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A177754
Partial sums of A047994.
5
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
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
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, May 12 2010
STATUS
approved