The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A290041 Set b(n) = A290040(n). Then a(n) is the least divisor d > 1 of b(n) with binomial(b(n)+d, d) == 1 mod b(n). 2
 10, 264, 10, 10, 55, 18, 20, 10, 52, 10, 18, 34, 10, 20, 34, 18, 10, 10, 18, 10, 2525, 10, 20, 10, 38, 18, 10, 10, 10, 20, 10, 55, 272, 10, 26, 10, 57, 10, 68, 18, 10, 10, 68, 18, 10, 18, 20, 10, 36, 10, 22, 10, 18, 33, 10, 38, 10, 18, 10, 10, 20, 10, 18, 10, 33, 10, 10, 10, 10, 18, 10, 34, 10, 50, 10, 20, 10, 33, 18, 10, 10 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Sondow (2017) shows that if d is any divisor of m > 0 with d > 1 and binomial(m+d, d) == 1 mod m, then d is composite. Can d be a prime power? The first term of A290040 for which more than one such d exists is A290040(165) = 101000, where d = 20 or d = 100. d cannot be a prime power p^r | m if p^(r+1) does not divide m. Can d = 6? - Jonathan Sondow, Dec 26 2017 LINKS Table of n, a(n) for n=1..81. C. Babbage, Demonstration of a theorem relating to prime numbers, Edinburgh Philosophical Journal, 1 (1819), 46-49. J. Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; arXiv:1812.07650 [math.NT], 2018. J. Sondow, Problem 12030, Amer. Math. Monthly, 125 (2018), 276. FORMULA binomial(A290040(n) + a(n), a(n)) == 1 mod A290040(n). EXAMPLE b(1) = 260 and binomial(260+10,10) = 479322759878148681 == 1 mod 260 gives the least d = 10, so a(1) = 10. MATHEMATICA M = Select[ Range[ 30000], (d = Divisors[#]; Product[ Mod[ Binomial[# + d[[k]], d[[k]]], #] - 1, {k, 2, Length[d]}] == 0) &]; Table[ Select[ Divisors[ M[[k]]], # > 1 && Mod[ Binomial[# + M[[k]], #], M[[k]]] == 1 &], {k, 1, Length[M]}] // Flatten PROG (PARI) terms(n) = my(i=0); for(k=1, oo, my(d=divisors(k)); for(x=2, #d, if(Mod(binomial(k+d[x], d[x]), k)==1, print1(d[x], ", "))); if(i==n, break)) /* Print initial 40 terms as follows */ terms(40) \\ Felix Fröhlich, Dec 26 2017 CROSSREFS Cf. A290040, A299115. Sequence in context: A268730 A217911 A054593 * A369423 A160481 A060608 Adjacent sequences: A290038 A290039 A290040 * A290042 A290043 A290044 KEYWORD nonn AUTHOR Jonathan Sondow, Jul 23 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 17 09:14 EDT 2024. Contains 373444 sequences. (Running on oeis4.)