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

Author

Rémy Sigrist, Nov 08 2018

Keywords

• nonn
• base

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.

Examples

			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.
		

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))

Crossrefs

Cf. A321392.

Programs

PARI

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