The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A173557 a(n) = Product_{primes p dividing n} (p-1). 84
 1, 1, 2, 1, 4, 2, 6, 1, 2, 4, 10, 2, 12, 6, 8, 1, 16, 2, 18, 4, 12, 10, 22, 2, 4, 12, 2, 6, 28, 8, 30, 1, 20, 16, 24, 2, 36, 18, 24, 4, 40, 12, 42, 10, 8, 22, 46, 2, 6, 4, 32, 12, 52, 2, 40, 6, 36, 28, 58, 8, 60, 30, 12, 1, 48, 20, 66, 16, 44, 24, 70, 2, 72, 36 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS This is A023900 without the signs. - T. D. Noe, Jul 31 2013 Numerator of c_n = Product_{odd p| n} (p-1)/(p-2). Denominator is A305444. The initial values c_1, c_2, ... are 1, 1, 2, 1, 4/3, 2, 6/5, 1, 2, 4/3, 10/9, 2, 12/11, 6/5, 8/3, 1, 16/15, ... [Yamasaki and Yamasaki]. - N. J. A. Sloane, Jan 19 2020 Kim et al. (2019) named this function the absolute Möbius divisor function. - Amiram Eldar, Apr 08 2020 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..65536 (first 1000 terms from T. D. Noe) Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Certain combinatoric convolution sums arising from Bernoulli and Euler Polynomials, Miskolc Mathematical Notes, No. 20, Vol. 1 (2019): pp. 311-330. Daeyeoul Kim, Umit Sarp, and Sebahattin Ikikardes, Iterating the Sum of Möbius Divisor Function and Euler Totient Function, Mathematics, Vol. 7, No. 11 (2019), pp. 1083-1094. Yamasaki, Yasuo, and Aiichi Yamasaki, On the Gap Distribution of Prime Numbers, Kyoto University Research Information Repository, October 1994. MR1370273 (97a:11141). FORMULA a(n) = A003958(n) iff n is squarefree. a(n) = |A023900(n)|. Multiplicative with a(p^e) = p-1, e >= 1. - R. J. Mathar, Mar 30 2011 a(n) = phi(rad(n)) = A000010(A007947(n)). - Enrique Pérez Herrero, May 30 2012 a(n) = A000010(n) / A003557(n). - Jason Kimberley, Dec 09 2012 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s) + p^(1-s)). The Dirichlet inverse is multiplicative with b(p^e) = (1 - p) * (2 - p)^(e - 1) = Sum_k A118800(e, k) * p^k. - Álvar Ibeas, Nov 24 2017 a(1) = 1; for n > 1, a(n) = (A020639(n)-1) * a(A028234(n)). - Antti Karttunen, Nov 28 2017 From Vaclav Kotesovec, Jun 18 2020: (Start) Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(2*s-2) * Product_{p prime} (1 - 2/(p + p^s)). Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A307868 = Product_{p prime} (1 - 2/(p*(p+1))) = 0.471680613612997868... (End) a(n) = (-1)^A001221(n)*A023900(n). - M. F. Hasler, Aug 14 2021 EXAMPLE 300 = 3*5^2*2^2 => a(300) = (3-1)*(2-1)*(5-1) = 8. MAPLE A173557 := proc(n) local dvs; dvs := numtheory[factorset](n) ; mul(d-1, d=dvs) ; end proc: # R. J. Mathar, Feb 02 2011 # second Maple program: a:= n-> mul(i[1]-1, i=ifactors(n)[2]): seq(a(n), n=1..80); # Alois P. Heinz, Aug 27 2018 MATHEMATICA a[n_] := Module[{fac = FactorInteger[n]}, If[n==1, 1, Product[fac[[i, 1]]-1, {i, Length[fac]}]]]; Table[a[n], {n, 100}] PROG (Haskell) a173557 1 = 1 a173557 n = product \$ map (subtract 1) \$ a027748_row n -- Reinhard Zumkeller, Jun 01 2015 (PARI) a(n) = my(f=factor(n)[, 1]); prod(k=1, #f, f[k]-1); \\ Michel Marcus, Oct 31 2017 (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 18 2020 (PARI) apply( {A173557(n)=vecprod([p-1|p<-factor(n)[, 1]])}, [1..77]) \\ M. F. Hasler, Aug 14 2021 (Scheme, with memoization-macro definec) (definec (A173557 n) (if (= 1 n) 1 (* (- (A020639 n) 1) (A173557 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017 (Magma) [EulerPhi(n)/(&+[(Floor(k^n/n)-Floor((k^n-1)/n)): k in [1..n]]): n in [1..100]]; // Vincenzo Librandi, Jan 20 2020 CROSSREFS Cf. A023900, A141564, A027748, A305444, A307868. Sequence in context: A300234 A070777 A173614 * A023900 A141564 A239641 Adjacent sequences: A173554 A173555 A173556 * A173558 A173559 A173560 KEYWORD nonn,easy,mult AUTHOR José María Grau Ribas, Feb 21 2010 EXTENSIONS Definition corrected by M. F. Hasler, Aug 14 2021 Incorrect formula removed by Pontus von Brömssen, Aug 15 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 10 02:09 EST 2022. Contains 358712 sequences. (Running on oeis4.)