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Search: id:A055653
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| A055653 |
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Sum of phi(d) [A000010] over all unitary divisors d of n (that is, GCD(d,n/d) = 1). |
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11
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| 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Phi-summation over d-s if runs over all divisors is n, so these values are not exceeding n. Compare also other "Phi-summations" like A053570, A053571, or distinct primes dividing n, etc.
a(n) is also the number of solutions of x^(k+1)=x mod n for some k>=1. - S. R. Finch (Steven.Finch(AT)inria.fr), Apr 11 2006
An integer a is called regular (mod n) if there is an integer x such that a^2 x == a (mod n). Then a(n) is also the number of regular integers a (mod n) such that 1 <= a <= n. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), Sep 04 2008]
Equals row sums of triangle A156361 and inverse Mobius transform (A051731) of A114810. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 28 2009]
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REFERENCES
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V. S. Joshi, Order-free integers (mod m), Number Theory (Mysore, 1981), Lect. Notes in Math. 938, Springer-Verlag, 1982, pp. 93-100.
J. Morgado, Inteiros regulares m\'odulo n, Gazeta de Matematica (Lisboa), 33 (1972), no. 125-128, 1-5. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), 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. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), Sep 04 2008]
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
S. R. Finch, Idempotents and Nilpotents Modulo n (arXiv:math.NT/0605019)
L. T\'oth, Regular integers modulo $n$, Annales Univ. Sci. Budapest., Sect. Comp., 29 (2008), 263-275. [From Laszlo Toth (ltoth(AT)ttk.pte.hu), Sep 04 2008]
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FORMULA
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If n = Product p_i^e_i, a(n) = Product (1+p_i^e_i-p_i^(e_i-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 19 2001
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EXAMPLE
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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.
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MAPLE
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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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CROSSREFS
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Cf. A000010, A053570, A053571, A000188, A006833, A055654.
A157361, A114810 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 28 2009]
Sequence in context: A052274 A085314 A085310 this_sequence A155918 A097248 A097247
Adjacent sequences: A055650 A055651 A055652 this_sequence A055654 A055655 A055656
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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Labos E. (labos(AT)ana.sote.hu), Jun 07 2000
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