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A169917
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Squares in carryless arithmetic mod 10 with addition and multiplication of digits both defined to be multiplication mod 10.
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3
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0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 100, 111, 144, 199, 166, 155, 166, 199, 144, 111, 400, 441, 464, 469, 446, 405, 446, 469, 464, 441, 900, 991, 964, 919, 946, 955, 946, 919, 964, 991, 600, 661, 644, 649, 666, 605, 666, 649, 644, 661, 500, 551, 504, 559, 506, 555, 506, 559, 504
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OFFSET
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0,3
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COMMENTS
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The rules of arithmetic used in A169916, A169917, A169918 have very strange consequences. Many of the familiar laws fail. For instance, the arithmetic in A169916 is not associative: 10*(9*2) = 10*1 = 21 != (10*9)*2 = 9*2 = 1.
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LINKS
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FORMULA
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a(n) = a(n') if the i-th digit of n' either equals the i-th digit of n or (10 - the i-th digit of n): e.g., a(12345) = a(18365), because the 2nd and 4th digit of 12345 equal 10-(the 2nd resp. 4th digit of 18365), and the other digits are the same. In particular, a(10k+5+m) = a(10k+5-m), for m=0,...,4. - M. F. Hasler, Mar 26 2015
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EXAMPLE
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a(24) = 24*24 = 446:
...24
...24
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...86
..48.
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..446
(The rule for "adding" the columns is to multiply mod 10: 8+8 = 8 * 8 mod 10 = 4.)
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PROG
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(PARI) A169917(n)={#n=digits(n); n=apply(d->n*d, n)%10; sum(i=0, 2*#n-2, prod(j=max(1, #n-i), min(2*#n-1-i, #n), n[2*#n-i-j][j])%10*10^i)} \\ M. F. Hasler, Mar 26 2015
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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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