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A290040 Numbers m > 0 that have a divisor d > 1 with binomial(m+d, d) == 1 mod m. 3
260, 1056, 1060, 3460, 3905, 4428, 5000, 5060, 5512, 5860, 6372, 6596, 7460, 8200, 8908, 9612, 9860, 10660, 11556, 12260, 12625, 13060, 14600, 14660, 14744, 14796, 15460, 16260, 17060, 17800, 17860, 18425, 18496, 18660, 19396, 20260, 21717, 21860, 22168, 22248, 22660, 24260, 24616, 25164, 26660, 27108, 27400, 27460, 28872, 29060, 29128, 29860 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Sondow (2017) shows that d must be composite. Can d be a prime power?

The corresponding sequence of smallest such d is A290041.

The first term a(n) for which more than one d exists is a(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

Giovanni Resta, Table of n, a(n) for n = 1..654 (terms < 400000)

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(a(n) + A290041(n), A290041(n)) == 1 mod a(n).

EXAMPLE

The first case is binomial(260+10,10) = 479322759878148681 == 1 mod 260, so a(1) = 260.

MATHEMATICA

Select[ Range[ 30000], (d = Divisors[#]; Product[ Mod[ Binomial[# + d[[k]], d[[k]]], #] - 1, {k, 2, Length[d]}] == 0) &]

PROG

(PARI) is(n) = my(d=divisors(n)); for(k=2, #d, if(Mod(binomial(n+d[k], d[k]), n)==1, return(1))); 0 \\ Felix Fröhlich, Dec 26 2017

CROSSREFS

Cf. A290041, A299115.

Sequence in context: A237631 A202585 A037219 * A061224 A108109 A235905

Adjacent sequences:  A290037 A290038 A290039 * A290041 A290042 A290043

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Jul 23 2017

STATUS

approved

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Last modified November 29 21:32 EST 2021. Contains 349416 sequences. (Running on oeis4.)