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a(n) is the least base b >= 2 where the sum of digits of n is a prime number.
2

%I #10 Jun 18 2022 08:41:50

%S 3,2,3,2,2,2,4,2,2,2,2,2,2,3,8,2,2,2,2,2,2,3,2,2,2,5,2,3,3,2,4,2,2,2,

%T 2,2,2,3,2,2,2,3,2,3,4,2,2,2,2,3,2,3,3,2,2,3,4,2,6,2,2,3,18,2,2,2,2,2,

%U 2,3,2,2,2,3,2,3,6,2,2,2,2,3,2,3,4,2,2

%N a(n) is the least base b >= 2 where the sum of digits of n is a prime number.

%C The sequence is well defined:

%C - a(2) = 3,

%C - for n >= 3, the expansion of n in base n-1 is "11", with sum of digits 2.

%H Rémy Sigrist, <a href="/A355034/b355034.txt">Table of n, a(n) for n = 2..10000</a>

%F a(n) = 2 iff n belongs to A052294.

%F a(n) <= n-1 for any n >= 3.

%e For n = 16:

%e - we have the following expansions and sum of digits:

%e b 16_b Sum of digits in base b

%e - ------- -----------------------

%e 2 "10000" 1

%e 3 "121" 4

%e 4 "100" 1

%e 5 "31" 4

%e 6 "24" 6

%e 7 "22" 4

%e 8 "20" 2

%e - so a(16) = 8.

%o (PARI) a(n) = for (b=2, oo, if (isprime(sumdigits(n,b)), return (b)))

%o (Python)

%o from sympy import isprime

%o from sympy.ntheory.digits import digits

%o def a(n):

%o b = 2

%o while not isprime(sum(digits(n, b)[1:])): b += 1

%o return b

%o print([a(n) for n in range(2, 89)]) # _Michael S. Branicky_, Jun 16 2022

%Y Cf. A052294, A216789, A355035 (corresponding prime numbers).

%K nonn,base

%O 2,1

%A _Rémy Sigrist_, Jun 16 2022