A177754 Partial sums of A047994.

1, 2, 4, 7, 11, 13, 19, 26, 34, 38, 48, 54, 66, 72, 80, 95, 111, 119, 137, 149, 161, 171, 193, 207, 231, 243, 269, 287, 315, 323, 353, 384, 404, 420, 444, 468, 504, 522, 546, 574, 614, 626, 668, 698, 730, 752, 798, 828, 876, 900, 932, 968, 1020, 1046, 1086

Offset

1,2

Comments

Partial sums of unitary totient (or unitary phi) function uphi(n). This is to A047994 as A002088 is to A000010. The subsequence of primes in the partial sum begins: 2, 7, 11, 13, 19, 137, 149, 193, 269, 353, 1523, 1543, 1609, 1657.

Links

Table of n, a(n) for n=1..55.

Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.

R. Sitaramachandrarao and D. Suryanarayana, On Sum_{n<=x} sigma*(n) and Sum_{n<=x} phi*(n), Proceedings of the American Mathematical Society, Vol. 41, No. 1 (1973), pp. 61-66.

Formula

a(n) = Sum_{i=1..n} A047994(i).

a(n) ~ alpha * n^2/2 + O(n*log^2(n)) where alpha = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). - Amiram Eldar, Dec 18 2018

Example

a(7) = 1 + 1 + 2 + 3 + 4 + 2 + 6 = 19.

Mathematica

uphi[1] = 1; uphi[n_] := Times @@ (-1 + Power @@@ FactorInteger[n]); s = 0; Accumulate[Array[uphi, 60]] (* Amiram Eldar, Dec 18 2018*)

Crossrefs

Cf. A000010, A002088, A003271, A047994, A065463.

Sequence in context: A370905 A362946 A345983 * A086795 A064690 A138766

Adjacent sequences: A177751 A177752 A177753 * A177755 A177756 A177757

Keyword

easy,nonn

Author

Jonathan Vos Post, May 12 2010

Status

approved