A069170 -id:A069170 - cp's OEIS frontend

A096172 Largest prime factor of n^4 + 1.

2, 17, 41, 257, 313, 1297, 1201, 241, 193, 137, 7321, 233, 14281, 937, 1489, 65537, 41761, 929, 3833, 160001, 97241, 3209, 139921, 331777, 11489, 26881, 6481, 614657, 353641, 3361, 1129, 61681, 6113, 1336337, 750313, 98801, 10529, 50857, 1156721

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

Mabkhout shows that a(n) >= 137 for n > 3. - Charles R Greathouse IV, Apr 07 2014

			a(1)=2 because 1^4 + 1 = 2;
a(2)=17: 2^4 + 1 = 17;
a(8)=241: 8^4 + 1 = 4097 = 17*241.

Mustapha Mabkhout, Minoration de P(x^4+1), Rendiconti del Seminario della Facoltà di Scienze dell'Università di Cagliari 63:2 (1993), pp. 135-148.

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
Marina Mureddu, A lower bound for P(x^4+1), Annales de la Faculté des sciences de Toulouse: Mathématiques, Serie 5, Vol. 8, No. 2 (1986-1987), pp. 109-119.

Cf. A000068, A002523, A014442, A037896, A006530, A069170, A096169, A096171.

FactorInteger[#^4+1][[-1,1]]&/@Range[40] (* Harvey P. Dale, Apr 30 2012 *)

a(n)=my(f=factor(n^4+1)[,1]); f[#f] \\ Charles R Greathouse IV, Apr 07 2014

a(n) = A006530(1+n^4) = A014442(n^2). - R. J. Mathar, Jan 28 2017
From Amiram Eldar, Oct 28 2024: (Start)
a(n) > 113 for n > 3 (Mureddu, 1986-1987).
a(n) >= 233 for n >= 11 (Luca, 2004). (End)

A069208 a(n) = Sum_{ d divides n } phi(n)/phi(d).

Original entry on oeis.org

1, 2, 3, 5, 5, 6, 7, 11, 10, 10, 11, 15, 13, 14, 15, 23, 17, 20, 19, 25, 21, 22, 23, 33, 26, 26, 31, 35, 29, 30, 31, 47, 33, 34, 35, 50, 37, 38, 39, 55, 41, 42, 43, 55, 50, 46, 47, 69, 50, 52, 51, 65, 53, 62, 55, 77, 57, 58, 59, 75, 61, 62, 70, 95, 65, 66, 67, 85, 69, 70, 71

Offset: 1

Vladeta Jovovic, Apr 10 2002

a(n) = n iff n is squarefree number (cf. A005117).
Conjecture: Let (f(n)), n > 0, be a multiplicative sequence. Then holds:
  (1) p(f; n) = Sum_{d powerful number (A001694) dividing n} f(d) is multiplicative;
  (2) p(f; n) equals inverse Moebius transform of A112526(n) * f(n). - Werner Schulte, Jan 23 2025
a(n) is also the number of conjugacy classes of the holomorph of the cyclic group of order n. Corollary: Let Rn be the dihedral quandle of order n. Then a(n) is the number of isomorphism classes of virtual quandles whose underlying quandle is isomorphic to Rn. - Luc Ta, Jun 16 2025

Ivan Neretin, Table of n, a(n) for n = 1..10000
Lực Ta, Enumeration of virtual quandles up to isomorphism, arXiv:2506.16536 [math.GT], 2025. See p. 2.

Cf. A000010, A000027, A005117, A005361, A069170, A112526.

Table[EulerPhi[n]*Total[1/EulerPhi@Divisors@n], {n, 71}] (* Ivan Neretin, Sep 20 2017 *)
f[p_, e_] := (p^(e + 1) - p^e + p^(e - 1) - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 14 2022 *)

a(n) = sumdiv(n, d, eulerphi(n)/eulerphi(d)) \\ Michel Marcus, Jun 17 2013

a(n) = my(f=factor(n)); prod(k=1, #f~, (f[k,1]^(f[k,2]-1) + (f[k,1]-1)*f[k,1]^f[k,2]-1) / (f[k,1]-1)); \\ Daniel Suteu, Nov 04 2018

Multiplicative with a(p^e) = (p^(e+1)-p^e+p^(e-1)-1)/(p-1).
a(n) = phi(n) * Sum_{k=1..n} 1/phi(n / gcd(n, k))^2. - Daniel Suteu, Nov 04 2018
a(n) = Sum_{k=1..n, gcd(n,k) = 1} tau(gcd(n,k-1)). - Ilya Gutkovskiy, Sep 24 2021
From Werner Schulte, Feb 27 2022: (Start)
Dirichlet convolution of A005361 and A000010.
Dirichlet convolution of A112526 and A000027.
Dirichlet g.f.: Sum_{n>0} a(n) / n^s = zeta(s-1) * zeta(2*s) * zeta(3*s) / zeta(6*s). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = 15015/(2764*Pi^2) = 0.550411... . - Amiram Eldar, Oct 22 2022
a(n) = Sum_{d powerful number (A001694) dividing n} n / d. - Werner Schulte, Jan 23 2025 (see Golomb link at A001694)

A096171 Numbers k such that k^4+1 is an odd semiprime.

Original entry on oeis.org

8, 10, 12, 14, 18, 22, 26, 30, 32, 36, 38, 40, 42, 50, 52, 58, 62, 68, 72, 78, 84, 86, 92, 94, 98, 100, 102, 108, 112, 114, 116, 120, 122, 124, 128, 130, 138, 146, 148, 152, 158, 162, 166, 170, 172, 176, 184, 186, 200, 212, 214, 216, 218, 222, 224, 226, 234, 250, 252

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

			a(1)=8 because 8^4 + 1 = 4097 = 17*241;
a(2)=10: 10^4 + 1 = 10001 = 73*137.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000068 (n^4+1 is prime), A037896 (primes of the form k^4+1), A096169 ((n^4+1)/2 is prime), A069170 (primes of the form (k^4+1)/2), A096172 (largest prime factor of n^4+1), A046388.

Select[Range[2,300,2],PrimeOmega[#^4+1]==2&] (* Harvey P. Dale, Dec 25 2021 *)

isA096171(n) = {local(m);m=n^4+1;(m%2==1)&&(bigomega(m)==2)} \\ Michael B. Porter, Feb 02 2010

cp's OEIS Frontend

A096172 Largest prime factor of n^4 + 1.

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A069208 a(n) = Sum_{ d divides n } phi(n)/phi(d).

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A096171 Numbers k such that k^4+1 is an odd semiprime.

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