A036907 -id:A036907 - cp's OEIS frontend

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

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;

A272524 Refactorable triangular numbers.

Original entry on oeis.org

1, 36, 136, 276, 1176, 2016, 2556, 2628, 3240, 4560, 11628, 12720, 12880, 18336, 18528, 25200, 32640, 32896, 51360, 64620, 73920, 86320, 89676, 100128, 114960, 115440, 126756, 131328, 148240, 166176, 248160, 253116, 265356, 270480, 294528, 295296, 320400, 345696, 373680, 380628, 400960, 401856, 438516

Offset: 1

Waldemar Puszkarz, May 01 2016

nonn

Intersection of A000217 and A033950.

			36 is a term as the number of divisors of 36 (see A000005) is 9 which divides 36.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000217 (triangular numbers), A033950 (refactorable numbers), A063440 (number of divisors of triangular numbers), A000005 (number of divisors), A036907 (refactorable squares).

Select[PolygonalNumber@Range@1000, Divisible[#, DivisorSigma[0,#]]&]

for (n=1, 1000, t=n*(n+1)/2; t%numdiv(t)==0 && print1(t ", " ))

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

cp's OEIS Frontend

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

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A272524 Refactorable triangular numbers.

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

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