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A295020
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Demlo numbers: concat(L,M,R) where M and L + R are repdigits using the same digit, see comments for details.
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0
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 55, 60, 61, 62, 63, 66, 70, 71, 72, 77, 80, 81, 88, 90, 99, 110, 111, 121, 132, 143, 154, 165, 176, 187, 198
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OFFSET
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1,2
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COMMENTS
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Demlo numbers have been invented and defined by D. R. Kaprekar as numbers of the form L.M.R (concatenated) where L + R = m*A002275(p) and M = m*A002275(k), with 1 <= m <= 9, k = length(M) >= 0, p = length(R) >= length(L) >= p-1, A002275(k) = (10^k-1)/9; zero k or p means that M resp. L and R is/are absent, not zero. In the paper it is shown that any such number can be written as (9L + m)*A002275(k+p).
Best known are the "Wonderful Demlo numbers" A002477(n) = A002275(n)^2 = 1...n...1, 1 <= n <= 9. (This expression does not yield a Demlo number for n=10, 19, 28, ...)
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LINKS
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K. R. Gunjikar and D. R. Kaprekar, Theory of Demlo numbers, J. Univ. Bombay, Vol. VIII, Part 3, Nov. 1939, pp. 3-9. [Annotated scanned copy]
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EXAMPLE
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Any repdigit number (cf. A010785) > 1, any one or two digit number L.R with digit sum m = L + R < 10, and any such number multiplied by a repunit 1...1, L.R*1...1 = L.M.R (where M = digit m repeated length(1...1)-1 times), satisfy the definition.
In Kaprekar's paper it is shown that all Demlo numbers (as defined in the comment) are of that form, cf. formula in comments.
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PROG
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(PARI) is_A295020(n, d=digits(n), N=#d)={for(r=!n, N, my(p=(1+N-r)\2); r>1 && #Set(d[N-p-r+1..N-p])>1 && return; (!p||((n%10^p>=10^(p-1)||p==1)&&(p==#p=digits(n\10^(p+r)+n%10^p))&&if(r, Set(p)==[d[N-#p]], #Set(p)==1))) && return(1))}
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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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