A319011 Let k = A064771(n) be the n-th pseudoperfect number such that {d(i)} is a unique subset of its proper divisors that sums to k, a(n) is the least number m such that k*d(i)*m + 1 is prime for all d(i) in this subset so their product is a Carmichael number.
1, 333, 2136, 14, 72765, 49, 9765, 5, 154, 490, 276, 55, 86, 104, 228195, 5, 25597845, 264, 220, 181, 24403740, 70, 226, 234, 199250835, 215, 358293, 13494274080, 49, 70, 14753835, 685, 35, 154, 60, 7307904366, 1, 570, 21792528, 154, 216, 145, 770, 228, 236
Offset: 1
Keywords
Examples
20 = 1 + 4 + 5 + 10 is the sum of a single subset of the proper divisors of 20. 333 is the least number such that 20*1*333 + 1 = 6661, 20*4*333 + 1 = 26641, 20*5*333 + 1 = 33301, and 20*10*333 + 1 = 66601 are all primes, therefore 6661*26641*33301*66601 = 393575432565765601 is a Carmichael number.
Links
- Jack Chernick, On Fermat's simple theorem, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.
- Bill Daly, Perfect numbers and Carmichael numbers - a hidden relation, transcription of a network conversation at David Eppstein's Egyptian Fractions web site.
- Erik Lieuwens, Fermat pseudo primes, Doctoral Thesis, Delft University of Technology, 1971, pp. 29-30.
- Andrzej Rotkiewicz, On some problems of W. Sierpinski, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.
Programs
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Mathematica
okQ[n_] := Module[{d = Most[Divisors[n]]}, SeriesCoefficient[Series[Product[1 + x^i, {i, d}], {x, 0, n}], n] == 1]; s = Select[Range[300], okQ]; divSubset[n_] := Module[{d = Most[Divisors[n]]}, divSets = Subsets[d]; ns = Length[divSets]; Do[divs = divSets[[k]]; If[Total[divs] == n, Break[]], {k, 1, ns}]; divs]; leastMultiplier[n_] := Module[{divs = divSubset[n]}, m = 1; While[! AllTrue[n*m*divs + 1, PrimeQ], m++]; m]; seq = {}; Do[s1 = s[[k]]; m = leastMultiplier[s1]; AppendTo[seq, m], {k, 1, Length[s]}]; seq (* after Harvey P. Dale at A064771 *)
Comments