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User: Walter Roscello

Walter Roscello has authored 2 sequences.

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 *)

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User: Walter Roscello

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

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