

A355017


a(n) is the number of bases in 2..n in which the sum of the digits of n is prime.


1



0, 1, 1, 3, 4, 4, 3, 5, 6, 7, 7, 8, 7, 8, 5, 11, 9, 10, 8, 13, 8, 12, 9, 13, 11, 12, 10, 15, 11, 16, 10, 17, 10, 20, 12, 20, 14, 18, 13, 21, 13, 22, 13, 20, 14, 25, 14, 22, 18, 22, 15, 26, 12, 29, 17, 25, 15, 27, 15, 30, 19, 26, 14, 32, 17, 33, 19, 27, 19, 31, 18, 34, 19, 29, 19, 37, 16, 33, 21, 30, 24, 39, 20, 38
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

2,4


COMMENTS

The graph of (n,a(n)) shows an interesting structure, somewhat resembling a comet with four tails. Starting at the bottom tail and going upwards:
Observations:
The bottom "tail" contains all n with both 2 and 3 as prime factors, i.e., numbers n in A008588 (1/6 of all n).
The second "tail" contains all n with 2 as a prime factor but not 3, i.e., numbers n in A047235 (1/3 of all n).
The third "tail" contains all n with 3 as a prime factor but 2, i.e., numbers n in A016945 (1/6 of all n).
The top "tail" contains all n with neither 2 nor 3 as a prime factor, i.e., numbers n in A007310 (1/3 of all n).
The bottom of each "tail" contains n with 5 as a prime factor. Moving up within each "tail," the prime factors of each n tend to increase.


LINKS



EXAMPLE

For n=7, express 7 in all bases from 2 to 7, then add the numbers, counting those which are prime:
base 2: 1 1 1 > 1+1+1=3 prime
base 3: 2 1 > 2+1=3 prime
base 4: 1 3 > 1+3=4 nonprime
base 5: 1 2 > 1+2=3 prime
base 6: 1 1 > 1+1=2 prime
base 7: 1 > 1=1 nonprime
The sum of the digits of the baseb expansion of 7 in 4 different bases b (2, 3, 5, and 6) from base 2 to 7 is prime, so a(7)=4.


MATHEMATICA

a[n_] := Count[Range[2, n], _?(PrimeQ[Plus @@ IntegerDigits[n, #]] &)]; Array[a, 84, 2] (* Amiram Eldar, Jun 17 2022 *)


PROG

(PARI) a(n) = sum(b=2, n, isprime(sumdigits(n, b))); \\ Michel Marcus, Jun 16 2022
(Python)
from sympy.ntheory import isprime, digits
def A355017(n): return sum(1 for b in range(2, n) if isprime(sum(digits(n, b)[1:]))) # Chai Wah Wu, Jun 17 2022


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



