A321392 -id:A321392 - cp's OEIS frontend

A321393 a(n) is the number of bases b > 1 such that n + digitsum(n, base b) is prime.

1, 1, 2, 2, 1, 1, 1, 3, 2, 2, 2, 3, 4, 4, 5, 4, 2, 4, 2, 3, 4, 3, 2, 5, 6, 5, 5, 5, 4, 6, 5, 5, 6, 6, 6, 7, 8, 7, 6, 7, 5, 7, 6, 5, 8, 7, 5, 10, 8, 6, 9, 10, 6, 9, 12, 8, 10, 11, 8, 10, 10, 8, 9, 12, 7, 12, 10, 9, 11, 11, 9, 11, 11, 10, 13, 11, 9, 12, 10, 9

Offset: 2

Rémy Sigrist, Nov 08 2018

For any n > 1 and b > n, n + digitsum(n, base b) equals 2*n and is composite, hence the sequence is well defined.
The sequence is not defined for n = 1 as 1 + digitsum(1, base b) equals 2 and is prime for any base b > 1.
In the scatterplot of the sequence, the points are separated into two beams according to whether n is divisible by 3 or not, then these beams are separated in two according to whether n is divisible by 5 or not, then similarly according to whether n is divisible by 7 or not; these separations seem to continue for each odd prime number; see scatterplot in Links section.

			For n = 9, we have:
  b     9 + sumdigits(9, base b)
  ----  ------------------------
     2   11 (prime)
     3   10
     4   12
     5   14
     6   13 (prime)
     7   12
     8   11 (prime)
     9   10
  >=10   18
Hence, a(9) = 3.

Rémy Sigrist, Table of n, a(n) for n = 2..10000
Rémy Sigrist, Colored scatterplot of the sequence for n = 2..250000
Rémy Sigrist, Colored scatterplot of (n, b) such that n + sumdigits(n, base b) is prime and 1 <= n <= 1000 and 2 <= b <= 1000 (where the color is function of n + sumdigits(n, base b))

Cf. A321392.

a(n) = sum(b=2, n, isprime(n + sumdigits(n, b)))

cp's OEIS Frontend

A321393 a(n) is the number of bases b > 1 such that n + digitsum(n, base b) is prime.

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