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A276765 Number of digits in n equals number of syllables in English name of n.

1, 2, 3, 4, 5, 6, 8, 9, 13, 14, 15, 16, 18, 19, 20, 30, 40, 50, 60, 80, 90, 100, 200, 300, 400, 500, 600, 800, 900, 7000, 70000, 700000, 800001, 800002, 800003, 800004, 800005, 800006, 800008, 800009, 800010, 800012, 801000, 802000, 803000, 804000, 805000

Offset: 1

Tyler Skywalker, Sep 17 2016

Numbers n such that A055642(n) = A075774(n). - Felix Fröhlich, Sep 30 2016

			From _Felix Fröhlich_, Sep 30 2016: (Start)
13 is a term, since the decimal expansion is two digits long and "thir-teen" has two syllables.
17 is not a term, since the decimal expansion is two digits long, but "se-ven-teen" has three syllables.
800001 is a term, since the decimal expansion is six digits long and "eight -hun-dred-thou-sand-one" has six syllables. (End)

Cf. A055642, A075774.

Edited and more terms from Felix Fröhlich, Sep 30 2016

A275465 a(n) = f^(n/f), where f is the smallest prime factor of n.

Original entry on oeis.org

2, 3, 4, 5, 8, 7, 16, 27, 32, 11, 64, 13, 128, 243, 256, 17, 512, 19, 1024, 2187, 2048, 23, 4096, 3125, 8192, 19683, 16384, 29, 32768, 31, 65536, 177147, 131072, 78125, 262144, 37, 524288, 1594323, 1048576, 41, 2097152, 43, 4194304, 14348907, 8388608, 47, 16777216

Offset: 2

Tyler Skywalker, Jul 28 2016

			For n = 12 = 2^2*3, the smallest prime factor of n is f = 2, so a(12) = f^(n/f) = 2^(12/2) = 2^6 = 64. - _Michael B. Porter_, Jul 31 2016

Chai Wah Wu, Table of n, a(n) for n = 2..1000

Cf. A020639, A032742.

a:= n-> (f-> f^(n/f))(min(numtheory[factorset](n))):
seq(a(n), n=2..50);  # Alois P. Heinz, Dec 11 2017

a[n_] := With[{f = FactorInteger[n][[1, 1]]}, f^(n/f)]; ; Array[a,50,2] (* JungHwan Min, Jul 29 2016 *)(* amended by Harvey P. Dale, Aug 12 2021 *)

a(n) = my(f=factor(n)[1, 1]); f^(n/f) \\ Felix Fröhlich, Jul 30 2016

from _future_ import division
from sympy import primefactors
def A275465(n):
    p = min(primefactors(n))
    return p**(n//p) # Chai Wah Wu, Jul 29 2016

a(p) = p, a(p^2) = p^p and a(p^m) = p^(p^(m-1)) for prime p. - Chai Wah Wu, Jul 29 2016
a(n) = A020639(n)^(n/A020639(n)). - Felix Fröhlich, Jul 30 2016
a(n) = A020639(n)^A032742(n). - Chai Wah Wu, Jul 30 2016

More terms from Chai Wah Wu, Jul 30 2016

A275203 Hyperoperations using consecutive integers.

Original entry on oeis.org

1, 2, 4, 64

Offset: 0

Tyler Skywalker, Jul 19 2016

nonn

a(4) is too large to display. The number of decimal digits in (the number of decimal digits in a(4)) is approximately 64^64*log[10](64), that is, about 7.1*10^115. - Robert Israel, Jul 19 2016

			a(0) = H_0(null,0) = 0+1 = 1.
a(1) = H_1(1,1) = 1+1 = 2.
a(2) = H_2(2,2) = 2*2 = 4.
a(3) = H_3(4,3) = 4^3 = 64.
a(4) = H_4(64,4) = 64^(64^(64^64)).

Wikipedia, Hyperoperation

Cf. A189896, A264930.

a(n) = H_n(a(n-1),n), where H_n is the n-th hyperoperation.

A275211 Numbers of the form p^^k, where p is prime, k > 1, and ^^ is the tetration operator: x^^y = x^x^...^x with y copies of x.

Original entry on oeis.org

4, 16, 27, 3125, 65536, 823543, 285311670611, 7625597484987, 302875106592253, 827240261886336764177, 1978419655660313589123979, 20880467999847912034355032910567

Offset: 1

Tyler Skywalker, Jul 19 2016

nonn

			a(1) = 2^^2 = 2^2 = 4.
a(2) = 2^^3 = 2^2^2 = 16.
a(3) = 3^^2 = 3^3 = 27.
a(4) = 5^^2 = 5^5 = 3125.

Charles R Greathouse IV, Table of n, a(n) for n = 1..79
Wikipedia, Tetration

Cf. A000040.

slogint(n,b)=if(nn<=lim, Set(v)) \\ Charles R Greathouse IV, Jul 19 2016

is(n)=my(p,e); e=isprimepower(n,&p); e && (e==p || (e%p==0 && is(e))) \\ Charles R Greathouse IV, Jul 19 2016

For any prime number, p, p tetrated x times, where x is any integer greater than 1, is a prime tetration.

a(5) inserted, a(10)-a(12) corrected by Charles R Greathouse IV, Jul 19 2016

A275120 List the least common multiples of {1, 2, ..., k} for k = 0, 1, ...; this sequence gives the length of the n-th block of consecutive equal numbers.

Original entry on oeis.org

2, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, 4, 2, 6, 2, 2, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 6, 6, 4, 2, 4, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 2, 8, 5, 1, 6, 6, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 6, 6, 4, 2, 4, 6, 2, 6, 6, 6, 4, 6

Offset: 1

Tyler Skywalker, Jul 18 2016

nonn

a(n) is the count of how many consecutive terms in A003418 are equal.

			lcm({}) = lcm({1}) = 1, so a(1) = 2.
lcm({1, 2}) = 2, so a(2) = 1.
lcm({1, 2, 3}) = 6, so a(3) = 1.
lcm({1, 2, 3, 4}) = 12, so a(4) = 1.
lcm({1, ..., 5}) = lcm({1, ..., 6}) = 60, so a(5) = 2.
lcm({1, ..., 7}) = 420, so a(6) = 1.
lcm({1, ..., 8}) = 840, so a(7) = 1.
lcm({1, ..., 9}) = lcm({1, ..., 10}) = 2520, so a(8) = 2.
lcm({1, ..., 11}) = lcm({1, ..., 12}) = 27720, so a(9) = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Frequency of given numbers using A003418.

Apart from the first term, same as A057820.

{2}~Join~Rest@ Most@ Map[Length, Split@ Table[LCM @@ Range@ n, {n, 396}]] (* Michael De Vlieger, Jul 18 2016 *)

do(lim)=my(v=List()); for(e=2,logint(lim\=1,2), forprime(p=2,sqrtnint(lim,e), listput(v,p^e))); v=Set(concat(Vec(v), primes([2,lim]))); concat(2, vector(#v-1,i,v[i+1]-v[i])) \\ Charles R Greathouse IV, Jul 18 2016

a(n) = A057820(n), n>1.

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User: Tyler Skywalker

A276765 Number of digits in n equals number of syllables in English name of n.

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A275465 a(n) = f^(n/f), where f is the smallest prime factor of n.

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A275203 Hyperoperations using consecutive integers.

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A275211 Numbers of the form p^^k, where p is prime, k > 1, and ^^ is the tetration operator: x^^y = x^x^...^x with y copies of x.

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A275120 List the least common multiples of {1, 2, ..., k} for k = 0, 1, ...; this sequence gives the length of the n-th block of consecutive equal numbers.

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Mathematica

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Formula