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A343481
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a(n) is the sum of all digits of n in every prime base 2 <= p <= n.
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1
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1, 3, 3, 6, 6, 10, 11, 11, 10, 15, 16, 22, 21, 21, 23, 30, 32, 40, 42, 42, 39, 48, 52, 53, 49, 52, 53, 63, 66, 77, 83, 82, 76, 77, 82, 94, 87, 85, 90, 103, 107, 121, 123, 129, 120, 135, 144, 147, 153, 150, 151, 167, 176, 178, 185, 181, 168, 185, 194, 212, 199
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OFFSET
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2,2
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LINKS
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FORMULA
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a(n) ~ (1-Pi^2/12)*n^2/log(n) + c*n^2/log(n)^2 + o(n^2/log(n)^2), where c = 1 - Pi^2/24 + zeta'(2)/2 = 1 - A222171 - (1/2)*A073002 = 0.1199923561... (Fissum, 2020).
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EXAMPLE
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a(5) = 6 since in the prime bases 2, 3 and 5 the representations of 5 are 101_2, 12_3 and 10_5, respectively, and (1 + 0 + 1) + (1 + 2) + (1 + 0) = 6.
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MATHEMATICA
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s[n_, b_] := Plus @@ IntegerDigits[n, b]; ps[n_] := Select[Range[n], PrimeQ]; a[n_] := Sum[s[n, b], {b, ps[n]}]; Array[a, 100, 2]
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PROG
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(PARI) a(n) = sum(b=2, n, if (isprime(b), sumdigits(n, b))); \\ Michel Marcus, Apr 17 2021
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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