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

2,3

Comments

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.

Links

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))

Example

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.

Prog

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

Crossrefs

Cf. A321392.

Sequence in context: A175685 A331182 A182591 * A058914 A238585 A123682

Adjacent sequences: A321390 A321391 A321392 * A321394 A321395 A321396

Keyword

nonn,base

Author

Rémy Sigrist, Nov 08 2018

Status

approved