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

1,2

Comments

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

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

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

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

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

Supersequence of A319008.

Links

Table of n, a(n) for n=1..45.

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.

Example

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.

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 *)

Crossrefs

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

Sequence in context: A372186 A043503 A202311 * A233030 A250835 A221827

Adjacent sequences: A319008 A319009 A319010 * A319012 A319013 A319014

Keyword

nonn

Author

Amiram Eldar, Sep 07 2018

Status

approved