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A355017 a(n) is the number of bases in 2..n in which the sum of the digits of n is prime.

%I A355017 #49 Jun 18 2022 19:56:32
%S A355017 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,
%T A355017 16,10,17,10,20,12,20,14,18,13,21,13,22,13,20,14,25,14,22,18,22,15,26,
%U A355017 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
%N A355017 a(n) is the number of bases in 2..n in which the sum of the digits of n is prime.
%C A355017 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:
%C A355017 Observations:
%C A355017 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).
%C A355017 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).
%C A355017 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).
%C A355017 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).
%C A355017 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.
%H A355017 Samuel Harkness, <a href="/A355017/b355017.txt">Table of n, a(n) for n = 2..10000</a>
%H A355017 Samuel Harkness, <a href="/A355017/a355017_2.jpg">Colored Scatterplot of the first 50000 terms, with n as multiples of 2 or 3</a>
%H A355017 Samuel Harkness, <a href="/A355017/a355017_3.jpg">Colored Scatterplot of the first 50000 terms, with the lowest factor of n among 5, 7, 11, or 13</a>
%H A355017 Rémy Sigrist, <a href="/A355017/a355017.png">Scatterplot of (n, b) such that the sum of digits of n in base b is prime (with n = 1..1000, b = 2..n)</a>.
%H A355017 Rémy Sigrist, <a href="/A355017/a355017_1.png">Colored scatterplot of the first 50000 terms</a> (where the color is function of gcd(n, 2*3*5)).
%e A355017 For n=7, express 7 in all bases from 2 to 7, then add the numbers, counting those which are prime:
%e A355017   base 2: 1 1 1 --> 1+1+1=3 prime
%e A355017   base 3: 2 1   --> 2+1=3   prime
%e A355017   base 4: 1 3   --> 1+3=4   nonprime
%e A355017   base 5: 1 2   --> 1+2=3   prime
%e A355017   base 6: 1 1   --> 1+1=2   prime
%e A355017   base 7: 1     --> 1=1     nonprime
%e A355017 The sum of the digits of the base-b expansion of 7 in 4 different bases b (2, 3, 5, and 6) from base 2 to 7 is prime, so a(7)=4.
%t A355017 a[n_] := Count[Range[2, n], _?(PrimeQ[Plus @@ IntegerDigits[n, #]] &)]; Array[a, 84, 2] (* _Amiram Eldar_, Jun 17 2022 *)
%o A355017 (PARI) a(n) = sum(b=2, n, isprime(sumdigits(n, b))); \\ _Michel Marcus_, Jun 16 2022
%o A355017 (Python)
%o A355017 from sympy.ntheory import isprime, digits
%o A355017 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
%Y A355017 Cf. A008588, A047235, A016945, A007310, A028834, A052294.
%K A355017 nonn,look
%O A355017 2,4
%A A355017 _Samuel Harkness_, Jun 15 2022