A037225 -id:A037225 - cp's OEIS frontend

A067229 Numbers n such that phi(2n+1) = sigma(n).

6, 7, 27, 38, 55, 85, 87, 127, 188, 236, 255, 266, 445, 451, 655, 717, 913, 1309, 1357, 1490, 1947, 2095, 2515, 2726, 3002, 3247, 3289, 4117, 4237, 5071, 5605, 7195, 8924, 12128, 12625, 12771, 12837, 13190, 13795, 15835, 16197, 16748, 17997, 20257

Offset: 1

Benoit Cloitre, Feb 20 2002

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000010, A000203, A037225, A067230.

Select[Range[21000],EulerPhi[2#+1]==DivisorSigma[1,#]&] (* Harvey P. Dale, Sep 06 2024 *)

isok(n) = eulerphi(2*n+1) == sigma(n); \\ Michel Marcus, Nov 21 2013

A072106 The number of nontotients (even and odd) in the reduced residue system of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 3, 2, 3, 4, 3, 5, 5, 4, 7, 8, 5, 9, 7, 5, 9, 11, 7, 10, 11, 10, 11, 15, 7, 16, 15, 11, 15, 13, 11, 20, 17, 14, 15, 23, 11, 24, 19, 15, 21, 26, 15, 23, 19, 19, 23, 30, 17, 23, 23, 21, 27, 33, 15, 34, 29, 22, 31, 27, 19, 38, 31, 28, 23, 41, 23, 42, 35, 26, 35, 37

Offset: 1

Labos Elemer, Jun 19 2002

nonn

			For n=113: the reduced residue system consists of 112 numbers: Card[OddNonTotients(113)] = 56 - 1, EvenNonTotients = {14,26,34,38,50,62,68,74,76,86,90,94,98}, i.e., 13 terms, therefore a(113) = 56 - 1 + 13 = 68.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A037225, A072451, A072455.

a(n) = sum(k = 1, n, gcd(n, k) == 1 && !istotient(k)); \\ Amiram Eldar, Nov 07 2024

A072454 Number of nontotients in the reduced residue system of 2n-1.

Original entry on oeis.org

0, 0, 1, 2, 2, 4, 5, 4, 8, 9, 5, 11, 10, 10, 15, 16, 11, 13, 20, 14, 23, 24, 15, 26, 23, 19, 30, 23, 21, 33, 34, 22, 27, 38, 28, 41, 42, 26, 37, 47, 35, 49, 37, 37, 53, 44, 38, 43, 59, 41, 62, 63, 32, 65, 66, 46, 68, 55, 46, 58, 69, 53, 64, 79, 55, 81, 65, 50, 85, 86, 60, 77, 72

Offset: 1

Labos Elemer, Jun 19 2002

nonn

			For n=105: phi(105) = 48 with 24 odd, 24 even terms in the reduced residue system, of which 9 even terms and (all but 1) odd term is nontotient: a((105+1)/2) = a(53) = 24-1+9 = 32.
For n=21: reduced residue system(21) = Union({1,5,11,13,17,19}, {2,4,8,16,20}) includes 6 odd and 5 even numbers. No even nontotients terms in the reduced residue system(21), so 6-1 = 5 odd terms give all nontotients, so a((21+1)/2) = a(11) = 5.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A002202, A005277, A037225, A072451, A072455.

Bisection of A072106.

a(n) = {my(m = 2*n-1); sum(k = 1, m, gcd(m, k) == 1 && !istotient(k));} \\ Amiram Eldar, Nov 07 2024

a(n) = A072106(2*n-1). - Amiram Eldar, Nov 07 2024

A062533 a(n) = A000010(A014076(n)).

Original entry on oeis.org

1, 6, 8, 12, 20, 18, 20, 24, 24, 24, 42, 32, 40, 36, 36, 48, 44, 40, 60, 54, 64, 56, 72, 60, 72, 60, 48, 72, 88, 72, 96, 110, 80, 100, 84, 108, 72, 92, 120, 112, 84, 96, 120, 104, 132, 80, 156, 108, 120, 116, 120, 144, 160, 108, 96, 132, 168, 160, 132, 180, 140, 168

Offset: 1

Jason Earls, Jul 10 2001

G. C. Greubel, Table of n, a(n) for n = 1..1000

Subset of the totients of the odds (A037225).

Cf. A000010, A014076.

A014076 := Select[Range[1, 350, 2], PrimeOmega[#] != 1 &]; Table[ EulerPhi[A014076[[n]]], {n, 1, 50}] (* G. C. Greubel, Sep 17 2017 *)

je=[]; forstep(n=1,301,2, if(isprime(n), n+1,je=concat(je,eulerphi(n)))); je

from sympy import primepi, totient
def A062533(n):
    if n == 1: return 1
    m, k = n-1, primepi(n) + n - 1 + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n - 1 + (k>>1)
    return totient(m) # Chai Wah Wu, Jul 31 2024

A089266 Rational knots of determinant 2n+1, counting chiral pairs twice.

Original entry on oeis.org

2, 3, 4, 4, 6, 7, 6, 9, 10, 8, 12, 11, 10, 15, 16, 12, 14, 19, 14, 21, 22, 14, 24, 22, 18, 27, 22, 20, 30, 31, 20, 26, 34, 24, 36, 37, 22, 32, 40, 28, 42, 34, 30, 45, 38, 32, 38, 49, 32, 51, 52, 28, 54, 55, 38, 57, 46, 38, 50, 56, 42, 51, 64, 44, 66, 56

Offset: 1

Ralf Stephan, Oct 30 2003

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
A. Stoimenow, Square numbers, spanning trees and invariants of rational knots (2000)

Cf. A078477, A078478, A018240, A051449, A051766.

a[n_] := (EulerPhi[2*n+1] + 2^PrimeNu[2*n+1])/2; Table[a[n], {n, 1, 66}] (* Jean-François Alcover, Oct 11 2013, after Pari *)

PARI
```
a(n)=(eulerphi(2*n+1)+2^omega(2*n+1))/2
```

a(n) = 1/2 * (A037225(n) + A034444(2*n+1)).

A177066 Determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(2i-1,2j-1) for 1 <= i,j <= n.

Original entry on oeis.org

1, 2, 8, 48, 288, 2880, 34560, 276480, 4423680, 79626240, 955514880, 21021327360, 420426547200, 7567677849600, 211894979788800, 6356849393664000, 127136987873280000, 3051287708958720000, 109846357522513920000

Offset: 1

John W. Layman, Dec 09 2010

nonn

It appears, but has not been proved, that the ratios a(n+1)/a(n) give phi(2n+1) (A037225).

See A001088, A059381, and A059382 for determinants of matrices M defined by M(i,j) = gcd(i,j), gcd(i^2,j^2), and gcd(i^3,j^3), respectively.

Cf. A001088, A037225, A059381, A059382.

A177066 := proc(n) M := Matrix(n) ; for i from 1 to n do for j from 1 to n do M[i,j] := igcd(2*i-1,2*j-1) ; end do: end do: LinearAlgebra[Determinant](M) ; end proc: # R. J. Mathar, Dec 10 2010

A250222 a(n) = phi(2n+1) - phi(2n), where phi is A000010.

Original entry on oeis.org

1, 2, 4, 2, 6, 8, 2, 8, 12, 4, 12, 12, 6, 16, 22, 4, 8, 24, 6, 24, 30, 4, 24, 26, 12, 28, 22, 12, 30, 44, 6, 16, 46, 12, 46, 48, 4, 24, 54, 22, 42, 40, 14, 48, 48, 16, 26, 64, 18, 60, 70, 0, 54, 72, 32, 64, 52, 16, 38, 78, 20, 40, 90, 20, 82, 68, 6, 72, 94, 44, 50, 64

Offset: 1

Eric Chen, Dec 24 2014

a(157) = -12 is the first negative number in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A037225, A057000, A062570, A185199.

a[n_] := EulerPhi[2*n+1] - EulerPhi[2*n]; Array[a, 100] (* Amiram Eldar, Nov 09 2024 *)
#[[2]]-#[[1]]&/@Partition[EulerPhi[Range[2,150]],2] (* Harvey P. Dale, Aug 02 2025 *)

a(n) = eulerphi(2*n+1) - eulerphi(2*n); \\ Amiram Eldar, Nov 09 2024

From Amiram Eldar, Nov 09 2024: (Start)
a(n) = A057000(2*n).
a(n) = A037225(n) - A062570(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). (End)

A290083 Odd bisection of A289626.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 8, 9, 10, 11, 12, 14, 10, 15, 16, 18, 19, 20, 19, 22, 23, 19, 24, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 27, 38, 39, 41, 42, 43, 44, 46, 47, 38, 48, 50, 38, 51, 52, 53, 54, 55, 48, 56, 57, 47, 58, 60, 61, 51, 62, 64, 65, 66, 48, 68, 69, 70, 71, 72, 64, 73, 74, 58, 71, 76, 77, 79, 80, 81, 82, 76, 66, 84, 71, 85, 86, 87, 71

Offset: 1

Antti Karttunen, Jul 19 2017

nonn

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

Cf. A037225, A278223, A289626.

(define (A290083 n) (A289626 (+ n n -1)))

a(n) = A289626(2n-1).

A363700 a(n) = phi(2*prime(n)+1).

Original entry on oeis.org

4, 6, 10, 8, 22, 18, 24, 24, 46, 58, 36, 40, 82, 56, 72, 106, 96, 80, 72, 120, 84, 104, 166, 178, 96, 168, 132, 168, 144, 226, 128, 262, 200, 180, 264, 200, 144, 216, 264, 346, 358, 220, 382, 252, 312, 216, 276, 296, 288, 288, 466, 478, 264, 502, 408, 480, 420, 360, 288, 562

Offset: 1

Alain Rocchelli, Jun 16 2023

nonn

2*prime(n)+1 is prime iff a(n) = 2*prime(n).

Cf. A000010, A000040, A008331, A037225, A072055.

a[n_] := EulerPhi[2*Prime[n] + 1]; Array[a, 100] (* Amiram Eldar, Jun 16 2023 *)

PARI
```
a(n)=eulerphi(2*prime(n)+1)
```

a(n) = A000010(A072055(n)).

a(n) = A037225(A000040(n)).

cp's OEIS Frontend

A067229 Numbers n such that phi(2n+1) = sigma(n).

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A072106 The number of nontotients (even and odd) in the reduced residue system of n.

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A072454 Number of nontotients in the reduced residue system of 2n-1.

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A062533 a(n) = A000010(A014076(n)).

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A089266 Rational knots of determinant 2n+1, counting chiral pairs twice.

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A177066 Determinant of the symmetric n X n matrix M defined by M(i,j) = gcd(2i-1,2j-1) for 1 <= i,j <= n.

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A250222 a(n) = phi(2n+1) - phi(2n), where phi is A000010.

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A290083 Odd bisection of A289626.

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A363700 a(n) = phi(2*prime(n)+1).

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