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

1,2

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

Links

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

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.

Example

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.

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}]

Prog

(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

Crossrefs

Cf. A006218. Row sums of A122191.

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

Sequence in context: A109534 A276163 A011341 * A116920 A116919 A270429

Adjacent sequences: A116474 A116475 A116476 * A116478 A116479 A116480

Keyword

easy,nonn

Author

Leroy Quet, Mar 18 2006

Extensions

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

Status

approved