A318662 -id:A318662 - cp's OEIS frontend

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

Labos Elemer, Jun 07 2000

Phi-summation over d-s if runs over all divisors is n, so these values do not exceed 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. - Steven Finch, 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. - Laszlo Toth, Sep 04 2008
Equals row sums of triangle A157361 and inverse Mobius transform of A114810. - Gary W. Adamson, Feb 28 2009
a(m) = m iff m is squarefree, a(A005117(n)) = A005117(n). - Reinhard Zumkeller, Mar 11 2012
Apostol & Tóth call this ϱ(n), i.e., varrho(n). - Charles R Greathouse IV, Apr 23 2013

			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.

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.

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.

Cf. A000010, A053570, A053571, A000188, A008833, A055654, A157361, A114810, A000010, A077610, A318661, A318662.

a055653 = sum . map a000010 . a077610_row
-- Reinhard Zumkeller, Mar 11 2012

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:

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 *)

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

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

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
Dirichlet convolution square of A318661(n)/A318662(n). - Antti Karttunen, Sep 03 2018
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

A318314 Denominators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

Original entry on oeis.org

1, 2, 1, 8, 1, 2, 1, 16, 2, 2, 1, 8, 1, 2, 1, 128, 1, 4, 1, 8, 1, 2, 1, 16, 2, 2, 2, 8, 1, 2, 1, 256, 1, 2, 1, 16, 1, 2, 1, 16, 1, 2, 1, 8, 2, 2, 1, 128, 2, 4, 1, 8, 1, 4, 1, 16, 1, 2, 1, 8, 1, 2, 2, 1024, 1, 2, 1, 8, 1, 2, 1, 32, 1, 2, 2, 8, 1, 2, 1, 128, 8, 2, 1, 8, 1, 2, 1, 16, 1, 4, 1, 8, 1, 2, 1, 256, 1, 4, 2, 16, 1, 2, 1, 16, 1

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

Note that A318314 differs from A318454 at exactly those n where A001227 differs from A068068, the numbers in A038838. - Antti Karttunen, Sep 07 2018

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A005187, A011371, A068068, A318313 (numerators), A318315.

Cf. also A006519, A046644, A299150, A317932, A317934, A317940, A318454, A318662.

a35[n_] := (1 - (-1)^n)/2;
a120[n_] := DigitCount[n, 2, 1];
a[n_] := Product[{p, e} = pe; 2^(((2 - a35[p])*e) - a120[e]), {pe, FactorInteger[n]}];
a /@ Range[100] (* Jean-François Alcover, Sep 19 2019 *)

up_to = 16384;
A068068(n) = (2^omega(n>>valuation(n, 2))); \\ From A068068
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA317937.
v318313_15 = DirSqrt(vector(up_to, n, A068068(n)));
A318313(n) = numerator(v318313_15[n]);
A318314(n) = denominator(v318313_15[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A068068(n) - Sum_{d|n, d>1, d 1.
a(n) = 2^A318315(n).
From Antti Karttunen, Sep 03-07 2018: (Start, conjectured formulas)
a(n) = A006519(n) * A317934(n), thus multiplicative with a(2^e) = 2^A005187(e), a(p^e) = 2^A011371(e) for odd primes p.
Equally, multiplicative with a(p^e) = 2^(((2-A000035(p))*e)-A000120(e)) for all primes p.
(End)

A318661 Numerators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 3, 19, 5, 11, 3, 13, 7, 15, 5, 17, 19, 19, 5, 21, 11, 23, 9, 59, 13, 95, 7, 29, 15, 31, 9, 33, 17, 35, 19, 37, 19, 39, 15, 41, 21, 43, 11, 95, 23, 47, 15, 123, 59, 51, 13, 53, 95, 55, 21, 57, 29, 59, 15, 61, 31, 133, 67, 65, 33, 67, 17, 69, 35, 71, 57, 73, 37, 177, 19, 77, 39, 79, 25, 2019, 41, 83, 21, 85, 43, 87, 33, 89

Offset: 1

Antti Karttunen, Sep 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A055653, A318662 (denominators).

up_to = 1+(2^16);
A055653(n) = sumdiv(n, d, if(gcd(n/d, d)==1, eulerphi(d))); \\ From A055653
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA055653(n)));
A318661(n) = numerator(v318661_62[n]);
A318662(n) = denominator(v318661_62[n]);
A318663(n) = valuation(A318662(n),2);

for(n=1, 100, print1(numerator(direuler(p=2, n, ((1 + X^2 - p*X^2 - X)/((1-X)*(1-p*X)))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 10 2025

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A055653(n) - Sum_{d|n, d>1, d 1.
From Vaclav Kotesovec, May 10 2025: (Start)
Let f(s) = Product_{primes p} (1 + 1/p^(2*s) - 1/p^(2*s-1) - 1/p^s).
Sum_{k=1..n} A318661(k) / A318662(k) ~ n^2 * sqrt(Pi*f(2)/(24*log(n))) * (1 - ((gamma - 1)/2 + f'[2]/(2*f(2)) + 3*zeta'(2)/Pi^2) / (2*log(n))), where
f(2) = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = A330523 = 0.5358961538283379998085026313185459506482223745141452711510108346133288119...
f'(2)/f(2) = Sum_{primes p} (p^2 + 2*p - 2) * log(p) / (p^4 - p^2 - p + 1) = 0.8249574883141571786856463180997569604486048593127391054584235479395133668...
and gamma is the Euler-Mascheroni constant A001620. (End)

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A318663 The 2-adic valuation of A318662.

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A055653 Sum of phi(d) [A000010] over all unitary divisors d of n (that is, gcd(d,n/d) = 1).

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A318314 Denominators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

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A318661 Numerators of the sequence whose Dirichlet convolution with itself yields A055653, sum of phi(d) over all unitary divisors d of n.

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