A267415 -id:A267415 - cp's OEIS frontend

A376279 Numbers k such that k^k is a cube.

0, 1, 3, 6, 8, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 64, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 125, 126, 129, 132, 135, 138, 141, 144, 147, 150, 153, 156, 159, 162, 165, 168

Offset: 1

Chai Wah Wu, Sep 18 2024

nonn

Strict subsequence of A267415. For instance, 76, 112, 172, 364, 427, 532 are not terms of this sequence, but are terms of A267415.

Cf. A267415, A371587.

Union of A000578 and A008585.

q:= n-> andmap(i-> irem(n*i[2], 3)=0, ifactors(n)[2]):
select(q, [$0..200])[];  # Alois P. Heinz, Sep 19 2024

Join[{0},Select[Range[170], IntegerQ[#^(#/3)] &]] (* Stefano Spezia, Sep 18 2024 *)

isok(k) = ispower(k^k, 3); \\ Michel Marcus, Sep 18 2024

from sympy import integer_nthroot
def A376279(n):
    def f(x): return n-1+x-x//3-integer_nthroot(x,3)[0]+integer_nthroot(x//27,3)[0]
    m, k = n-1, f(n-1)
    while m != k: m, k = k, f(k)
    return m

from itertools import count, islice
from sympy import integer_nthroot
def A376279_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda k:not k%3 or integer_nthroot(k,3)[1],count(max(startvalue,0)))
A376279_list = list(islice(A376279_gen(),30))

k is a term if and only if k is a multiple of 3 or k is a cube.

A376291 Numbers k such that k^k is not a cube and can be expressed as (a^3 + b^3)/2 for positive integers a, b.

Original entry on oeis.org

76, 112, 172, 364, 427, 532

Offset: 1

Chai Wah Wu, Sep 19 2024

Sequence is equal to all terms in A267415 that are not in A376279.

If a or b are allowed to be 0, then 2, 4, 128, 256, 686, 1372, 2000, 4000, 4394, ... are also terms.

			76^76 is not a cube and is equal to (523974089123227128080087214816032969930445946880^3 + 314384453473936276848052328889619781958267568128^3)/2.
112^112 is not a cube and is equal to (39739105680019344543609706294181022974041418385894471812303329593638887882752^3 + 13246368560006448181203235431393674324680472795298157270767776531212962627584^3)/2.
172^172 is not a cube and is equal to (186302856478727791003189416646781404088735216286873615701287403506960347135763572314025783484358072432973760558663310530664464384^3 + 26614693782675398714741345235254486298390745183839087957326771929565763876537653187717969069194010347567680079809044361523494912^3)/2

Cf. A267415, A376279.

A273546 Integers n such that n^n is the average of a nonzero square and a positive cube.

Original entry on oeis.org

0, 1, 4, 6, 8, 9, 12, 13, 14, 18, 24, 25

Offset: 1

Altug Alkan, May 26 2016

Integers n such that n^n = (x^2 + y^3)/2 where x, y > 0, is soluble.

			13 is a term because 13^13 = (24134045^2 + 28561^3)/2.

Cf. A000312, A055394, A267415.

isA055394(n)=for(k=1, sqrtnint(n-1, 3), if(issquare(n-k^3), return(1))); 0
is(n)=isA055394(2*n^n) \\ Charles R Greathouse IV, Jun 08 2016

a(10)-a(12) from Giovanni Resta, May 26 2016

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A376279 Numbers k such that k^k is a cube.

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A376291 Numbers k such that k^k is not a cube and can be expressed as (a^3 + b^3)/2 for positive integers a, b.

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A273546 Integers n such that n^n is the average of a nonzero square and a positive cube.

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