A081381 -id:A081381 - cp's OEIS frontend

A235524 Primitive refactorable (or tau) numbers: refactorable numbers which are not part of any family.

1, 2, 8, 9, 12, 18, 72, 80, 96, 108, 128, 288, 448, 625, 720, 864, 972, 1152, 1200, 1250, 1620, 1944, 2000, 2025, 2560, 4032, 4050, 5000, 5625, 6144, 6561, 6912, 7500, 7776, 9408, 10800, 11250, 11264, 12960, 13122, 16200, 18000, 18432, 19440, 20412, 21952

Offset: 1

Walter Roscello, Jan 11 2014

To be "primitive", the set of prime factors of N and of d(N) must be identical, otherwise any prime only in N is arbitrary and this defines a family of refactorable numbers.  These are referred to as generators in the Zelinsky reference.
This sequence is therefore the intersection of the refactorable numbers (A033950) and those numbers with identical sets of prime factors for N and d(N) (A081381).
The first numbers in A081381 which are not in this sequence are 486, 768, 8748, and 303750.  This sequence is A235525.

			720 is in the sequence since 720 = 2^4 * 3^2 * 5^1, therefore the prime decomposition of d(720) is 5 * 3 * 2 and each prime in 720 is required to make it refactorable.

Walter Roscello and Giovanni Resta, Table of n, a(n) for n = 1..1008 (terms < 10^11, first 459 terms from Walter Roscello)
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8

Cf. A000005, A033950, A081381, A235525.

A235525 Numbers which have identical primes in n and d(n) but are not refactorable.

Original entry on oeis.org

486, 768, 8748, 303750, 354294, 393216, 480000, 506250, 984150, 1179648, 1228800, 1417176, 3906250, 5467500, 6635520, 9841500, 18750000, 24504606, 25312500, 35156250, 47829690, 57177414, 57395628, 83886080, 90354432, 123018750, 153600000, 154140672, 156243654, 201326592, 210937500, 221433750, 245760000, 258280326, 382637520, 460800000, 492075000, 600000000

Offset: 1

Walter Roscello, Jan 11 2014

nonn

Numbers in A081381 that are not in A033950.

Although the set of primes in d(n) and n are identical, there is at least one prime occurring with a higher power in d(n) than in n.

			486 = 2^1 * 3^5 therefore d(486) = 2 * 6 = 2^2 * 3^1
768 = 2^8 * 3^1 therefore d(768) = 9 * 2 = 2^1 * 3^2
Each has the same set of primes in n and d(n) but has too many of one of the primes in d(n) to be refactorable.

Walter Roscello and Giovanni Resta, Table of n, a(n) for n = 1..100 (first 50 terms from Walter Roscello)

A000005, A033950, A081381, A235524

Select[Range[10^6], Mod[#, t = DivisorSigma[0, #]] > 0 && First /@ FactorInteger[#] == First /@ FactorInteger[t] &] (* Giovanni Resta, Jan 11 2014 *)

A290498 Numbers m such that the set of distinct prime divisors of the number of divisors of Fibonacci(m) is equal to the set of distinct prime divisors of m.

Original entry on oeis.org

1, 4, 8, 16, 24, 32, 60, 64, 72, 96, 128, 192, 256, 300, 336, 512, 576, 648, 900, 1008, 1024, 1080, 1250

Offset: 1

Altug Alkan, Aug 04 2017

Thanks to squarefree terms of A058635, numbers of the form 2^k appear in this sequence for k > 1. However it is not proven yet whether it is always true.
From Jon E. Schoenfield, Aug 05 2017: (Start)
The difficulty in extending this sequence is that it becomes hard to obtain the complete prime factorization of Fibonacci(m) as m increases. However, since every number having an odd number of divisors is a square, and the largest Fibonacci number that is also a square is Fibonacci(12) = 144, we can confine the search for terms > 12 to even numbers only.
Even for values of m for which we are unable to completely factorize Fibonacci(m), we can determine with a high degree of confidence whether m is in the sequence by considering only the multiplicities of the smaller primes in those factorizations, because multiplicities greater than 1 in the prime factorizations of Fibonacci numbers rarely occur among the larger prime factors. If, in place of the actual complete factorization of Fibonacci(m) for each examined value of m, we were to use only the multiplicities of the prime factors of Fibonacci(m) that are less than 10000 (which are quickly and easily counted using trial division), the terms we would obtain for this sequence would begin with 1, 4, 8, 16, 24, 32, 60, 64, 72, 96, 128, 192, 256, 300, 336, 512, 576, 648, 900, 1008, 1024, 1080, 1250, 1500, 1536, 1620, 1920, 2048, 2352, 2500, 2592, 2700, 4096, 4608, 5000, 5184, 5400, 5832, 7500, 8100, 8192, 8448, 8640, 9072, 9600, 10000, 13608, 15000, ...
Perhaps surprisingly, we would get the same terms (up through at least a(141) = 960000) if, instead of the multiplicities of prime factors <= 10000, we were to use the multiplicities of just the prime factors <= 13. (End)

			72 is a term because d(Fibonacci(2^3*3^2)) = 2^9*3.
300 is a term because d(Fibonacci(2^2*3*5^2)) = 2^15*3^2*5.

Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000005, A000045, A022307, A038575, A058635, A063375, A081381, A086597, A135939.

Select[Range[12]~Join~Range[14, 300, 2], Apply[SameQ, Map[FactorInteger[#][[All, 1]] &, {#, DivisorSigma[0, Fibonacci@ #]}]] &] (* Michael De Vlieger, Aug 07 2017 *)

is(n) = factor(numdiv(fibonacci(n)))[,1]==factor(n)[,1] \\ David A. Corneth, Aug 04 2017

a(16)-a(19) from David A. Corneth, Aug 04 2017
a(20)-a(22) from Jon E. Schoenfield, Aug 05 2017
a(23) from Amiram Eldar, Oct 14 2019

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A235524 Primitive refactorable (or tau) numbers: refactorable numbers which are not part of any family.

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A235525 Numbers which have identical primes in n and d(n) but are not refactorable.

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A290498 Numbers m such that the set of distinct prime divisors of the number of divisors of Fibonacci(m) is equal to the set of distinct prime divisors of m.

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