A065572 -id:A065572 - cp's OEIS frontend

A065557 Solutions k of the equation phi(k) = phi(k-1) + phi(k-2). Also known as Phibonacci numbers.

3, 5, 7, 11, 17, 23, 37, 41, 47, 101, 137, 233, 257, 857, 1037, 1297, 1541, 1601, 2017, 4337, 6527, 9179, 14401, 16097, 30497, 55387, 61133, 62801, 65537, 72581, 77617, 110177, 152651, 179297, 244967, 299651, 603461, 619697, 686737, 1876727

Offset: 1

Joseph L. Pe, Nov 28 2001

nonn

All terms listed here are squarefree. (Thanks to Vladeta Jovovic for this observation.) The first two nonsquarefree terms are 72340252337 and 179115011177. There are 205 terms < 5*10^11. Most early terms are prime but later most terms are composite. - Jud McCranie, Feb 21 2012
There are 233 terms < 10^12. - Jud McCranie, Jan 02 2013
Bagers (1981) named these numbers Phibonacci numbers and asked about the existence of composite terms. According to the solution, P. J. Weinberg found 70 terms below 2*10^8, of which 46 are composite. The existence of an even term was discussed, and if it exists, it exceeds 10^1600. - Amiram Eldar, Mar 01 2020

			phi(23) = phi(22) + phi(21) (22=10+12), so 23 is in the sequence.
phi(101) = phi(100) + phi(99) (100=40+60), so 101 is in the sequence.

A. H. Beiler, Recreations in the Theory of Numbers, Dover Pub., NY, 1966.
Stanley J. Bezuszka and Margaret Kenney, Number Treasury: A Sourcebook of Problems for Calculators and Computers, Dale Seymour Publications, 1982, pp. 126 and 179.
Mihai Caragiu, Sequential Experiments with Primes, Springer, 2017, chapter 4, p. 152.
Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 117, entry 1037.
József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 224.

Jud McCranie, Table of n, a(n) for n = 1..321 (first 85 terms from Harry J. Smith)
Anders Bagers, Problem E2833, The American Mathematical Monthly, Vol. 87, No. 5 (1980), p. 404, solution, ibid., Vol. 88, No. 8 (1981), p. 622.
Florian Luca and Pantelimon Stanica, Linear equations with the Euler totient function, Acta Arithmetica, Vol. 128 (2007), pp. 135-147.
J. L. Pe, On Solutions of phi(n) = phi(n-1) + phi(n-2): A Problem Proposal

Cf. A000010. A065572 gives nonprime solutions.

Select[ Range[3, 10^6], EulerPhi[ # ] == EulerPhi[ # - 1] + EulerPhi[ # - 2] & ]

for(n=3,10^8, if(eulerphi(n)==eulerphi(n-1)+eulerphi(n-2),print1(n,",")))

{ n=0; e1=eulerphi(2); e2=eulerphi(1); for (m=3, 10^9, e=eulerphi(m); if (e==e2 + e1, write("b065557.txt", n++, " ", m); if (n==100, return)); e2=e1; e1=e ) } \\ Harry J. Smith, Oct 22 2009

More terms from Jason Earls, Robert G. Wilson v and Dean Hickerson, Nov 30 2001

A065604 a(n) = smallest k satisfying the equation phi(k) = phi(k-1) + phi(k-2) and having just n prime factors.

Original entry on oeis.org

3, 1037, 619697, 218688017, 32617225577

Offset: 1

Robert G. Wilson v, Dec 01 2001

			a(1) = 3 which is prime and is the first term in A065557, a(2) = 1037 = 17*61 which is the first term in A065572, a(3) = 619697=13*73*653

Cf. A065557 and A065572.

a = Table[0, {4}]; x = y = 1; Do[ z = EulerPhi[n]; If[z == x + y, If[l = Length[ FactorInteger[ n]]; a[[l]] == 0, a[[l]] = n; Print[n]]]; x = y; y = z, {n, 3, 10^7 } ]; a

a(5) from Donovan Johnson, Feb 05 2010

A266164 Primes p such that phi(p) = phi(p-2) + phi(p-1); Phibonacci primes.

Original entry on oeis.org

3, 5, 7, 11, 17, 23, 37, 41, 47, 101, 137, 233, 257, 857, 1297, 1601, 2017, 4337, 14401, 16097, 30497, 62801, 65537, 77617, 686737, 18800897, 255080417, 12885295097, 12918324737, 96052225601, 516392008697, 7026644072737

Offset: 1

Jaroslav Krizek, Dec 22 2015

Primes from A065557; complement of A065572 with respect to A065557.

The first 5 known Fermat primes from A019434 are in sequence.

			17 is in this sequence because phi(17) = phi(15) + phi(16); 16 = 8 + 8.

Cf. A000010, A065557, A065572.

[n: n in [3..5*10^7] | IsPrime(n) and EulerPhi(n) eq EulerPhi(n-2)+ EulerPhi(n-1)]

select(t -> isprime(t) and t-1 = numtheory:-phi(t-1) + numtheory:-phi(t-2), [seq(i,i=3..10^6,2)]); # Robert Israel, Dec 22 2015

Select[Prime[Range[56000]],EulerPhi[#]==EulerPhi[#-2]+EulerPhi[#-1]&] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, Aug 22 2025 *)

cp's OEIS Frontend

A065557 Solutions k of the equation phi(k) = phi(k-1) + phi(k-2). Also known as Phibonacci numbers.

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A065604 a(n) = smallest k satisfying the equation phi(k) = phi(k-1) + phi(k-2) and having just n prime factors.

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A266164 Primes p such that phi(p) = phi(p-2) + phi(p-1); Phibonacci primes.

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