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A194269
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Numbers j such that Sum_{i=1..k} d(i)^i = j+1 for some k where d(i) is the sorted list of divisors of j.
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1
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4, 9, 25, 49, 68, 121, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 17500, 18769, 19321, 22201, 22801, 24649, 26569, 27889
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OFFSET
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1,1
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COMMENTS
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The majority of these numbers are squares.
The sequence of numbers j such that Sum_{i=1..k} d(i)^i = j generates the numbers 1, 130, 135, 288, 5083, 8064, 10130, ... (no more terms through 10^8).
All prime squares p^2 (A001248) are terms because the partial sum 1^1 + p^2 satisfy the condition.
Up to 10^8, the terms that are not squares are: 68, 17500, 5053176.
(End)
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LINKS
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EXAMPLE
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The divisors of 68 are 1, 2, 4, 17, 34, 68; 1^1 + 2^2 + 4^3 = 69, so 68 is a term.
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MAPLE
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isA194269 := proc(n) local dgs , i, k; dgs := sort(convert(numtheory[divisors](n), list)) ; for k from 1 to nops(dgs) do if add(op(i, dgs)^i, i=1..k) = n+1 then return true; end if; end do; false ; end proc:
for n from 1 to 30000 do if isA194269(n) then print(n); end if; end do: # R. J. Mathar, Aug 27 2011
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PROG
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(PARI) isok(n) = {my(d=divisors(n), s=0); for(k=1, #d, s += d[k]^k; if (s == n+1, return (1)); if (s > n+1, break); ); } \\ Michel Marcus, Mar 25 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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