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).

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

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