A065557 -id:A065557 - 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

A292033 Unitary phibonacci numbers: solutions k of the equation uphi(k) = uphi(k-1) + uphi(k-2), where uphi(k) is the unitary totient function (A047994).

Original entry on oeis.org

3, 4, 7, 23, 9179, 244967, 14307856, 24571871, 128199059, 140830367, 401767631, 420567856, 468190439, 525970979, 780768167, 886434647, 1597167647, 4046753951, 4473784823, 5364666167, 5515718207, 11175736336, 14408460167, 18026319712, 20106993887, 20357733131

Offset: 1

Amiram Eldar, Sep 07 2017

nonn

The unitary version of A065557. Common terms are 3, 7, 23, 9179, 244967, ... Terms that are not in A065557 are 4, 14307856, 420567856, ...

			uphi(14307856) = uphi(14307855) + uphi(14307854) (3366080 = 7102080 + 6264000), so 14307856 is in the sequence.

Cf. A047994, A065557.

uphi[n_]:=If[n == 1, 1, (Times@@(Table[#[[1]]^#[[2]]-1,{1}] & /@ FactorInteger[n]))[[1]]]; Select[ Range[3, 10^6], uphi[#] == uphi[#-1] + uphi[#-2] &]

uphi(n) = my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2]-1);
isok(n) = uphi(n)==uphi(n-1)+uphi(n-2); \\ Altug Alkan, Sep 08 2017

a(18)-a(26) from Amiram Eldar, Mar 01 2020

A065572 Composite numbers k such that phi(k) = phi(k-1) + phi(k-2).

Original entry on oeis.org

1037, 1541, 6527, 9179, 55387, 61133, 72581, 110177, 152651, 179297, 244967, 299651, 603461, 619697, 1876727, 2841917, 3058211, 3971321, 4110653, 4316441, 4397317, 6008861, 10076627, 10667801, 10835441, 11561597, 24571871, 36521777, 45981377

Offset: 1

Len Smiley and Robert G. Wilson v, Nov 30 2001

nonn

619697 = 13*73*653 is the smallest solution not of the form p or p*q for distinct primes p and q.

218688017 is the first term that has four prime factors and 32617225577 is the first term with five prime factors. 72340252337 and 179115011177 are the first two that are not squarefree. There are 175 terms less than 5*10^11. - Jud McCranie, Feb 20 2012

Harry J. Smith and Jud McCranie, Table of n, a(n) for n = 1..175 (first 50 terms from Harry J. Smith)

Cf. A065557 (includes prime solutions).

Select[ Range[3, 10^7], !PrimeQ[ # ] && EulerPhi[ # ] == EulerPhi[ # - 1] + EulerPhi[ # - 2] & ]

{ n=0; e1=eulerphi(2); e2=eulerphi(1); for (m=3, 10^9, e=eulerphi(m); if (!isprime(m) && e==e2 + e1, write("b065572.txt", n++, " ", m); if (n==100, return)); e2=e1; e1=e ) } \\ Harry J. Smith, Oct 23 2009

More terms from Jud McCranie, Feb 21 2012

A066232 Numbers k such that phi(k) = phi(k-2) - phi(k-1).

Original entry on oeis.org

195, 3531, 9339, 27231, 46795, 78183, 90195, 112995, 135015, 437185, 849405, 935221, 1078581, 1283601, 1986975, 2209585, 2341185, 2411175, 2689695, 2744145, 3619071, 3712545, 4738185, 5132985, 6596121, 7829031, 8184715, 12176109

Offset: 1

Joseph L. Pe, Dec 18 2001

nonn

As in A065557, all terms listed here are odd. Problem: Prove that this holds in general.

			Phi(195) = 96 = 192-96 = phi(193)-phi(194).

Harry J. Smith and Jud McCranie, Table of n, a(n) for n = 1..289 (first 55 terms from Harry J. Smith)

Cf. A065557, A066231, A067701, A197112, A220160.

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

isok(k) = { k > 2 && eulerphi(k) == eulerphi(k - 2) - eulerphi(k - 1) } \\ Harry J. Smith, Feb 07 2010

a(n) = A220160(n) + 1 = A197112(n) + 2. - Andrew Howroyd, Dec 19 2024

a(13)-a(28) from Harry J. Smith, Feb 07 2010

A066231 Numbers n such that phi(n) = phi(n-1) - phi(n-2).

Original entry on oeis.org

6, 8, 26, 78, 218, 306, 3666, 4646, 5066, 8816, 12206, 12546, 19878, 20436, 24236, 29546, 37736, 47996, 60116, 72086, 73026, 77046, 87476, 121146, 126056, 129246, 149756, 190268, 234636, 247856, 273296, 275724, 419366, 531236, 553476, 621726

Offset: 1

Joseph L. Pe, Dec 18 2001

nonn

Question: Are all terms of this sequence even? (Compare A065557, whose terms could be all odd and squarefree.)

			phi(8) = 4 = 6-2 = phi(7) - phi(6).

Harry J. Smith and Jud McCranie, Table of n, a(n) for n = 1..494 (first 117 terms from Harry J. Smith)

Cf. A066232, A067701.

Flatten[Position[Partition[EulerPhi[Range[630000]],3,1],?(#[[3]] == #[[2]]- #[[1]]&),1,Heads->False]]+2 (* _Harvey P. Dale, Aug 19 2018 *)

n=0; for (m=3, 10^9, if (eulerphi(m) == eulerphi(m - 1) - eulerphi(m - 2), write("b066231.txt", n++, " ", m); if (n==117, return)) ) \\ Harry J. Smith, Feb 06 2010

a(24)-a(36) from Harry J. Smith, Feb 06 2010

A229552 Numbers k such that phi(k) = phi(k+2) - phi(k+1).

Original entry on oeis.org

1, 3, 5, 9, 15, 21, 35, 39, 45, 99, 135, 231, 255, 855, 1035, 1295, 1539, 1599, 2015, 4335, 6525, 9177, 14399, 16095, 30495, 55385, 61131, 62799, 65535, 72579, 77615, 110175, 152649, 179295, 244965, 299649, 603459, 619695, 686735, 1876725, 2841915, 3058209

Offset: 1

Vincenzo Librandi, Sep 27 2013

nonn

Donovan Johnson and Giovanni Resta, Table of n, a(n) for n = 1..324 (terms < 10^13, first 100 terms from Donovan Johnson)

Cf. A092404, A197112, A057000, A000010, A065557.

Select[Range[10^6], EulerPhi[#] == EulerPhi[# + 2] - EulerPhi[# + 1] &]
Position[Partition[EulerPhi[Range[31*10^5]],3,1],?(#[[3]]-#[[2]] == #[[1]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Nov 29 2017 *)

a(n) = A065557(n) - 2. - Amiram Eldar, Dec 09 2022

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

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A292033 Unitary phibonacci numbers: solutions k of the equation uphi(k) = uphi(k-1) + uphi(k-2), where uphi(k) is the unitary totient function (A047994).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions

A065572 Composite numbers k such that phi(k) = phi(k-1) + phi(k-2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A066232 Numbers k such that phi(k) = phi(k-2) - phi(k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A066231 Numbers n such that phi(n) = phi(n-1) - phi(n-2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A229552 Numbers k such that phi(k) = phi(k+2) - phi(k+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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).

Views