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A331787
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T(b,n) is the largest m such that there exists N such that none of S(N), S(N+1), ..., S(N+m-1) is divisible by n, where S(N) is the sum of digits of N in base b. Square array read by ascending diagonals.
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4
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0, 0, 2, 0, 1, 6, 0, 2, 4, 14, 0, 1, 2, 7, 30, 0, 2, 4, 6, 16, 62, 0, 1, 4, 3, 14, 25, 126, 0, 2, 2, 6, 8, 14, 52, 254, 0, 1, 4, 5, 4, 13, 30, 79, 510, 0, 2, 4, 6, 8, 10, 28, 62, 160, 1022, 0, 1, 2, 3, 8, 5, 22, 23, 62, 241, 2046, 0, 2, 4, 6, 8, 10, 12, 34, 48, 126, 484, 4094
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OFFSET
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2,3
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COMMENTS
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Write n = (b-1)*s + t, 1 <= t <= b-1. The smallest N_0 such that none of S(N_0), S(N_0+1), ..., S(N_0+m-1) is divisible by n is given by N_0 = b^(u_0) - b^s*(t-gcd(t,b-1)+1) + 1, where u_0 is the smallest nonnegative solution to (b-1)*u == -gcd(t,b-1) (mod n). See my link below for more detailed information.
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LINKS
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FORMULA
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If n = (b-1)*s + t, 1 <= t <= b-1, then T(b,n) = b^s*(2*t-gcd(t,b-1)+1) - 2. See my link for a proof of the formula.
T(b,n) = T(b,n-1) + b*T(b,n-b+1) - b*T(b,n-b) for b >= 2, n >= b+1.
T(b,n) = O(b^(n/(b-1))).
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EXAMPLE
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Table begins
b\n 1 2 3 4 5 6 7 8 9 10
2 0 2 6 14 30 62 126 254 510 1022
3 0 1 4 7 16 25 52 79 160 241
4 0 2 2 6 14 14 30 62 62 126
5 0 1 4 3 8 13 28 23 48 73
6 0 2 4 6 4 10 22 34 46 34
7 0 1 2 5 8 5 12 19 26 47
8 0 2 4 6 8 10 6 14 30 46
9 0 1 4 3 8 9 12 7 16 25
10 0 2 2 6 8 8 12 14 8 18
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PROG
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(PARI) T(b, n) = my(s=(n-1)\(b-1), t=(n-1)%(b-1)+1); b^s*(2*t-gcd(t, b-1)+1)-2
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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