A272412 -id:A272412 - cp's OEIS frontend

A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

1, 3, 5, 6, 15, 21, 28, 140, 182, 496, 546, 672, 918, 1890, 2016, 4005, 4590, 24384, 52780, 55860, 68200, 84812, 90090, 105664, 145782, 186992, 204600, 381654, 728910, 907680, 1655400, 2302344, 2862405, 3828009, 3926832, 5959440, 21059220, 33550336, 33839988, 42325920

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

Terms that also Fibonacci numbers are 1, 3, 5, 21, and no more below Fibonacci(300).

			3 is a term since the harmonic mean of its divisors is 3/2 = Fibonacci(4)/Fibonacci(3).
15 is a term since the harmonic mean of its divisors is 5/2 = Fibonacci(5)/Fibonacci(3).

Cf. A000045, A099377, A099378.

Similar sequences: A074266, A123193, A272412, A272440, A348659.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := fibQ[Numerator[(hn = h[n])]] && fibQ[Denominator[hn]]; Select[Range[1000], q]

from itertools import islice
from sympy import integer_nthroot, gcd, divisor_sigma
def A348658(): # generator of terms
    k = 1
    while True:
        a, b = divisor_sigma(k), divisor_sigma(k,0)*k
        c = gcd(a,b)
        n1, n2 = 5*(a//c)**2-4, 5*(b//c)**2-4
        if (integer_nthroot(n1,2)[1] or integer_nthroot(n1+8,2)[1]) and (integer_nthroot(n2,2)[1] or integer_nthroot(n2+8,2)[1]):
            yield k
        k += 1
A348658_list = list(islice(A348658(),10)) # Chai Wah Wu, Oct 28 2021

A272440 Numbers n such that the average of the positive divisors of n is a Fibonacci number.

Original entry on oeis.org

1, 3, 5, 6, 21, 41, 45, 65, 67, 68, 78, 96, 109, 382, 497, 517, 527, 658, 682, 705, 759, 805, 930, 966, 1155, 1557, 1973, 3211, 3653, 4563, 5167, 5620, 9037, 10027, 10117, 13279, 17353, 28856, 35174, 35534, 45459, 56072, 154555, 175151, 177721, 181561, 183181, 184201, 184421, 184601, 185466, 226666

Offset: 1

Altug Alkan, Apr 29 2016

1, 3, 5 and 21 are Fibonacci numbers. Are there other Fibonacci numbers in this sequence?

For a similar question and related proof attempt see the paper in the links section of A272412.

			3 is a term because 3 is divisible by 1 and 3. Average of 3 and 1 is 2 that is a Fibonacci number.

Charles R Greathouse IV, Table of n, a(n) for n = 1..734

Cf. A000045, A003601, A272412.

s = Array[Fibonacci, {28}]; Select[Range@ Max@ s, MemberQ[s, Mean@ Divisors@ #] &] (* Michael De Vlieger, Apr 29 2016 *)

isFibonacci(n)=my(k=n^2); k+=((k + 1) << 2); issquare(k) || (n > 0 && issquare(k-8))
is(n)=my(f=factor(n),s=sigma(f),d=numdiv(f)); s%d==0 && isFibonacci(s/d) \\ Charles R Greathouse IV, May 02 2016

A275789 Least k such that sigma(n) divides Fibonacci(k) (k > 0).

Original entry on oeis.org

1, 4, 6, 8, 12, 12, 6, 20, 7, 12, 12, 24, 24, 12, 12, 30, 12, 28, 30, 24, 24, 12, 12, 60, 30, 24, 30, 24, 60, 12, 24, 24, 12, 36, 12, 56, 18, 60, 24, 60, 24, 24, 30, 24, 84, 12, 12, 30, 36, 60, 12, 168, 36, 60, 12, 60, 60, 60, 60, 24, 30, 24, 42, 128, 24, 12, 18, 24, 24, 12, 12

Offset: 1

Altug Alkan, Aug 09 2016

nonn

			a(5) = 12 because Fibonacci(12) = 144 is divisible by sigma(5) = 6.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000045, A001177, A000203, A272412.

Table[k = 1; While[! Divisible[Fibonacci@k, DivisorSigma[1, n]], k++]; k, {n, 120}] (* Michael De Vlieger, Aug 11 2016 *)

a(n)=my(k=1); while(fibonacci(k) % sigma(n), k++); k;

a(n)=my(s=sigma(n),a=Mod(1,s),b=a,k=1); while(a, [a,b]=[b,a+b]; k++); k \\ Charles R Greathouse IV, Aug 12 2016

use ntheory ":all"; sub a275789 { my($sigma,$k)=(divisor_sum(shift), 1); return 1 if $sigma==1; $k++ while (lucas_sequence($sigma,1,-1,$k))[0]; $k; } # Dana Jacobsen, Aug 15 2016

a(n) = A001177(A000203(n)). - Robert Israel, Aug 09 2016

log n << a(n) << n log log n. - Charles R Greathouse IV, Aug 12 2016

A280545 Numbers k such that sigma(sigma(k)) is a Fibonacci number.

Original entry on oeis.org

1, 4, 13228, 19289, 187838, 201255, 281771, 308591, 7391381, 9041581, 9124081, 9589141, 193690242024, 254218445292, 306676216538, 375274847812, 402512534123, 423082506591, 425477274579, 443127377343, 456962010177, 458339382147, 476155178309, 480858185253, 484213195281

Offset: 1

Altug Alkan, Jan 05 2017

nonn

19289, 7391381, 9041581, 9124081, 9589141 are prime numbers. See A280555 for other prime terms of this sequence.

			19289 is a term because sigma(19289) = 19290 and sigma(19290) = 46368 is a Fibonacci number.

Cf. A051027, A272412, A280555.

isFibonacci(n)=my(k=n^2); issquare(k+=(k+1)<<2) || (n>0 && issquare(k-8));
is(n)=isFibonacci(sigma(sigma(n)));

Numbers k such that A051027(k) is in A000045. - Wesley Ivan Hurt, Apr 05 2023

More terms from Giovanni Resta, Jan 08 2017

A280555 Primes p such that sigma(sigma(p)) is a Fibonacci number.

Original entry on oeis.org

19289, 7391381, 9041581, 9124081, 9589141, 645617593711, 786881099503, 793374393583, 188950298985689, 215446003400539, 228846950929339, 257138974382029, 265666386165589, 276918720321829, 280481623844131, 323331286115017, 326905876894417

Offset: 1

Altug Alkan, Jan 05 2017

nonn

Is this sequence infinite?

			Prime number 7391381 is a term because sigma(sigma(7391381)) = 14930352 is a Fibonacci number.

Cf. A008333, A051027, A272412, A280545.

isFibonacci(n)=my(k=n^2); issquare(k+=(k+1)<<2) || (n>0 && issquare(k-8));
is(n)=isFibonacci(sigma(n+1))&&isprime(n);

Terms confirmed by Giovanni Resta, Jan 07 2017

A329660 Numbers m such that sigma(m) is a Lucas number (A000032), where sigma(m) is the sum of divisors of m (A000203).

Original entry on oeis.org

1, 2, 3, 4, 10, 17, 688, 1075, 103681, 7860997, 10749957121, 115561578124838522881, 488296733939737583689, 489501979450313254561, 3628116960730713370000, 8784132317383836036997, 8784200214538920269317, 50755107290462736080376601, 94426187701102977738552612783157

Offset: 1

Amiram Eldar, Nov 18 2019

nonn

Prime numbers of the form L(k)-1, where L(k) is the k-th Lucas number, are in this sequence. The terms 2, 3, 17, 103681, and 10749957121 are primes of this form (with k = 2, 3, 6, 24, 48). Also in the sequence is the prime L(96) - 1 = 115561578124838522881.

			4 is in the sequence since sigma(4) = 7 is a Lucas number.

Daniel Suteu, Table of n, a(n) for n = 1..225

Cf. A000032, A000203, A272412.

f = LucasL @ Range[1, 40]; Select[Range[10^6], MemberQ[f, DivisorSigma[1, #]] &]  (* after Giovanni Resta at A272412 *)

a(12)-a(19) from Giovanni Resta, Nov 18 2019

A333677 Numbers whose divisors can be partitioned into two disjoint sets whose sums are consecutive Fibonacci numbers.

Original entry on oeis.org

1, 2, 66, 70, 18084, 19180, 24934, 26715, 5346390, 8197798, 8424178, 9088863, 1874967204, 1988601580, 2585182054, 2769837915

Offset: 1

Amiram Eldar, Apr 01 2020

Since the sum of divisors of each term is also a Fibonacci number, this sequence is a subsequence of A272412.

Cf. A000045, A000203, A083207, A272412.

fibs = Fibonacci @ Range[2, 40]; seqQ[n_] := MemberQ[fibs, DivisorSigma[1, n]] && Module[{d = Divisors[n], s}, s = Round[Plus @@ d/GoldenRatio]; c = CoefficientList[Product[1 + x^i, {i, d}], x]; c[[1 + s]] > 0]; Select[Range[10^5], seqQ]

66 is a term since its divisors {1, 2, 3, 6, 11, 22, 33, 66} can be partitioned into the two disjoint sets, {2, 3, 6, 11, 33} and {1, 22, 66}, whose sums, 55 and 89, are 2 consecutive Fibonacci numbers.

a(13)-a(16) from Giovanni Resta, Apr 02 2020

A272589 Numbers n such that the equation F(n) = sigma(F(i) + F(j)) has a solution with i >= 1 and j >= 0, where F(k) = A000045(k) represents the k-th Fibonacci number.

Original entry on oeis.org

1, 2, 4, 6, 7, 12, 24

Offset: 1

Altug Alkan, May 03 2016

Corresponding distinct F(n) values for listed terms are 1, 3, 8, 13, 144, 46368.
Corresponding F(i) + F(j) values are for listed terms are 1, 2, 7, 9, 94, 28678.
It is known that for almost all positive integers n, the sum of divisors of Fibonacci(n) is not a Fibonacci number (see A272412). This sequence focuses on the sums of two Fibonacci numbers for a similar question. Since A000045 is obvious subsequence of A084176 by definition of Fibonacci numbers, the reason of existence of this sequence can be seen as a generalized version of question that is motivation of A272412.

			7 is a term because Fibonacci(7) = 13 = sigma(1 + 8) and 1, 8 are Fibonacci numbers.

Cf. A000203, A000045, A084176, A272412.

A279359 Even numbers whose sum of proper divisors is a Fibonacci number.

Original entry on oeis.org

2, 4, 10, 18, 36, 62, 90, 902, 1214, 2936, 3436, 325726, 378646, 646840, 5432174, 273222344, 285775216, 1411668430, 1416411790, 2586905784, 34620518734, 564515976926, 811882646208, 1326864647898, 1548008754928, 2063984735284, 28933238284712, 29040521400152, 31260018191912, 31586954074712, 160595113092766, 164736642645358, 180312970059936

Offset: 1

Altug Alkan, Dec 16 2016

nonn

What is the asymptotic behavior of this sequence?

			18 is a term because 1 + 2 + 3 + 6 + 9 = 21 is a Fibonacci number.

Cf. A000045, A001065, A235710, A272412.

isFibonacci(n)=my(k=n^2); issquare(k+=(k+1)<<2) || (n>0 && issquare(k-8));
is(n)=isFibonacci(sigma(n)-n)&&n%2==0

a(18)-a(20) from Amiram Eldar, Oct 18 2019

a(21)-a(33) from Max Alekseyev, May 23 2025

A290936 a(n) is the least k such that sigma(k) is a Fibonacci number when k is the product of n distinct primes, or 0 if no such k exists.

Original entry on oeis.org

2, 94, 66, 19290, 2000006490, 247917529768610, 276320525457530886869600795810

Offset: 1

Altug Alkan, Aug 14 2017

			a(4) = 19290 since 19290 = 2*3*5*643, sigma(19290) = 3*4*6*644 = 46368 = Fibonacci(24) and 19290 is the least number with this property.
a(5) = 2000006490 since 2000006490 = 2*3*5*643*103681, sigma(2000006490) = 3*4*6*644*103682 = 4807526976 = Fibonacci(48) and 2000006490 is the least number with this property.

Cf. A000045, A000203, A272412.

isFibonacci(n)=my(k=n^2); issquare(k+=(k+1)<<2) || (n>0 && issquare(k-8));
a(n) = {my(k = 1); while (!issquarefree(k) || bigomega(k) != n || !isFibonacci(sigma(k)), k++); k;} \\ after Charles R Greathouse IV at A272412

a(5)-a(6) confirmed by Giovanni Resta, Aug 16 2017

a(7) from Daniel Suteu, Nov 23 2019

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A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

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A272440 Numbers n such that the average of the positive divisors of n is a Fibonacci number.

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A275789 Least k such that sigma(n) divides Fibonacci(k) (k > 0).

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A280545 Numbers k such that sigma(sigma(k)) is a Fibonacci number.

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A280555 Primes p such that sigma(sigma(p)) is a Fibonacci number.

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A329660 Numbers m such that sigma(m) is a Lucas number (A000032), where sigma(m) is the sum of divisors of m (A000203).

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A333677 Numbers whose divisors can be partitioned into two disjoint sets whose sums are consecutive Fibonacci numbers.

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A272589 Numbers n such that the equation F(n) = sigma(F(i) + F(j)) has a solution with i >= 1 and j >= 0, where F(k) = A000045(k) represents the k-th Fibonacci number.

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