A055653 Sum of phi(d) [A000010] over all unitary divisors d of n (that is, gcd(d,n/d) = 1).
1, 2, 3, 3, 5, 6, 7, 5, 7, 10, 11, 9, 13, 14, 15, 9, 17, 14, 19, 15, 21, 22, 23, 15, 21, 26, 19, 21, 29, 30, 31, 17, 33, 34, 35, 21, 37, 38, 39, 25, 41, 42, 43, 33, 35, 46, 47, 27, 43, 42, 51, 39, 53, 38, 55, 35, 57, 58, 59, 45, 61, 62, 49, 33, 65, 66, 67, 51, 69, 70, 71, 35, 73
Offset: 1
Examples
n=1260 has 36 divisors of which 16 are unitary ones: {1,4,5,7,9,20,28,35,36,45,63,140,180,252,315,1260}.
EulerPhi values of these divisors are: {1,2,4,6,6,8,12,24,12,24,36,48,48,72,144,288}.
The sum is 735, thus a(1260)=735.
Or, 1260=2^2*3^2*5*7, thus a(1260) = (1 + 2^2 - 2)*(1 + 3^2 - 3)*(1 + 5 - 5^0)*(1 + 7 - 7^0) = 735.
References
- J. Morgado, Inteiros regulares módulo n, Gazeta de Matematica (Lisboa), 33 (1972), no. 125-128, 1-5. [From Laszlo Toth, Sep 04 2008]
- J. Morgado, A property of the Euler phi-function concerning the integers which are regular modulo n, Portugal. Math., 33 (1974), 185-191.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from T. D. Noe)
- Osama Alkam and Emad Abu Osba, On the regular elements in Zn, Turk J Math, 32 (2008), 31-39.
- B. Apostol and L. Petrescu, Extremal Orders of Certain Functions Associated with Regular Integers (mod n), Journal of Integer Sequences, 2013, # 13.7.5.
- Brăduţ Apostol and László Tóth, Some remarks on regular integers modulo n, arXiv:1304.2699 [math.NT], 2013.
- Klaus Dohmen, On the Number of Regular Elements in Zn, arXiv:2304.02471 [math.CO], 2023.
- S. R. Finch, Idempotents and nilpotents modulo n, arXiv:1304.2699 [math.NT], 2013.
- V. S. Joshi, Order-free integers (mod m), Number Theory (Mysore, 1981), Lect. Notes in Math. 938, Springer-Verlag, 1982, pp. 93-100.
- Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^2 / 12) for n = 1..100000
- Sagar Mandal, Divisibility and Sequence Properties of sigma+ and phi+, arXiv:2508.11660 [math.GM], 2025.
- József Sándor and Krassimir Atanassov, Some new arithmetic functions, Notes on Number Theory and Discrete Mathematics, Volume 30, 2024, Number 4, Pages 851-856. See phi+ function.
- L. Tóth, Regular integers modulo n, arXiv:0710.1936 [math.NT], 2007-2008; Annales Univ. Sci. Budapest., Sect. Comp., 29 (2008), 263-275.
- L. Tóth, A gcd-sum function over regular integers modulo n, JIS 12 (2009) 09.2.5.
Crossrefs
Programs
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Haskell
a055653 = sum . map a000010 . a077610_row -- Reinhard Zumkeller, Mar 11 2012
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Maple
A055653 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ] [ 1 ]^ifactors(n)[ 2 ] [ i ] [ 2 ]-ifactors(n)[ 2 ][ i ] [ 1 ]^(ifactors(n)[ 2 ] [ i ] [ 2 ]-1)): od: RETURN(ans) end:
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Mathematica
a[n_] := Total[EulerPhi[Select[Divisors[n], GCD[#, n/#] == 1 &]]]; Array[a, 73] (* Jean-François Alcover, May 03 2011 *) f[p_, e_] := p^e - p^(e-1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)
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PARI
a(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ Charles R Greathouse IV, Feb 19 2013, corrected by Antti Karttunen, Sep 03 2018
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PARI
a(n)=my(f=factor(n));prod(i=1,#f[,1],f[i,1]^f[i,2]-f[i,1]^(f[i,2]-1)+1) \\ Charles R Greathouse IV, Feb 19 2013
Formula
If n = product p_i^e_i, a(n) = product (1+p_i^e_i-p_i^(e_i-1)). - Vladeta Jovovic, Apr 19 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)*product_{primes p} (1+p^(-2s)-p^(1-2s)-p^(-s)). - R. J. Mathar, Oct 24 2011
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.535896... - Vaclav Kotesovec, Dec 17 2019
Comments