A072884
3rd-order digital invariants: the sum of the cubes of the digits of n equals some number k and the sum of the cubes of the digits of k equals n.
Original entry on oeis.org
1, 136, 153, 244, 370, 371, 407, 919, 1459
Offset: 1
136 is included because 1^3 + 3^3 + 6^3 = 244 and 2^3 + 4^3 + 4^3 = 136.
244 is included because 2^3 + 4^3 + 4^3 = 136 and 1^3 + 3^6 + 6^3 = 244.
- J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique Des Nombres, Problem 257 pp. 41; 185 Ellipses Paris 2004.
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, London, England, 1997, pp. 124-125.
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f[n_] := Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[n]^3]]^3]; Select[ Range[10^7], f[ # ] == # &]
Select[Range[10000], Plus@@(IntegerDigits[Plus@@(IntegerDigits[ # ]^3)]^3)== #&]
A072897
Least n-th order digital invariant which is not an Armstrong number (A005188), or 0 if no such term exists.
Original entry on oeis.org
136, 2178, 58618, 63804, 2755907, 0, 144839908, 304162700, 4370652168, 0, 0, 0, 0, 0, 21914086555935085, 187864919457180831, 0, 13397885590701080090, 0, 0, 0, 19095442247273220984552, 1553298727699254868304830, 1539325689516673750004702, 242402817739393059296681797
Offset: 3
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, London, England, 1997, pp. 124, 155.
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Do[k = 1; While[ !(Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[k]^n]]^n] == k && Apply[Plus, IntegerDigits[k]^n] != k), k++ ]; Print[k], {n, 3, 7}]
A072896
5th-order digital invariants: the sum of the 5th powers of the digits of n equals some number k and the sum of the 5th powers of the digits of k equals n.
Original entry on oeis.org
1, 4150, 4151, 54748, 58618, 76438, 89883, 92727, 93084, 157596, 194979
Offset: 1
58618 is included because 5^5 + 8^5 + 6^5 + 1^5 + 8^5 = 76438 and 7^5 + 6^5 + 4^5 + 3^5 + 8^5 = 58618.
- David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, London, England, 1997, pp. 157, 168.
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f[n_] := Apply[Plus, IntegerDigits[Apply[Plus, IntegerDigits[n]^5]]^5]; Select[ Range[10^7], f[ # ] == # &]
di5Q[n_]:=Module[{k=Total[IntegerDigits[n]^5]},Total[ IntegerDigits[k]^5] == n]; Select[Range[200000],di5Q] (* Harvey P. Dale, Nov 26 2014 *)
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