Floor[n/a[[i]]], {i, 1, Length[a]}], {n, 1, 60}]

PARI

A116477(n) = sum(k=1,n,(gcd(k,n)==1)*floor(n/k)) \\ Michael B. Porter, Mar 01 2010

PARI

A006218(n)=sum(k=1, sqrtint(n), n\k)*2-sqrtint(n)^2
a(n,f=factor(n))=my(K=1,N); for(i=1,#f~, if(f[i,2]>1, K*=f[i,1]^(f[i,2]-1); f[i,2]=1)); sumdiv(N=n/K,k,moebius(N/k)*A006218(k*K)) \\ Charles R Greathouse IV, Nov 30 2021

Formula

a(n) is also Sum_{k|n} mu(n/k) (Sum_{j=1..k} d(j)) and Sum_{k=1..n} phi(n,n/k), where mu() is the Mobius (Moebius) function, d(j) is the number of positive divisors of j and phi(n,x) is the number of positive integers which are <= x and are coprime to n.

Extensions

More terms from Emeric Deutsch and Stefan Steinerberger, Apr 01 2006