A319011 - OEIS

%I #14 Sep 27 2018 05:04:20

%S 1,333,2136,14,72765,49,9765,5,154,490,276,55,86,104,228195,5,

%T 25597845,264,220,181,24403740,70,226,234,199250835,215,358293,

%U 13494274080,49,70,14753835,685,35,154,60,7307904366,1,570,21792528,154,216,145,770,228,236

%N 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.

%C Product_{i} (k*d(i)*m + 1) is a Carmichael number.

%C Chernick proved that (6m + 1)*(12m + 1)*(18m + 1) is a Carmichael number, if all the 3 factors are prime (A033502, A046025).

%C Lieuwens generalized it to Product_{i} (k*d(i)*m + 1), for k a perfect number (A000396), e.g., A067199 for k = 28.

%C Rotkiewicz generalized it to any number k with a subset of its proper divisors that sums to k.

%C The corresponding generated Carmichael numbers are 1729, 393575432565765601, 599966117492747584686619009, 17167430884969, 11744090279809908081796578516491199598397832961, 3680409480386689, 617027029751094776871101828064081267143041, 7622722964881, 700705956080852569, 90694625332467786841, 24182595473200959889, 553229304821570521, 3915654940974324169, 9215447790472998049, 4890416189580986381506017143209122707839833885365268481, 1746281192537521, ...

%C Supersequence of A319008.

%H Jack Chernick, <a href="http://projecteuclid.org/euclid.bams/1183501763">On Fermat's simple theorem</a>, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.

%H Bill Daly, <a href="https://www.ics.uci.edu/~eppstein/numth/egypt/carmichael.html">Perfect numbers and Carmichael numbers - a hidden relation</a>, transcription of a network conversation at David Eppstein's Egyptian Fractions web site.

%H Erik Lieuwens, <a href="https://repository.tudelft.nl/islandora/object/uuid:becf731b-0b3c-4b2f-8479-9d4f205659d9?collection=research">Fermat pseudo primes</a>, Doctoral Thesis, Delft University of Technology, 1971, pp. 29-30.

%H Andrzej Rotkiewicz, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa21/aa21137.pdf">On some problems of W. Sierpinski</a>, Acta Arithmetica, Vol. 21 (1972), pp. 251-259.

%e 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.

%t 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];

%t   Do[divs = divSets[[k]]; If[Total[divs] == n, Break[]], {k, 1, ns}]; divs]; leastMultiplier[n_] := Module[{divs = divSubset[n]}, m = 1;

%t   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 *)

%Y Cf. A000396, A002997, A033502, A046025, A064771, A067199, A319008.

%K nonn

%O 1,2

%A _Amiram Eldar_, Sep 07 2018