A235524 -id:A235524 - cp's OEIS frontend

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

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 *)

A281495 Least k > 1 such that k^n is a refactorable number.

Original entry on oeis.org

2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 9, 15, 2, 17, 6, 19, 10, 21, 8, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 9, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78, 79

Offset: 1

Altug Alkan, Jan 22 2017

nonn

Theorem: There are infinitely many n-th power refactorable numbers for any given value of n > 1.
For proof see Alkan link.
Numbers n such that a(n) is not equal to A007947(n+1) are 13, 21, 40, 85, 121, 171, 182, 208, 312, 341, 364, 514, 562, 585, 661, 665, 781, ...
Primes p such that a(p-1) is not equal to p are 41, 313, 563, 1013, 1201, 1823, ....

			a(4) = 5 because 625 = 5^4 is the least fourth power refactorable number that is greater than 1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Altug Alkan, Proof of Theorem
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.
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. A001597, A007947, A033950, A036907, A235524.

isA033950(n) = n % numdiv(n) == 0;
a(n) = my(k=2); while (!isA033950 (k^n), k++); k;

A281389 Least k > 1 such that refactorable number k is an n-th power.

Original entry on oeis.org

2, 9, 8, 625, 7776, 117649, 128, 6561, 1000000000, 25937424601, 362797056, 23298085122481, 2541865828329, 29192926025390625, 32768, 48661191875666868481, 16926659444736, 104127350297911241532841, 10000000000000000000, 278218429446951548637196401

Offset: 1

Altug Alkan, Jan 21 2017

nonn

			a(4) = 625 because 625 = 5^4 is the least fourth power refactorable number that is greater than 1.

S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999.
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. A001597, A007947, A033950, A036907, A235524, A281495.

isA033950(n) = n % numdiv(n) == 0;
a(n) = {my(k=2); while (!isA033950 (k^n), k++); k^n; }

a(n) = A281495(n)^n.

More terms from Giovanni Resta, Jan 22 2017

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

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A281495 Least k > 1 such that k^n is a refactorable number.

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A281389 Least k > 1 such that refactorable number k is an n-th power.

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