A292787 -id:A292787 - cp's OEIS frontend

A292788 For n > 1, a(n) = least positive k, not a power of n, such that the digital sum of k in base n equals the digital sum of k^3 in base n.

56953, 13, 2, 3, 20, 2, 6, 3, 8, 5, 1110, 3, 65, 8, 4, 7, 86, 9, 2374, 4, 8, 12, 114, 3, 99, 12, 135, 15, 3567, 4, 185, 15, 11, 16, 6, 19, 73, 20, 12, 5, 81, 6, 85, 23, 19, 24, 93, 7, 97, 24, 18, 27, 796, 28, 44, 7, 19, 28, 413, 4, 365, 32, 8, 31, 26, 21, 200

Offset: 2

Rémy Sigrist, Sep 23 2017

The term a(10) = 8 belongs to A070276.
For any n > 1, a(n^2) <= n.
Is this sequence defined for any n > 1 ?
Apparently, a(k) < k for any odd k > 3.
Among the first 99 999 terms, the digital sum of a(n) in base n is > n for n = 2, 12, 20, 30.
The scatterplot of the sequence shows beams on the upper part, which correspond to clusters of close points for which a(n) = k*n + (n-k-e) for some k > 0 and e in { 0, 2 }.
See also A292787 for a similar sequence involving squares instead of cubes.
The least positive k, not a power of 2, such that the hamming weight of k equals the hamming weight of k^4 is 34225258495.

			For n = 3:
- let d_3 denote the digital sum in base 3 (d_3 = A053735),
- 1 is a power of 3,
- d_3(2) = 2 and d_3(2^3) = 4,
- 3 is a power of 3,
- d_3(4) = 2 and d_3(4^3) = 4,
- d_3(5) = 3 and d_3(5^3) = 7,
- d_3(6) = 2 and d_3(6^3) = 4,
- d_3(7) = 3 and d_3(7^3) = 5,
- d_3(8) = 4 and d_3(8^3) = 8,
- 9 is a power of 3,
- d_3(10) = 2 and d_3(10^3) = 4,
- d_3(11) = 3 and d_3(11^3) = 9,
- d_3(12) = 2 and d_3(12^3) = 4,
- d_3(13) = 3 and d_3(13^3) = 3,
- hence a(3) = 13.

Rémy Sigrist, Table of n, a(n) for n = 2..10000
Rémy Sigrist, Colorized scatterplot of the sequence for n=2..100000

Cf. A053735, A070276, A292787.

With[{kk = 10^5}, Table[SelectFirst[Complement[Range[2, kk], n^Range@ Floor@ Log[n, kk]], Total@ IntegerDigits[#, n] == Total@ IntegerDigits[#^3, n] &] /. k_ /; MissingQ@ k -> -1, {n, 2, 68}]] (* Michael De Vlieger, Sep 24 2017 *)

a(n) = my (p=1); for (k=1, oo, if (k==p, p*=n, if (sumdigits(k,n) == sumdigits(k^3,n), return (k))))

A292832 a(n) is the least positive k, not a power of 2, such that the Hamming weight of k equals the Hamming weight of k^n, or 0 if no such k exists.

Original entry on oeis.org

3, 3, 56953, 34225258495

Offset: 1

Rémy Sigrist, Sep 24 2017

The Hamming weight of a number n is given by A000120(n).
Is this sequence positive for every positive integer?
a(2) = A292787(2).
a(3) = A292788(2).

			For n = 2:
- 1 is a power of 2,
- 2 is a power of 2,
- A000120(3) = 2 and A000120(3^2) = 2,
- hence a(2) = 3.

Cf. A000120, A292787, A292788, A363836.

a(n) = forstep(k=3, oo, 2, if (hammingweight(k) == hammingweight(k^n), return (k)))

A321366 a(n) is the least integer k greater than 1 such that n divides binomial(k, 2) = A000217(k-1).

Original entry on oeis.org

2, 4, 3, 8, 5, 4, 7, 16, 9, 5, 11, 9, 13, 8, 6, 32, 17, 9, 19, 16, 7, 12, 23, 16, 25, 13, 27, 8, 29, 16, 31, 64, 12, 17, 15, 9, 37, 20, 13, 16, 41, 21, 43, 33, 10, 24, 47, 33, 49, 25, 18, 40, 53, 28, 11, 49, 19, 29, 59, 16, 61, 32, 28, 128, 26

Offset: 1

Kevin Long, Aug 27 2019

Kevin Long, Table of n, a(n) for n = 1..10000
Kevin Long, Python code for sequence

Cf. A000217, A011772, A292787.

a(n) = {my(s=1, k=2); while(s%n, s+=k; k++); k} \\ Andrew Howroyd, Aug 27 2019
(Python 3.8+)
from itertools import combinations
from math import prod
from sympy import factorint, divisors
from sympy.ntheory.modular import crt
def A321366(n):
    plist = [p**q for p, q in factorint(2*n).items()]
    if len(plist) == 1:
        return int((2 - plist[0] % 2)*n)
    return 1+int(min(min(crt([m,2*n//m],[0,-1])[0],crt([2*n//m,m],[0,-1])[0]) for m in (prod(d) for l in range(1,len(plist)//2+1) for d in combinations(plist,l)))) # Chai Wah Wu, Jun 03 2021

a(n) = A011772(n) + 1.
a(2^n) = 2^(n+1).
a(p^n) = p^n for odd primes p.

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A292788 For n > 1, a(n) = least positive k, not a power of n, such that the digital sum of k in base n equals the digital sum of k^3 in base n.

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A292832 a(n) is the least positive k, not a power of 2, such that the Hamming weight of k equals the Hamming weight of k^n, or 0 if no such k exists.

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A321366 a(n) is the least integer k greater than 1 such that n divides binomial(k, 2) = A000217(k-1).

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