A188531 -id:A188531 - cp's OEIS frontend

A188533 Intersection of A089633 and A188531.

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

Offset: 1

Vladimir Shevelev, Apr 03 2011

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

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

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.

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 *)

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

A188532 Numbers which contain only the digit 5 in their base-6 representation, with at most one exception. If the exception is the most-significant digit, it must be the digit 1, 2, 3, or 4, otherwise the exception must be the digit 4.

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 17, 23, 29, 34, 35, 71, 107, 143, 179, 209, 214, 215, 431, 647, 863, 1079, 1259, 1289, 1294, 1295, 2591, 3887, 5183, 6479, 7559, 7739, 7769, 7774, 7775, 15551, 23327, 31103, 38879, 45359, 46439, 46619, 46649, 46654, 46655, 93311, 139967

Offset: 1

Vladimir Shevelev, Apr 03 2011

The sequence lists the positive binomial coefficient predictors in base 6. For definition, see paper in link.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
V. Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8

Cf. A089633, A188341, A188499, A188529, A188531.

a:= proc(n) option remember; local i, l, m, t;
      m:= `if`(n=1, 0, a(n-1));
      l:=NULL;
      for t while m>0 do l:=l, irem(m, 6, 'm') od;
      l:= array([l, 0]);
      for i while l[i]=5 do od;
      if l[i]<4 then l[i]:= l[i]+1
                else l[i]:= 5;
                     if i>1 then l[i-1]:= 4 fi
      fi;
      add(l[i] *6^(i-1), i=1..t)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 03 2011

A188534 Intersection of A089633 and A188532.

Original entry on oeis.org

1, 2, 3, 5, 11, 23, 29

Offset: 1

Vladimir Shevelev, Apr 03 2011

nonn

Binomial coefficient predictors in both bases 2 and 6 (for definition, see paper in link).

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

Cf. A089633, A188341, A188499, A188529, A188531, A188532, A188533.

A188535 Intersection of A188341 and A188532.

Original entry on oeis.org

1, 2, 5, 17, 23, 71, 215, 647, 6479

Offset: 1

Vladimir Shevelev, Apr 03 2011

nonn

Binomial coefficient predictors in both bases 3 and 6 (for definition, see paper in link).

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

Cf. A089633, A188341, A188499, A188529, A188531, A188532, A188533, A188534.

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A188533 Intersection of A089633 and A188531.

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A188532 Numbers which contain only the digit 5 in their base-6 representation, with at most one exception. If the exception is the most-significant digit, it must be the digit 1, 2, 3, or 4, otherwise the exception must be the digit 4.

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A188534 Intersection of A089633 and A188532.

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A188535 Intersection of A188341 and A188532.

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