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A078606
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Constant c(p) used in determining divisibility by the n-th prime, p=A000040(n), for n>=4.
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3
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-2, -1, 4, -5, 2, 7, 3, -3, -11, -4, 13, -14, 16, 6, -6, -20, -7, 22, 8, 25, 9, -29, -10, 31, -32, 11, 34, -38, -13, -41, 14, 15, -15, -47, 49, -50, 52, 18, -18, -19, 58, -59, 20, -21, 67, -68, 23, 70, 24, -24, -25, -77, 79, 27, -27, -83, -28, 85, 88, -92, -31, 94, -95, -33, -101, -104, 35, 106, 36, -110, 112, 38
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OFFSET
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4,1
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COMMENTS
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To determine if N is divisible by p: Write N=10*M+D, where D=ones digit of N and M=N without ones digit. Then N is divisible by p iff M+c(p)*D is.
c(p) is the multiplicative inverse of 10 (mod p) with the smallest absolute value. (But note that the link below uses c(13)=9 rather than -4.)
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LINKS
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EXAMPLE
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The first few terms are c(7)=-2, c(11)=-1, c(13)=4. To find out if a number is divisible by 7, take the last digit, double it and subtract the result from the rest of the number. If you get an answer divisible by 7 (including 0), then the original number is divisible by 7. If you do not know the new number's divisibility, you can apply the rule again. Example: If you had 203, you would subtract 2*3=6 from 20 to get 14; since 14 is divisible by 7, so is 203.
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MATHEMATICA
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c[p_] := If[(v=PowerMod[10, -1, p])>p/2, v-p, v]; c/@Prime/@Range[4, 100]
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PROG
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(Python)
import sympy
[(pow(10, -1, p))-p*(p%10%6==1) for p in sympy.primerange(7, 300)]
(PARI) a(n) = centerlift(Mod(1, prime(n))/10); \\ Kevin Ryde, Feb 18 2023
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CROSSREFS
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KEYWORD
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sign,base
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AUTHOR
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Devora Rahav (rahav-d(AT)inter.net.il), Dec 09 2002
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EXTENSIONS
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STATUS
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approved
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