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A145388 Sum of (k,n)_* for k=1,2,...,n, where (k,n)_* is the greatest divisor of k which is a unitary divisor of n. 4
1, 3, 5, 7, 9, 15, 13, 15, 17, 27, 21, 35, 25, 39, 45, 31, 33, 51, 37, 63, 65, 63, 45, 75, 49, 75, 53, 91, 57, 135, 61, 63, 105, 99, 117, 119, 73, 111, 125, 135, 81, 195, 85, 147, 153, 135, 93, 155, 97, 147, 165, 175, 105, 159 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A unitary analog of Pillai's function A018804; another unitary analog of A018804 is A089912.

The sequence is the row sums of the following triangle of (k,n)_* with rows n and columns 1 <= k <= n (R. J. Mathar, Jun 01 2011):

  1;

  1,  2;

  1,  1,  3;

  1,  1,  1,  4;

  1,  1,  1,  1,  5;

  1,  2,  3,  2,  1,  6;

  1,  1,  1,  1,  1,  1,  7;

  1,  1,  1,  1,  1,  1,  1,  8;

  1,  1,  1,  1,  1,  1,  1,  1,  9;

  1,  2,  1,  2,  5,  2,  1,  2,  1, 10;

  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 11;

  1,  1,  3,  4,  1,  3,  1,  4,  3,  1,  1, 12;

  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 13;

  1,  2,  1,  2,  1,  2,  7,  2,  1,  2,  1,  2,  1, 14;

Sum_{k<=x} a(n) = Ax^2 log x + O(x^2) with A = Product(1 - 1/(p+1)^2) * 3/Pi^2 = 0.23584030... where the product is over the primes. That is, the average value of a(n) is A n log n. - Charles R Greathouse IV, Mar 21 2012

LINKS

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

S. Chen, W. Zhai, Reciprocals of the Gcd-Sum Functions, J. Int. Seq. 14 (2011) # 11.8.3.

László Tóth, The unitary analogue of Pillai's arithmetical function, Collectanea Mathematica 40:1 (1989), pp. 19-30.

László Tóth, The unitary analogue of Pillai's arithmetical function II, Notes Number Theory Discrete Math. 2 (1996), no 2, 40-46.

László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009) 09.5.2.

László Tóth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010) # 10.8.1.

FORMULA

Multiplicative: a(p^e) = 2*p^e - 1 for every prime power p^e.

a(n) = Sum_{k=1..n} A034444(n/gcd(n,k)) = Sum_{d|n} A000010(d) * A034444(d). - Daniel Suteu, May 26 2019

a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * uphi(n/d), where uphi is A047994. - Amiram Eldar, May 29 2020

MAPLE

A145388 := proc(n) option remember; local pf, p ; if n = 1 then 1; else pf := ifactors(n)[2] ; if nops(pf) = 1 then 2*n-1 ; else mul(procname(op(1, p)^op(2, p)), p=pf) ; end if; end if; end proc:

seq(A145388(n), n=1..70) ; # R. J. Mathar, Jan 07 2011

MATHEMATICA

f[p_, e_] := 2*p^e - 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 29 2020 *)

PROG

(PARI) a(n)=n=factor(n); prod(i=1, #n[, 1], 2*n[i, 1]^n[i, 2]-1) \\ Charles R Greathouse IV, Mar 21 2012

CROSSREFS

Cf. A000010, A018804, A034444, A047994, A089912.

Sequence in context: A275254 A029608 A211135 * A268496 A121820 A258159

Adjacent sequences:  A145385 A145386 A145387 * A145389 A145390 A145391

KEYWORD

mult,nonn

AUTHOR

Laszlo Toth, Oct 10 2008

STATUS

approved

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Last modified August 8 08:27 EDT 2020. Contains 336293 sequences. (Running on oeis4.)