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A173290 Partial sums of the Dedekind psi function (A001615). 2
1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, 132, 156, 180, 204, 222, 258, 278, 314, 346, 382, 406, 454, 484, 526, 562, 610, 640, 712, 744, 792, 840, 894, 942, 1014, 1052, 1112, 1168, 1240, 1282, 1378, 1422, 1494, 1566, 1638, 1686, 1782, 1838, 1928, 2000, 2084 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Partial sums of Dedekind psi function. There no primes in the sequence. The subsequence of semiprimes begins: 4, 14, x, 278, 314, 346, 382, 454, 526, 562, 1282, 1838, 2138, 2246. The subsequence of squares begins: 1, 4, 64, 484.

REFERENCES

W Hürlimann, Dedekind's arithmetic function and primitive four squares counting functions, Journal of Algebra, Number Theory: Advances and Applications, Volume 14, Number 2, 2015, Pages 73-88; http://scientificadvances.co.in; DOI: http://dx.doi.org/10.18642/jantaa_7100121599

LINKS

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

E. Pérez Herrero, Recycling Hardy & Wright, Average Order of Dedekind Psi Function, Psychedelic Geometry Blogspot.

FORMULA

a(n) = SUM[i=1..n] A001615(i) = SUM[i=1..n] (n * Product_{p|n, p prime} (1 + 1/p)).

a(n) = 15*n^2/(2*Pi^2) + O(n*log(n)). - Enrique Pérez Herrero, Jan 14 2012

a(n) = sum_{i=1..n} A063659(i) * floor(n/i). - Enrique Pérez Herrero, Feb 23 2013

EXAMPLE

a(69) = 1 + 3 + 4 + 6 + 6 + 12 + 8 + 12 + 12 + 18 + 12 + 24 + 14 + 24 + 24 + 24 + 18 + 36 + 20 + 36 + 32 + 36 + 24 + 48 + 30 + 42 + 36 + 48 + 30 + 72 + 32 + 48 + 48 + 54 + 48 + 72 + 38 + 60 + 56 + 72 + 42 + 96 + 44 + 72 + 72 + 72 + 48 + 96 + 56 + 90 + 72 + 84 + 54 + 108 + 72 + 96 + 80 + 90 + 60 + 144 + 62 + 96 + 96 + 96 + 84 + 144 + 68 + 108 + 96.

PROG

(Sage)

def A173290(n) :

    return add(k*mul(1+1/p for p in prime_divisors(k)) for k in (1..n))

[A173290(n) for n in (1..52)]  # Peter Luschny, Jun 10 2012

CROSSREFS

Cf. A001615, A063659.

Cf. A082020.

Cf. A175836 (partial products of the Dedekind psi function).

Sequence in context: A265284 A055507 A121896 * A131937 A183857 A088804

Adjacent sequences:  A173287 A173288 A173289 * A173291 A173292 A173293

KEYWORD

nonn

AUTHOR

Jonathan Vos Post, Feb 15 2010

STATUS

approved

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Last modified August 18 04:17 EDT 2017. Contains 290684 sequences.