A051404 Numbers k such that neither 4 nor 9 divides binomial(2k-1,k) (almost certainly finite).

1, 2, 3, 4, 6, 9, 10, 12, 18, 33, 34, 36, 40, 64, 66, 192, 256, 264, 272, 513, 514, 516, 576, 768, 1026, 1056, 2304, 16392, 65664, 81920, 532480, 545259520

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• changed

Comments

Complete up to 2^64 = 18446744073709551616.

Complete up to 2^30000. - Don Reble, Oct 27 2013

A number n is in the sequence if and only if the following inequalities hold s_2(n) <= 2 and s_3(n) + s_3(n-1) - s_3(2*n-1) <= 2, where s_m(n) is sum of digits of n in base m. - Vladimir Shevelev, Oct 30 2013

Equivalently, a number n is in the sequence if and only if there is at most 1 "carry" when adding n and n-1 in both base-2 arithmetic and base-3 arithmetic. - Tom Edgar, Oct 31 2013

Examples

			For n = 64 we have s_2(64) = 1, s_3(n) = 4, s_3(64-1) = 3, s_3(2*64-1) = 5 and 4+3-5 = 2. So 64 is in the sequence. - _Vladimir Shevelev_, Oct 30 2013
		

References

• Adrien-Marie Legendre, Théorie de Nombres, Firmin Didot Frères, Paris, 1830.

Links

• E. E. Kummer, Uber die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew Math. 44 (1852), 93-146.
• Don Reble, A051404, SeqFan Post, Oct 30 2013.
• Vladimir Shevelev, Binomial coefficient predictors, J. of Integer Sequences, Vol. 14 (2011), Article 11.2.8.
• Vladimir Shevelev, Re: A051404, SeqFan Post, Oct 30 2013.
• Wikipedia, Kummer's Theorem.

Crossrefs

Cf. A000120, A001700, A053735, A110556.

Programs

Mathematica

s[n_] :=DigitSum[n, 3]; With[{emax = 30}, Select[Flatten@ Table[2^e1 + If[e2 < 0, 0, 2^e2], {e1, 0, emax}, {e2, -1, e1-1}], s[#] + s[#-1] - s[2*#-1] <= 2 &]] (* Amiram Eldar, Aug 26 2025 *)

PARI

isok(k) = my(b=binomial(2*k-1,k)); (b%4) && (b%9); \\ Michel Marcus, Jan 22 2025

PARI

s(n) = sumdigits(n, 3);
list(emax = 30) = {my(k); for(e1 = 0, emax, for(e2 = -1, e1-1, k = 1 << e1 + if(e2 >= 0, 1 << e2); if(s(k) + s(k-1) - s(2*k-1) <= 2, print1(k, ", "))));} \\ Amiram Eldar, Aug 26 2025