A368245 Numbers k such that there is a positive integer r for which k^(1/r) = digsum(k) - r.
4, 25, 64, 125, 196, 216, 289, 343, 2744, 3375, 4096, 3518743761, 13537086546263552, 15633814156853823
Offset: 1
Examples
The terms 4, 25, 64, 196, and 289 are obtained for r = 2: 4^(1/2) = sqrt(4) = 2 = digsum(4) - 2; 25^(1/2) = sqrt(25) = 5 = digsum(25) - 2 = 7 - 2; 64^(1/2) = sqrt(64) = 8 = digsum(64) - 2 = 10 - 2; 196^(1/2) = sqrt(196) = 14 = digsum(196) - 2 = 16 - 2; 289^(1/2) = sqrt(289) = 17 = digsum(289) - 2 = 19 - 2. The terms 125, 216, 343, 2744, 3375, and 4096 are obtained for r = 3: 125^(1/3) = 5 = digsum(125) - 3 = 8 - 3; 216^(1/3) = 6 = digsum(216) - 3 = 9 - 3; 343^(1/3) = 7 = digsum(343) - 3 = 10 - 7; 2744^(1/3) = 14 = digsum(2744) - 3 = 17 - 3; 3375^(1/3) = 15 = digsum(3375) - 3 = 18 - 3; 4096^(1/3) = 16 = digsum(4096) - 3 = 19 - 3. The term 3518743761 is obtained for r = 6: 3518743761^(1/6) = 39 = digsum(3518743761) - 6 = 45 - 6.
Programs
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Mathematica
min = 0; max = 10^8; list=Union@ Flatten@ Table[ n^expo, {expo, Prime@ Range@ PrimePi@ Log2@ max}, {n, Floor[1 + min^(1/expo)], max^(1/expo)}]; (* A001597 *) a = {}; For[k = 1, k <=Length[list], k++, For[r = 2, r <= Ceiling[Log2[Part[list,k]]], r++, If[Part[list,k]^(1/r) == Total[IntegerDigits[Part[list,k]]]-r, AppendTo[a, Part[list,k]]]]]; a
Extensions
a(13)-a(14) from Chai Wah Wu, Jan 30 2024
Comments