A145469 -id:A145469 - 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

A145470 A positive integer k is included if d(k) = d(k+1) + d(k+2), where d(k) is the number of divisors of k.

Original entry on oeis.org

12, 30, 42, 45, 56, 66, 112, 220, 261, 282, 294, 297, 308, 342, 364, 390, 416, 477, 492, 513, 516, 532, 536, 555, 567, 572, 580, 620, 621, 632, 651, 700, 705, 768, 777, 786, 795, 812, 832, 836, 860, 880, 884, 891, 897, 906, 957, 975, 976, 981, 992, 1000, 1005

Offset: 1

Leroy Quet, Oct 11 2008

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A145469, A175116.

Transpose[Select[Partition[Range[1010], 3, 1], DivisorSigma[0, First[#]] == DivisorSigma[0, Last[#]] + DivisorSigma[0, #[[2]]]&]][[1]] (* Amiram Eldar, Jul 17 2019 after Harvey P. Dale at A175116 *)

Extended by Ray Chandler, Oct 27 2008

A175115 An integer k, k >=3, is included if d(k) = d(k-1)*d(k-2), where d(k) is the number of divisors of k.

Original entry on oeis.org

3, 24, 120, 150, 195, 216, 294, 312, 399, 440, 459, 525, 540, 558, 570, 615, 630, 663, 693, 696, 744, 759, 774, 858, 999, 1032, 1095, 1125, 1152, 1176, 1239, 1350, 1455, 1470, 1494, 1608, 1624, 1659, 1664, 1710, 1725, 1734, 1785, 1860, 1880, 1896, 1914

Offset: 1

Leroy Quet, Feb 13 2010

nonn

Terms calculated by M. F. Hasler.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A175116, A145469.

Transpose[Select[Partition[Range[2000], 3, 1], DivisorSigma[0, Last[#]] == DivisorSigma[0, First[#]] * DivisorSigma[0, #[[2]]]&]][[1]] + 2 (* Amiram Eldar, Jul 17 2019 after Harvey P. Dale at A175116 *)

A332973 Solutions k of the equation usigma(k) = usigma(k-1) + usigma(k-2) where usigma(k) is the sum of unitary divisors of k (A034448).

Original entry on oeis.org

3, 42, 188970, 998670, 51670374, 91397154, 236280786, 259172826, 792554574, 1106710914, 1468869930, 1957827498, 2467823442, 2496238590, 3324585210, 4055970282, 4183629690, 4384566870, 13479861630, 20681058270, 29343074178, 43449285210, 68705958690, 71418085926

Offset: 1

Amiram Eldar, Mar 04 2020

nonn

			42 is a term since s(42) = 96 and s(40) + s(41) = 54 + 42 = 96.

Giovanni Resta, Table of n, a(n) for n = 1..64 (terms < 10^13)

The unitary version of A065900.

Cf. A034448, A065557, A075565, A076136, A145469, A291126, A291176, A292033, A294995, A332974.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[3, 10^8], usigma[#] == usigma[# - 1] + usigma[# - 2] &]

usigma(k) = sumdivmult(k, d, if(gcd(d, k/d)==1, d)); \\ A034448
isok(k) = usigma(k) == usigma(k-1) + usigma(k-2); \\ Jinyuan Wang, Mar 08 2020

Terms a(22) and beyond from Giovanni Resta, Mar 10 2020

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] &]

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

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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A145470 A positive integer k is included if d(k) = d(k+1) + d(k+2), where d(k) is the number of divisors of k.

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A175115 An integer k, k >=3, is included if d(k) = d(k-1)*d(k-2), where d(k) is the number of divisors of k.

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A332973 Solutions k of the equation usigma(k) = usigma(k-1) + usigma(k-2) where usigma(k) is the sum of unitary divisors of k (A034448).

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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.