A076251 -id:A076251 - cp's OEIS frontend

A291126 Psibonacci numbers: solutions n of the equation psi(n) = psi(n-1) + psi(n-2), where psi is the Dedekind psi function (A001615).

3, 6, 210, 88200, 101970, 193290, 289680, 993990, 11264550, 59068230, 72776970, 98746230, 122460690, 126500910, 132766770, 234150930, 514442214, 531391650, 638082390, 650428020, 790769790, 1249160790, 3727074450, 4775972850, 8299675650, 9530202210

Offset: 1

Amiram Eldar, Aug 19 2017

nonn

Analogous to phibonacci numbers (A065557) and other sequences (see crossrefs).

			psi(210) = 576 = 240 + 336 = psi(209) + psi(208), therefore 210 is in the sequence.

Giovanni Resta, Table of n, a(n) for n = 1..70 (terms < 4*10^12)

Cf. A001615, A065557, A065900, A076136, A076251, A145469.

psi[n_]:=If[n < 1, 0, n Sum[ MoebiusMu[ d]^2 / d, {d, Divisors @ n}]];
Select[Range[10^6], psi[#]==psi[#-1]+psi[#-2] &]

a(21)-a(26) from Giovanni Resta, Aug 26 2018

A291176 Numbers k such that s(k) = s(k-1) + s(k-2), where s(k) is the sum of proper divisors of k (A001065).

Original entry on oeis.org

3, 8, 20, 146139, 584835, 44814015, 1436395095, 9988999095, 25997557299, 193861767939, 2105722150095, 3921293253003, 8234992646643

Offset: 1

Amiram Eldar, Aug 19 2017

a(14) > 10^13. - Giovanni Resta, Feb 25 2020

			s(146139) = 76581 = 75802 + 779 = s(146138) + s(146137), therefore 146139 is in the sequence.

Cf. A001065, A065557, A065900, A076136, A076251, A145469.

s[n_]:=DivisorSigma[1,n]-n; Select[Range[10^6], s[#]==s[#-1]+s[#-2] &]

a(7)-a(10) from Giovanni Resta, Aug 29 2017

a(11)-a(13) from Giovanni Resta, Feb 25 2020

A076252 Integers k such that omega(k) = omega(k-1) + omega(k-2) + omega(k-3), where omega(n) is the number of distinct prime factors of n.

Original entry on oeis.org

2310, 3990, 4290, 6090, 6270, 10010, 11550, 12810, 13650, 17094, 17940, 18270, 19380, 21930, 22110, 22770, 23100, 24990, 25410, 27300, 28644, 30090, 32214, 32604, 34034, 34314, 35340, 35880, 37310, 38190, 38570, 38640, 39270, 39780

Offset: 1

Joseph L. Pe, Nov 04 2002

nonn

			omega(2310) = 5 = 1 + 2 + 2 = omega(2309) + omega(2308) + omega(2307), so 2310 belongs to the sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A001221, A006049, A076251, A076253.

omega[n_] := Length[FactorInteger[n]]; a = {}; Do[If[omega[n] == omega[n - 1] + omega[n - 2] + omega[n - 3], a = Append[a, n]], {n, 1, 10^5}]; a
Flatten[Position[Partition[PrimeNu[Range[40000]],4,1],?(#[[4]] == Total[ Take[ #,3]]&), {1}, Heads->False]]+3 (* _Harvey P. Dale, Oct 31 2016 *)

A076253 a(n) = the least positive integer solution of the "n-th omega recurrence" omega(k) = omega(k-1) + ... + omega(k-n), if such k exists; = 0 otherwise. (omega(n) denotes the number of distinct prime factors of n.)

Original entry on oeis.org

3, 3, 2310, 746130, 601380780, 89419589469210, 489423552293946270

Offset: 1

Joseph L. Pe, Nov 04 2002

Question: Is a(n) > 0 for all n, i.e. can the n-th omega recurrence be solved for all n?

Note that 601380780 is not squarefree. Using primorials, I easily found candidates up to a(8). - Lambert Klasen (lambert.klasen(AT)gmx.net), Nov 05 2005

			k=3 is the smallest solution of omega(k)=omega(k-1), so a(1)=3.
k=3 is the smallest solution of omega(k)=omega(k-1)+omega(k-2), so a(2)=3.
k=2310 is the smallest solution of omega(k)=omega(k-1)+omega(k-2)+omega(k-3), so a(3)=2310.

Cf. A001221, A006049, A076251, A076252.

(*code to find a(4)*) omega[n_] := Length[FactorInteger[n]]; ub = 2*10^6; For[i = 2, i <= ub, i++, a[i] = omega[i]]; start = 5; For[j = start, j <= ub, j++, If[a[j] == a[j - 1] + a[j - 2] + a[j - 3] + a[j - 4], Print[j]]]

/* find a(5) */ v=[0,0,0,0,0]; s=0;for(i=1,5,v[i]=omega(i);s+=v[I])
for(i=6,10^10,o=omega(i);if(o==s,print(i);break);s-=v[i%5+1];s+=o;v[i%5+1]=o) \\ Lambert Klasen (lambert.klasen(AT)gmx.net), Nov 05 2005

a(5) from Lambert Klasen (lambert.klasen(AT)gmx.net), Nov 05 2005

a(6)-a(7) from Donovan Johnson, Feb 07 2009

A332975 Solutions k of the equation isigma(k) = isigma(k-1) + isigma(k-2) where isigma(k) is the sum of the infinitary divisors of k (A049417).

Original entry on oeis.org

3, 24, 360, 5016, 28440, 42066, 50568, 60456, 187176, 998670, 1454706, 12055512, 14365608, 25726728, 27896424, 51670374, 91702962, 141084774, 236280786, 249854952, 386668344, 439362504, 792554574, 1115866152, 1931976696, 2467823442, 2496238590, 2655297558, 2715505440

Offset: 1

Amiram Eldar, Mar 04 2020

nonn

			24 is a term since isigma(24) = 60 and isigma(22) + isigma(23) = 36 + 24 = 60.

Amiram Eldar, Table of n, a(n) for n = 1..38

The infinitary version of A065900.

Cf. A049417, A065557, A075565, A076136, A076251, A145469, A291126, A291176, A292033, A294995, A332976.

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]; Select[Range[3, 10^5], isigma[#] == isigma[# - 1] + isigma[# - 2] &]

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

Original entry on oeis.org

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] &]

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] &]

cp's OEIS Frontend

A291126 Psibonacci numbers: solutions n of the equation psi(n) = psi(n-1) + psi(n-2), where psi is the Dedekind psi function (A001615).

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A291176 Numbers k such that s(k) = s(k-1) + s(k-2), where s(k) is the sum of proper divisors of k (A001065).

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A076252 Integers k such that omega(k) = omega(k-1) + omega(k-2) + omega(k-3), where omega(n) is the number of distinct prime factors of n.

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A076253 a(n) = the least positive integer solution of the "n-th omega recurrence" omega(k) = omega(k-1) + ... + omega(k-n), if such k exists; = 0 otherwise. (omega(n) denotes the number of distinct prime factors of n.)

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A332975 Solutions k of the equation isigma(k) = isigma(k-1) + isigma(k-2) where isigma(k) is the sum of the infinitary divisors of k (A049417).

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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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Mathematica

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

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Mathematica

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).