A188533 Intersection of A089633 and A188531.

1, 2, 3, 14, 23, 119, 123

Offset: 1

Author

Vladimir Shevelev, Apr 03 2011

Keywords

• nonn
• base

Comments

Binomial coefficient predictors in both bases 2 and 5 (for definition, see paper in link).

a(8) > 10^50000 (if it exists). - Pontus von Brömssen, Jul 06 2025

All terms > a(4) = 14 must have all base 5 digits equal to 4 except for exactly one digit 3 which cannot be the initial digit. Indeed, numbers with only 4s in their base-5 expansion are of the form 5^n-1, n > 0, but since 5^n-1 == 1-1 == 0 (mod 4), the binary expansion of such numbers ends in '00'. If the exception is the first digit, we have a number of the form N = (d+1)*5^m-1, where 1 <= d <= 3 is the first digit and m is the number of subsequent digits 4, in base 5. But if d = 1, then N = 2*5^m-1 == 1 (mod 8), since 2*5^m == 2*(-3)^m == 2*(-3 or 1) == 2 (mod 8). That means, N's binary expansion ends in '001'. If d = 2, then N = 3*5^m-1 is even, so it ends in a bit 0, and N/2 has another bit 0 at position p = A001511(m+3)+1 = valuation(m+3, 2)+2 from the right (i.e., the binary digit with value 2^p). If d = 3, then N = 4*5^m-1 == 3 (mod 16), so its binary expansion ends in '0011'. - M. F. Hasler, Jun 28 2025

Links

• Vladimir Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8.

Crossrefs

Cf. A089633 (numbers having at most one bit 0), A188531 (only 4s in base 5 with at most one exception), A188341 (no 0s and at most one 1 in base 3), A188499, A188529, A188532.

Programs

Mathematica

aQ[n_]:= DigitCount[n, 2, 0] < 2 && Module[{d=IntegerDigits[n,5]}, s=Select[d, #!=4 &]; s=={} || s =={3} || (d[[1]]<3 && s=={d[[1]]})]; Select[Range[100000], aQ] (* Amiram Eldar, Dec 14 2018 *)

Extensions

Incorrect term 4 deleted by Pontus von Brömssen, Jun 28 2025