A065900 -id:A065900 - cp's OEIS frontend

A291175 Numbers k such that lambda(k) = lambda(k-1) + lambda(k-2), where lambda(k) is Carmichael lambda function (A002322).

3, 5, 7, 11, 13, 22, 46, 371, 717, 1379, 1436, 1437, 3532, 5146, 12209, 35652, 45236, 58096, 93932, 130170, 263589, 327095, 402056, 680068, 808303, 814453, 870689, 991942, 1178628, 1670065, 1686526, 2041276, 2319102, 2324004, 3869372, 4290742, 4449280

Offset: 1

Amiram Eldar, Aug 19 2017

nonn

			lambda(717) = 238 = 178 + 60 = lambda(716) + lambda(715), therefore 717 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..250

Cf. A002322, A065557, A065900, A076136, A076251, A145469.

Select[Range[10^6], CarmichaelLambda[#]==CarmichaelLambda[#-1]+CarmichaelLambda[#-2]&]
Flatten[Position[Partition[CarmichaelLambda[Range[45*10^5]],3,1],?(#[[1]]+#[[2]] == #[[3]]&),1,Heads->False]]+2 (* _Harvey P. Dale, Sep 02 2024 *)

from sympy import reduced_totient
A291175_list, a, b, c, n = [], 1, 1, 2, 3
while n < 10**6:
    if c == a + b:
        A291175_list.append(n)
        print(len(A291175_list),n)
    n += 1
    a, b, c = b, c, reduced_totient(n) # Chai Wah Wu, Aug 31 2017

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

Amiram Eldar, Mar 04 2020

nonn

			7 is a term since iphi(7) = 6 and iphi(5) + iphi(6) = 4 + 2 = 6.

Amiram Eldar, Table of n, a(n) for n = 1..70

An infinitary version of A065557.

Cf. A065900, A075565, A076136, A076251, A091732, A145469, A291126, A291176, A292033, A294995.

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

Amiram Eldar, Mar 04 2020

nonn

			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.

Amiram Eldar, Table of n, a(n) for n = 1..176

Cf. A051953, A065557, A065900, A075565, A076136, A076251, A145469, A291126, A291176, A292033, A294995.

cotot[n_] := n - EulerPhi[n]; Select[Range[3, 10^6], cotot[#] == cotot[# - 1] + cotot[# - 2] &]

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

Amiram Eldar, Mar 04 2020

a(17) > 10^13. - Giovanni Resta, May 09 2020

			21 is a term since s(21) = 11 and s(19) + s(20) = 1 + 10 = 11.

The unitary version of A291176.

Cf. A063919, A065557, A065900, A075565, A076136, A145469, A291126, A292033, A294995, A332973.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); s[n_] := usigma[n] - n; Select[Range[3, 6*10^5], s[#] == s[# - 1] + s[# - 2] &]

a(12)-a(16) from Giovanni Resta, May 09 2020

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

Amiram Eldar, Mar 04 2020

			8 is a term since s(8) = 7 and s(6) + s(7) = 6 + 1 = 7.

The infinitary version of A291176.

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] &]

A226753 Numbers n such that sigma(n) + sigma(n+1) + sigma(n+2) = sigma(n+3) + sigma(n+4) + sigma(n+5).

Original entry on oeis.org

52, 56, 2199, 17312, 92444, 31768040, 41159079, 52236776, 79563436, 683321810, 689969123, 757808252, 881421986, 910016369, 22184779055

Offset: 1

Alex Ratushnyak, Jun 16 2013

The sequence of corresponding sums begins: 272, 290, 10820, 86514, 473568, 162349956, 203060920, 258797664, 398693936, 3418531584, 3544794576, 3812186016, 4066082208, 4374966816.

			Sigma(52) + sigma(53) + sigma(54) = 98 + 54 + 120 = 272.
Sigma(55) + sigma(56) + sigma(57) = 72 + 120 + 80 = 272.
So 52 is in the sequence.

Cf. A000203, A055574, A065900, A073500, A099632, A226361, A226475.

a(15) from Giovanni Resta, Jun 17 2013

cp's OEIS Frontend

A291175 Numbers k such that lambda(k) = lambda(k-1) + lambda(k-2), where lambda(k) is Carmichael lambda function (A002322).

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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).

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A332972 Solutions k of the equation cototient(k) = cototient(k-1) + cototient(k-2) where cototient(k) is A051953.

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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).

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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).

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A226753 Numbers n such that sigma(n) + sigma(n+1) + sigma(n+2) = sigma(n+3) + sigma(n+4) + sigma(n+5).

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