A332971
Infinitary phibonacci numbers: solutions k of the equation iphi(k) = iphi(k-1) + iphi(k-2) where iphi(k) is an infinitary analog of Euler's phi function (A091732).
Original entry on oeis.org
3, 4, 7, 23, 121, 2857, 5699, 6377, 9179, 46537, 63209, 244967, 654497, 1067873, 1112009, 3435929, 3831257, 6441593, 7589737, 7784507, 8149751, 14307856, 22434089, 24007727, 24571871, 44503417, 44926463, 56732729, 128199059, 140830367, 190145936, 401767631, 403152737
Offset: 1
7 is a term since iphi(7) = 6 and iphi(5) + iphi(6) = 4 + 2 = 6.
-
f[p_, e_] := p^(2^(-1 + Position[Reverse@IntegerDigits[e, 2], 1])); iphi[1] = 1; iphi[n_] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) - 1); Select[Range[3, 10^5], iphi[#] == iphi[# - 1] + iphi[# - 2] &]
A332972
Solutions k of the equation cototient(k) = cototient(k-1) + cototient(k-2) where cototient(k) is A051953.
Original entry on oeis.org
3, 4, 105, 165, 195, 2205, 2835, 38805, 131145, 407925, 936495, 1025505, 1231425, 1276905, 1788255, 1925565, 2521695, 2792145, 2847585, 3289935, 5003745, 5295885, 5710089, 6315309, 6986889, 13496385, 17168085, 19210065, 20171385, 22348365, 26879685, 27798705
Offset: 1
3 is a term since cototient(3) = 1 and cototient(1) + cototient(2) = 0 + 1 = 1.
105 is a term since cototient(105) = 57 and cototient(103) + cototient(104) = 1 + 56 = 57.
Cf.
A051953,
A065557,
A065900,
A075565,
A076136,
A076251,
A145469,
A291126,
A291176,
A292033,
A294995.
A332974
Solutions k of the equation s(k) = s(k-1) + s(k-2) where s(k) = usigma(k) - k is the sum of proper unitary divisors of k (A063919).
Original entry on oeis.org
3, 21, 321, 1257, 3237, 146139, 268713, 584835, 26749089, 9988999095, 25997557299, 54449485353, 935628578283, 2105722150095, 3921293253003, 8234992646643
Offset: 1
21 is a term since s(21) = 11 and s(19) + s(20) = 1 + 10 = 11.
A332976
Solutions k of the equation s(k) = s(k-1) + s(k-2) where s(k) = isigma(k) - k is the sum of proper infinitary divisors of k (A126168).
Original entry on oeis.org
3, 8, 10, 21, 3237, 7377, 146139, 584835, 9988999095, 25997557299
Offset: 1
8 is a term since s(8) = 7 and s(6) + s(7) = 6 + 1 = 7.
Cf.
A065557,
A065900,
A075565,
A076136,
A076251,
A126168,
A145469,
A291126,
A292033,
A294995,
A332975.
-
fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; s[n_] := isigma[n] - n; Select[Range[3, 6*10^5], s[#] == s[# - 1] + s[# - 2] &]
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
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
-
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
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
17 is in this sequence because phi(17) = phi(15) + phi(16); 16 = 8 + 8.
-
[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 *)
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