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A316650
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Result when n is divided by the sum of its digits and the resulting integer is concatenated to the remainder.
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7
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10, 10, 10, 10, 10, 10, 10, 10, 10, 100, 51, 40, 31, 24, 23, 22, 21, 20, 19, 100, 70, 52, 43, 40, 34, 32, 30, 28, 27, 100, 73, 62, 53, 46, 43, 40, 37, 35, 33, 100, 81, 70, 61, 54, 50, 46, 43, 40, 310, 100, 83, 73, 65, 60, 55, 51, 49, 46, 43, 100, 85, 76, 70, 64, 510, 56, 52, 412, 49, 100, 87
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OFFSET
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1,1
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COMMENTS
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When the remainder is zero, this 0 is still concatenated to the result (12/3 becomes 40).
All terms of A052224 are fixed points; more generally, if n has digital sum 10^k for some k > 0 and the remainder of n when divided by 10^k has k decimal digits, then n is a fixed point (corrected by Rémy Sigrist, Jul 10 2018).
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LINKS
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FORMULA
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a(10^k) = 10^(k+1) for any k >= 0. - Rémy Sigrist, Jul 10 2018
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EXAMPLE
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1 divided by 1 is 1 with the remainder 0, thus a(1) = 10;
2 divided by 2 is 1 with the remainder 0, thus a(2) = 10;
3 divided by 3 is 1 with the remainder 0, thus a(3) = 10;
4 divided by 4 is 1 with the remainder 0, thus a(4) = 10;
...
10 divided by (1+0) is 10 with the remainder 0, thus a(10) = 100;
11 divided by (1+1) is 5 with the remainder 1, thus a(11) = 51;
12 divided by (1+2) is 4 with the remainder 0, thus a(12) = 40;
13 divided by (1+3) is 3 with the remainder 1, thus a(13) = 31;
...
2018 divided by (2+0+1+8) is 183 with the remainder 5, thus a(2018) = 1835.
Etc.
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MATHEMATICA
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Array[FromDigits@ Flatten[IntegerDigits@ # & /@ QuotientRemainder[#, Total[IntegerDigits@ #]]] &, 71] (* Michael De Vlieger, Jul 10 2018 *)
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PROG
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(PARI) a(n, base=10) = my (ds=sumdigits(n, base), q=n\ds, r=n%ds); q * base^max(1, #digits(r, base)) + r \\ Rémy Sigrist, Jul 10 2018
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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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