A216789 -id:A216789 - cp's OEIS frontend

A355034 a(n) is the least base b >= 2 where the sum of digits of n is a prime number.

3, 2, 3, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 3, 8, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 5, 2, 3, 3, 2, 4, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 4, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 4, 2, 6, 2, 2, 3, 18, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 6, 2, 2, 2, 2, 3, 2, 3, 4, 2, 2

Offset: 2

Rémy Sigrist, Jun 16 2022

The sequence is well defined:
- a(2) = 3,
- for n >= 3, the expansion of n in base n-1 is "11", with sum of digits 2.

			For n = 16:
- we have the following expansions and sum of digits:
     b  16_b     Sum of digits in base b
     -  -------  -----------------------
     2  "10000"                        1
     3    "121"                        4
     4    "100"                        1
     5     "31"                        4
     6     "24"                        6
     7     "22"                        4
     8     "20"                        2
- so a(16) = 8.

Rémy Sigrist, Table of n, a(n) for n = 2..10000

Cf. A052294, A216789, A355035 (corresponding prime numbers).

a(n) = for (b=2, oo, if (isprime(sumdigits(n,b)), return (b)))

from sympy import isprime
from sympy.ntheory.digits import digits
def a(n):
    b = 2
    while not isprime(sum(digits(n, b)[1:])): b += 1
    return b
print([a(n) for n in range(2, 89)]) # Michael S. Branicky, Jun 16 2022

a(n) = 2 iff n belongs to A052294.

a(n) <= n-1 for any n >= 3.

A352671 a(n) is the number of nonnegative numbers k < n such that for any base b >= 2, the sum of digits of n and k in base b are different.

Original entry on oeis.org

1, 1, 2, 2, 2, 1, 3, 3, 3, 2, 5, 2, 2, 3, 4, 3, 4, 3, 4, 2, 2, 1, 7, 2, 2, 3, 6, 3, 5, 1, 4, 4, 3, 3, 7, 3, 3, 3, 7, 3, 4, 1, 4, 4, 2, 2, 8, 3, 4, 4, 3, 4, 6, 4, 7, 3, 2, 1, 9, 2, 2, 3, 7, 5, 8, 3, 4, 3, 5, 2, 7, 4, 4, 3, 5, 4, 4, 1, 8, 4, 4, 3, 7, 3, 2, 2, 6

Offset: 1

Rémy Sigrist, Mar 28 2022

See A352740 for the corresponding k's.

			The first terms, alongside the corresponding k's, are:
  n   a(n)  k's
  --  ----  --------------
   1     1  0
   2     1  0
   3     2  0, 2
   4     2  0, 3
   5     2  0, 4
   6     1  0
   7     3  0, 4, 6
   8     3  0, 6, 7
   9     3  0, 7, 8
  10     2  0, 8
  11     5  0, 6, 8, 9, 10
  12     2  0, 11

Rémy Sigrist, Scatterplot of (x, y) such that x, y <= 1000 and for any base b >= 2, the sum of digits of x and y in base b are different

Cf. A216789, A352740.

a(n) = { my (v=0); for (k=0, n-1, my (ok=1); for (b=2, max(2, n+1), if (sumdigits(n,b)==sumdigits(k,b), ok=0; break)); v+=ok); v }

A352740 Irregular table T(n, k) read by rows; the n-th row contains, in ascending order, the numbers k < n such that for any base b >= 2, the sum of digits of n and k in base b are different.

Original entry on oeis.org

0, 0, 0, 2, 0, 3, 0, 4, 0, 0, 4, 6, 0, 6, 7, 0, 7, 8, 0, 8, 0, 6, 8, 9, 10, 0, 11, 0, 12, 0, 10, 12, 0, 8, 12, 14, 0, 13, 15, 0, 14, 15, 16, 0, 15, 16, 0, 16, 17, 18, 0, 19, 0, 20, 0, 0, 12, 16, 18, 20, 21, 22, 0, 23, 0, 24, 0, 18, 24, 0, 14, 21, 24, 25, 26

Offset: 1

Rémy Sigrist, Mar 31 2022

A352671 gives row lengths.

			irregular table begins:
     1:    [0]
     2:    [0]
     3:    [0, 2]
     4:    [0, 3]
     5:    [0, 4]
     6:    [0]
     7:    [0, 4, 6]
     8:    [0, 6, 7]
     9:    [0, 7, 8]
    10:    [0, 8]

Rémy Sigrist, Table of n, a(n) for n = 1..10302 (rows for n = 1..2000, flattened)
Rémy Sigrist, Scatterplot of (x, y) such that x, y <= 1000 and for any base b >= 2, the sum of digits of x and y in base b are different

Cf. A216789, A352671 (row lengths).

row(n) = { my (v=[]); for (k=0, n-1, my (ok=1); for (b=2, max(2, n+1), if (sumdigits(n, b)==sumdigits(k, b), ok=0; break)); if (ok, v=concat(v,k))); v }

T(n, 1) = 0.

A355035 Consider the least base b >= 2 where the sum of digits of n is a prime number; a(n) corresponds to this prime number.

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 5, 2, 3, 3, 3, 3, 3, 2, 5, 2, 2, 2, 3, 2, 3, 3, 3, 2, 3, 3, 5, 3, 3, 7, 5, 2, 3, 3, 5, 3, 7, 2, 5, 3, 3, 7, 5, 5, 5, 5, 3, 13, 2, 2, 3, 2, 3, 3, 7, 2, 3, 3, 5, 3, 7, 3, 5, 2, 3, 3, 3, 3, 3, 5, 5, 3

Offset: 2

Rémy Sigrist, Jun 16 2022

			For n = 16:
- we have the following expansions and sum of digits:
     b  16_b     Sum of digits in base b
     -  -------  -----------------------
     2  "10000"                        1
     3    "121"                        4
     4    "100"                        1
     5     "31"                        4
     6     "24"                        6
     7     "22"                        4
     8     "20"                        2
- so a(16) = 2.

Rémy Sigrist, Table of n, a(n) for n = 2..10000

Cf. A216789, A355034 (corresponding b's).

a(n) = my (s); for (b=2, oo, if (isprime(s=sumdigits(n,b)), return (s)))

from sympy import isprime
from sympy.ntheory.digits import digits
def s(n, b): return sum(digits(n, b)[1:])
def a(n):
    b = 2
    while not isprime(s(n, b)): b += 1
    return s(n, b)
print([a(n) for n in range(2, 89)]) # Michael S. Branicky, Jun 16 2022

a(n) = A216789(n, A355034(n)).

cp's OEIS Frontend

A355034 a(n) is the least base b >= 2 where the sum of digits of n is a prime number.

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A352671 a(n) is the number of nonnegative numbers k < n such that for any base b >= 2, the sum of digits of n and k in base b are different.

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A352740 Irregular table T(n, k) read by rows; the n-th row contains, in ascending order, the numbers k < n such that for any base b >= 2, the sum of digits of n and k in base b are different.

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A355035 Consider the least base b >= 2 where the sum of digits of n is a prime number; a(n) corresponds to this prime number.

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Python

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