A178193 Smith numbers of order 4.
3777, 7773, 17418, 30777, 53921, 66111, 97731, 111916, 119217, 122519, 128131, 133195, 135488, 138878, 145229, 178814, 180174, 198581, 257376, 269636, 281179, 296396, 317686, 358256, 362996, 366514, 394114, 435777, 457377, 469552, 475856, 502960, 513833
Offset: 1
Examples
3777 = 3*1259 is composite; sum of 4th power of the digits is 3^4 + 7^4 + 7^4 + 7^4 = 7284. Sum of 4th power of the digits of the prime factors 3, 1259 is 3^4 + 1^4 + 2^4 + 5^4 + 9^4 = 7284. The sums are equal, so 3777 is in the sequence. 17418 = 2*3*2903 is composite; sum of 4th power of the digits is 1^4 + 7^4 + 4^4 + 1^4 + 8^4 = 6755. Sum of 4th power of the digits of the prime factors 2, 3, 2903 is 2^4 + 3^4 + 2^4 + 9^4 + 0^4 + 3^4 = 6755. The sums are equal, so 17418 is in the sequence. 269636 = 2*2*67409 is composite; sum of 4th power of the digits is 2^4 + 6^4 + 9^4 + 6^4 + 3^4 + 6^4 = 10546. Sum of 4th power of the digits of the prime factors 2, 2, 67409 (with multiplicity) is 2^4 + 2^4 + 6^4 + 7^4 + 4^4 + 0^4 + 9^4 = 10546. The sums are equal, so 269636 is in the sequence.
Links
- Donovan Johnson, Table of n, a(n) for n = 1..1000
- Patrick Costello, A new largest Smith number, Fibonacci Quarterly 40(4) (2002), 369-371.
- Underwood Dudley, Smith numbers, Mathematics Magazine 67(1) (1994), 62-65.
- S. S. Gupta, Smith Numbers, Mathematical Spectrum 37(1) (2004/5), 27-29.
- S. S. Gupta, Smith Numbers.
- Eric Weisstein's World of Mathematics, Smith number.
- Wikipedia, Smith number.
- A. Wilansky, Smith Numbers, Two-Year College Math. J. 13(1) (1982), p. 21.
- Amin Witno, Another simple construction of Smith numbers, Missouri J. Math. Sci. 22(2) (2010), 97-101.
- Amin Witno, Smith multiples of a class of primes with small digital sum, Thai Journal of Mathematics 14(2) (2016), 491-495.
Programs
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Mathematica
fQ[n_] := Block[{id = Sort@ IntegerDigits@ n, fid = Sort@ Flatten[ IntegerDigits@ Table[#[[1]], {#[[2]]}] & /@ FactorInteger@ n]}, While[ id[[1]] == 0, id = Drop[id, 1]]; While[ fid[[1]] == 0, fid = Drop[fid, 1]]; id != fid && Plus @@ (id^4) == Plus @@ (fid^4)]; k = 1; lst = {}; While[k < 10^6, If[f Q@ k, AppendTo[lst, k]; Print@ k]; k++]; lst
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