A063986 -id:A063986 - cp's OEIS frontend

A073128 Integer quotients arising in A063986.

0, 1, 1, 5, 5, 49, 118, 121, 2406, 2698, 4182, 4946, 153627, 3087192, 8203485, 38394376, 487844934, 2822741576, 4140154385, 4397137572, 8583966231

Offset: 1

Labos Elemer, Jul 16 2002

			n=15, A063986(15)=41846733, sum of first 41846733 cototients is s=343289046464505, a(15)=s/41846733. Only in knowledge of either the quotients or of partial sums of cototients is it possible to continue A063986 without recomputing previous terms!

Cf. A051953, A000010, A002088, A048920, A000217, A050226, A063986.

a(n)=Sum[cototient[j], j=1..A063986(n)]/A063986(n)

Changed A063896 to A063986 and a(16)-a(21) from Donovan Johnson, May 11 2010

A073129 Partial sums of cototients arising in A063986.

Original entry on oeis.org

0, 4, 5, 120, 125, 12201, 70800, 74657, 29526432, 37132574, 89210424, 124777688, 120392102955, 48617257062792, 343289046464505, 7519663359716376, 1214022599940709056, 40644839476190305216, 87437200646372849005, 98628693371623948080, 375871306587181970568

Offset: 1

Labos Elemer, Jul 16 2002

nonn

			n=15, a(15)=343289046464505, sum of first 41846733 cototients A063986(15)*A073128(15)=a(15) To continue A063986, A073128 or A073129 without recomputing previous terms, corresponding entries from 2 of above sequences is required.

Cf. A051953, A000010, A002088, A048920, A000217, A050226, A073128, A063986.

a(n)=Sum[cototient[j], j=1..A063986(n)]

Changed A063896 to A063986 and a(16)-a(21) from Donovan Johnson, May 11 2010

A051953 Cototient(n) := n - phi(n).

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 4, 3, 6, 1, 8, 1, 8, 7, 8, 1, 12, 1, 12, 9, 12, 1, 16, 5, 14, 9, 16, 1, 22, 1, 16, 13, 18, 11, 24, 1, 20, 15, 24, 1, 30, 1, 24, 21, 24, 1, 32, 7, 30, 19, 28, 1, 36, 15, 32, 21, 30, 1, 44, 1, 32, 27, 32, 17, 46, 1, 36, 25, 46, 1, 48, 1, 38, 35, 40, 17, 54, 1, 48, 27

Offset: 1

Labos Elemer, Dec 21 1999

Unlike totients, cototient(n+1) = cototient(n) never holds -- except 2-phi(2) = 3 - phi(3) = 1 -- because cototient(n) is congruent to n modulo 2. - Labos Elemer, Aug 08 2001
Theorem (L. Redei): b^a(n) == b^n (mod n) for every integer b. - Thomas Ordowski and Robert Israel, Mar 11 2016
Let S be the sum of the cototients of the divisors of n (A001065). S < n iff n is deficient, S = n iff n is perfect, and S > n iff n is abundant. - Ivan N. Ianakiev, Oct 06 2023

			n = 12, phi(12) = 4 = |{1, 5, 7, 11}|, a(12) = 12 - phi(12) = 8, numbers not exceeding 12 and not coprime to 12: {2, 3, 4, 6, 8, 9, 10, 12}.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Browkin and A. Schinzel, On integers not of the form n-phi(n), Colloq. Math., 68 (1995), 55-58.
R. E. Jamison, The Helly bound for singular sums, Discrete Math., 249 (2002), 117-133.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, Trans. Amer. Math. Soc. Ser. B 3 (2016), 1-26. For Richard Guy on his 99th birthday. May his sequence be unbounded.
Carl Pomerance and Hee-Sung Yang, Variant of a theorem of Erdős on the sum-of-proper-divisors function, Math. Comp. 83 (2014), 1903-1913.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Eric Weisstein's World of Mathematics, Cototient

Cf. A000010, A001065 (inverse Möbius transform), A005278, A001274, A083254, A098006, A049586, A051612, A053579, A054525, A062790 (Möbius transform), A063985 (partial sums), A063986, A290087.
Records: A065385, A065386.
Number of zeros in the n-th row of triangle A054521. - Omar E. Pol, May 13 2016
Cf. A063740 (number of k such that cototient(k) = n). - M. F. Hasler, Jan 11 2018

a051953 n = n - a000010 n  -- Reinhard Zumkeller, Jan 21 2014

with(numtheory); A051953 := n->n-phi(n);

Table[n - EulerPhi[n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

A051953(n) = n - eulerphi(n); \\ Michael B. Porter, Jan 28 2010

from sympy.ntheory import totient
print([i - totient(i) for i in range(1, 101)]) # Indranil Ghosh, Mar 17 2017

a(n) = n - A000010(n).
Equals Mobius transform (A054525) of A001065. - Gary W. Adamson, Jul 11 2008
a(A006881(n)) = sopf(A006881(n)) - 1; a(A000040(n)) = 1. - Wesley Ivan Hurt, May 18 2013
G.f.: sum(n>=1, A000010(n)*x^(2*n)/(1-x^n) ). - Mircea Merca, Feb 23 2014
From Ilya Gutkovskiy, Apr 13 2017: (Start)
G.f.: -Sum_{k>=2} mu(k)*x^k/(1 - x^k)^2.
Dirichlet g.f.: zeta(s-1)*(1 - 1/zeta(s)). (End)
From Antti Karttunen, Sep 05 2018 & Apr 29 2022: (Start)
Dirichlet convolution square of A317846/A046644 gives this sequence + A063524.
a(n) = A003557(n) * A318305(n).
a(n) = A000010(n) - A083254(n).
a(n) = A318325(n) - A318326(n).
a(n) = Sum_{d|n} A062790(d) = Sum_{d|n, dA007431(d)*(A000005(n/d)-1).
a(n) = A048675(A318834(n)) = A276085(A353564(n)). [These follow from the formula below]
a(n) = Sum_{d|n, dA000010(d).
a(n) = A051612(n) - A001065(n).
(End)

A048290 Numbers m that divide Sum_{k=1..m} phi(k).

Original entry on oeis.org

1, 2, 5, 6, 16, 25, 36, 249, 617, 1296, 13763, 76268, 189074, 783665, 1102394, 3258466, 3808854, 7971034, 15748051, 27746990, 41846733, 153673168, 195853251, 302167272, 402296412, 732683468, 807656448, 844492262, 848152352, 1122039882, 2258200198, 2438160726

Offset: 1

David J. Rusin

The odd terms of this sequence and A063986 are the same. - Jud McCranie, Jun 26 2005

			Euler sums are *1*, *2*, 4, 6, *10*, *12*, ..., *80*, ..., *510624*,... for n=1, 2, 3, 4, 5, 6, ..., 16, ...., 1296, ...

Donovan Johnson, Table of n, a(n) for n = 1..37 (terms < 10^12)
Edward A. Bender, Oren Patashnik and Howard Rumsey, Jr., Pizza Slicing, Phi's and the Riemann Hypothesis, American Mathematical Monthly, Vol. 101 (1994), pp. 307-317.
D. Rusin, Euler phi function

Cf. A000010, A002088. See A063986 for n divides Sum_{k=1..n} k-phi(k).

s = 0; Do[s = s + EulerPhi[n]; If[IntegerQ[s/n], Print[n]], {n, 1, 10^8}]

list(lim)=my(v=List(),s); for(k=1,lim, s+=eulerphi(k); if(s%k==0, listput(v, k))); Vec(v) \\ Charles R Greathouse IV, Feb 07 2017

Sum_{k=1..m} phi(k) is about (3/Pi^2)*m^2 [cf. A002088, first formula].

Not obviously infinite; rough heuristics predict about 3/2 log(N) terms less than N, log(N) even ones and log(N)/2 odd ones.

10 more terms computed by Dean Hickerson
One more term from Robert G. Wilson v, Sep 07 2001
More terms from Naohiro Nomoto, Mar 22 2002
5 more terms from Jud McCranie, Jun 21 2005

A064610 Places k where A064608(k) (partial sums of unitary tau) is divisible by k.

Original entry on oeis.org

1, 35, 37, 1015, 27417, 27421, 27449, 27453, 19774739, 530743781, 530743799, 530743807, 530743813

Offset: 1

Labos Elemer, Sep 24 2001

The corresponding quotients are 1, 3, 3, 5, 7, 7, 7, 7, 11, 13, 13, 13, 13, ...

a(14) > 7.5*10^10, if it exists. - Amiram Eldar, Jun 04 2021

			For n = 37, the sum A064608(37) = 1+2+2+2+2+4+2+...+4+4+4+2 = 111 = 3*37, so 37 is in the sequence.

Cf. A064608.

Analogous "integer-mean" sequences for various arithmetical functions are A050226, A056650, A064605, A064606, A064607, A048290, A063986, A063971, A064911, A062982, A045345.

s[1] = 1; s[n_] := s[n] = s[n - 1] + 2^PrimeNu[n]; Select[Range[30000], Divisible[s[#], #] &] (* Amiram Eldar, Jun 04 2021 *)

{n: A064608(n) == 0 (mod n)}.

a(10)-a(13) from Donovan Johnson, Jul 20 2012

cp's OEIS Frontend

A073128 Integer quotients arising in A063986.

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A073129 Partial sums of cototients arising in A063986.

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A051953 Cototient(n) := n - phi(n).

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A048290 Numbers m that divide Sum_{k=1..m} phi(k).

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A064610 Places k where A064608(k) (partial sums of unitary tau) is divisible by k.

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