A083255 -id:A083255 - cp's OEIS frontend

A083254 a(n) = 2*phi(n) - n.

1, 0, 1, 0, 3, -2, 5, 0, 3, -2, 9, -4, 11, -2, 1, 0, 15, -6, 17, -4, 3, -2, 21, -8, 15, -2, 9, -4, 27, -14, 29, 0, 7, -2, 13, -12, 35, -2, 9, -8, 39, -18, 41, -4, 3, -2, 45, -16, 35, -10, 13, -4, 51, -18, 25, -8, 15, -2, 57, -28, 59, -2, 9, 0, 31, -26, 65, -4, 19, -22, 69, -24, 71, -2, 5, -4, 43, -30, 77, -16, 27, -2, 81, -36, 43, -2, 25

Offset: 1

Labos Elemer, May 08 2003

Möbius transform of A033879, deficiency of n. - Antti Karttunen, Dec 26 2017

			Case 1# - totient(x)-cototient[x] = 0 if x is a power of 2;
Case 2# - totient(x)>cototient[x] gives odd primes and also A067800, (= A014076 except probably A036798); e.g. n = 33: a(33) = 2.20-33 = 7; n = p prime: a(p) = p-2;
Case 3# - totient(x)<cototient[x] gives even numbers without powers of 2 and most probably A036798; e.g. n = 20: a(20) = -4; n = 105: a(105) = 2.48-105 = 96-105 = -9.

R. J. Mathar, Table of n, a(n) for n = 1..10000

Cf. A000010, A051953, A000079, A014076, A033879, A067800, A036798, A083255, A115405, A293516, A297114, A297115, A297153, A297154.

A083254 := proc(n)
    2*numtheory[phi](n)-n ;
end proc: # R. J. Mathar, Jan 13 2014

Mathematica
```
Table[2*EulerPhi[w]-w, {w, 1, 1000}]
```

a(n)=2*eulerphi(n)-n \\ Charles R Greathouse IV, Feb 21 2013

a(n) = totient(n) - cototient(n) = A000010(n) - A051953(n).
From Antti Karttunen, Dec 26 2017: (Start)
a(n) = A065620(A297153(n)) = A117966(A297154(n)).
a(n) = A297114(n) + A297115(n).
a(2n) = A297114(2n).
For all n >= 1, -a(A000010(n)) = A293516(n).
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where  c = 6/Pi^2 - 1/2 = 0.107927... . - Amiram Eldar, Sep 07 2023

A083260 a(n) = gcd(A046523(n), A071364(n)).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 6, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 6, 6, 12, 2, 6, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, 12, 6, 30, 2, 72, 2, 6, 6, 12, 6, 30, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24, 2, 30, 6

Offset: 1

Labos Elemer, May 09 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A046523, A071364, A083255, A083256, A083257, A083258, A083259, A083261.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A071364(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = prime(i)); factorback(f); }; \\ This function from Michel Marcus, Jun 13 2014
A083260(n) = gcd(A046523(n), A071364(n)); \\ Antti Karttunen, Aug 24 2017

A083258 a(n) = gcd(A046523(n), n).

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 3, 16, 1, 6, 1, 4, 3, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 48, 1, 2, 3, 4, 1, 6, 1, 8, 3, 2, 1, 60, 1, 2, 3, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 3, 8, 1, 30, 1, 4, 3, 2, 1, 96, 1, 2

Offset: 1

Labos Elemer, May 09 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A046523, A071364, A083255-A083260, A083261.

Table[GCD[n, Times @@ MapIndexed[Prime[First[#2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]], {n, 98}] (* Michael De Vlieger, May 21 2017 *)

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A083258(n) = gcd(n,A046523(n)); \\ Antti Karttunen, May 21 2017

A083259 a(n) = gcd(n, A071364(n)), where A071364(n) is the smallest number with same sequence of exponents in canonical prime factorization as n.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 3, 16, 1, 18, 1, 4, 3, 2, 1, 24, 1, 2, 1, 4, 1, 30, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 3, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 60, 1, 2, 3, 64, 1, 6, 1, 4, 3, 10, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 1, 2, 1, 12, 1, 2, 3, 8, 1, 90, 1, 4, 3, 2, 1, 96, 1

Offset: 1

Labos Elemer, May 09 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A007395, A002808, A046523, A071364, A083255-A083260, A322361, A330749.

A071364(n) = { my(f=factor(n)); for(i=1, #f~, f[i, 1] = prime(i)); factorback(f); }; \\ From A071364
A083259(n) = gcd(A071364(n),n); \\ Antti Karttunen, Jan 22 2020

A083261 a(n) = gcd(A046523(n+1), A046523(n)).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 6, 6, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 6, 6, 6, 2, 2, 6, 6, 2, 2, 2, 2, 12, 6, 2, 2, 4, 4, 6, 6, 2, 2, 6, 6, 6, 6, 2, 2, 2, 2, 6, 4, 2, 6, 2, 2, 6, 6, 2, 2, 2, 2, 6, 12, 6, 6, 2, 2, 16, 2, 2, 2, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 2, 2, 12, 12, 2, 2, 2, 2

Offset: 1

Labos Elemer, May 09 2003

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A046523, A071364, A083255, A083256, A083257, A083258, A083259, A083260, A286255.

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A083261(n) = gcd(A046523(n),A046523(1+n)); \\ From Antti Karttunen, May 21 2017

A233466 Numbers k such that phi(k) = (k-5)/2.

Original entry on oeis.org

165, 64005, 6992962170388485, 18446744047939747845

Offset: 1

Farideh Firoozbakht, Dec 26 2013

According to the following theorem I discovered recently, the 20-digit number 18446744047939747845 is in the sequence.
Theorem: If k and m are integers, k < 6, and p = 2^2^k + m is a prime such that p does not divide 2^2^k-1, then p*(2^2^k-1) is a solution to the equation phi(x) = (x+m)/2.
Note that m cannot be -1 and for k < 6, 2^2^k-1 is the product of the first k Fermat primes.
Take m=-5; since 2^2^2-5, 2^2^3-5 and 2^2^5-5 are prime we get three terms of the sequence.
Take m=1; since 2^2^0+1, 2^2^1+1, 2^2^2+1, 2^2^3+1 and 2^2^4+1 are prime (Fermat primes) we get five terms of the sequence A050474.
Conjecture (i): There is no solution to the equation phi(x) = (x-1)/2.
Conjecture (ii): The sequence has only three terms and a(3) = (2^2^5-5) * (2^2^5-1) = 18446744047939747845.
Conjecture (i) is a part of Lehmer's totient problem. Conjecture (ii) is disproved with the term a(3) = 6992962170388485 = 3 * 5 * 17 * 353 * 929 * 83623931. - Max Alekseyev, Oct 28 2023
a(5) <= 202317618492499837497376768005 = 3 * 5 * 17 * 257 * 65951 * 10414721 * 4494603392933. - Max Alekseyev, Oct 30 2023

			phi(165) = 80 = (165-5)/2.

Wikipedia, Lehmer's totient problem.

Subsequence of A083255. - R. J. Mathar, Jan 13 2014

Cf. A000010, A050474.

Do[If[EulerPhi[n]==1/2(n-5),Print[n]],{n,1,70001,4}]

is(n)=eulerphi(n)==(n-5)/2 \\ Charles R Greathouse IV, Jan 20 2014

from itertools import islice, count
from sympy import totient
def A233466gen(): return filter(lambda n:2*totient(n) == n-5,count(1,2))
A233466_list = list(islice(A233466gen(),2)) # Chai Wah Wu, Dec 15 2021

Missing term a(3) inserted, a(4) confirmed by Max Alekseyev, Oct 28 2023

cp's OEIS Frontend

A083254 a(n) = 2*phi(n) - n.

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A083260 a(n) = gcd(A046523(n), A071364(n)).

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A083258 a(n) = gcd(A046523(n), n).

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A083259 a(n) = gcd(n, A071364(n)), where A071364(n) is the smallest number with same sequence of exponents in canonical prime factorization as n.

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A083261 a(n) = gcd(A046523(n+1), A046523(n)).

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A233466 Numbers k such that phi(k) = (k-5)/2.

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