A067199 -id:A067199 - cp's OEIS frontend

A319008 Let k = A000396(n) be the n-th perfect number, a(n) is the least number m such that kdm + 1 is prime for all of the proper divisors d of k so their product is a Carmichael number.

1, 2136, 13494274080, 216818853118725

Offset: 1

Amiram Eldar, Sep 07 2018

Chernick proved that (6m + 1)*(12m + 1)*(18m + 1) is a Carmichael number, if all the 3 factors are primes (A033502, A046025).
Lieuwens generalized it to Product_{i} (k*d(i)*m + 1), for k a perfect number.
a(1) corresponds to 6. It was found by Jack Chernick in 1939.
a(2) corresponds to 28. It was found by Dubner in 1996. Lieuwens evaluated that the least corresponding Carmichael number > 10^27.
a(3) corresponds to 496. It was found by Jim Fougeron in 2002 (Dubner found a larger value: 474382033125).
a(4) corresponds to 8128. It was found by Phil Carmody in 2002.
The corresponding Carmichael numbers are 1729, 599966117492747584686619009, 1.631... * 10^126, 4.559... * 10^260, ...

			28 = 1 + 2 + 4 + 7 + 14 is the second perfect number. 2136 is the least number m such that 28*1*333 + 1 = 59809, 28*2*2136 + 1 = 119617, 28*4*2136 + 1 = 239233, 28*7*2136 + 1 =  418657 and 28*14*2136 + 1 = 837313 are all primes, therefore 59809*119617*239233*418657*837313 = 599966117492747584686619009 is a Carmichael number.

Harold Davenport, The Higher Arithmetic, Cambridge University Press, 7th ed., 1999, exercise 8.4.
Harvey Dubner, Carmichael numbers and Egyptian fractions, Mathematica japonicae, Vol. 43, No. 2 (1996), pp. 411-419.

Jack Chernick, On Fermat's simple theorem, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.
Claude Goutier, De l'utilité d'une vieille curiosité grecque, Crux Mathematicorum, Vol. 46, No. 8 (2020), pp. 397-403.
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 59996...19009 (27-digits).
Erik Lieuwens, Fermat pseudo primes, Doctoral Thesis, Delft University of Technology, 1971, pp. 29-30.
Carlos Rivera, Puzzle 171. Perfect & Carmichael numbers, The Prime Puzzles & Problems Connection.

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

ms = {2, 3, 5, 7, 13}; ns = Length[ms]; M[p_] := 2^(p - 1)*(2^p - 1); L[m_] := Module[{}, d = Most[Divisors[m]]*m; aQ[n_] := AllTrue[d*n + 1, PrimeQ]; n=1; While[!aQ[n], n++];n]; s={}; Do[m = M[ms[[k]]]; b = L[m]; AppendTo[s, b], {k, 1, ns}]; s

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 kd(i)m + 1 is prime for all d(i) in this subset so their product is a Carmichael number.

Original entry on oeis.org

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

Amiram Eldar, Sep 07 2018

nonn

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.

			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.

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.

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

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 *)

cp's OEIS Frontend

A319008 Let k = A000396(n) be the n-th perfect number, a(n) is the least number m such that kdm + 1 is prime for all of the proper divisors d of k so their product is a Carmichael number.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

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 kd(i)m + 1 is prime for all d(i) in this subset so their product is a Carmichael number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

cp's OEIS Frontend

A319008 Let k = A000396(n) be the n-th perfect number, a(n) is the least number m such that k*d*m + 1 is prime for all of the proper divisors d of k so their product is a Carmichael number.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A319008 Let k = A000396(n) be the n-th perfect number, a(n) is the least number m such that kdm + 1 is prime for all of the proper divisors d of k so their product is a Carmichael number.

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 kd(i)m + 1 is prime for all d(i) in this subset so their product is a Carmichael number.