A188341 -id:A188341 - cp's OEIS frontend

A188499 Intersection of A089633 and A188341.

Original entry on oeis.org

1, 2, 5, 7, 23, 239

Offset: 0

Vladimir Shevelev, Apr 02 2011

Binomial coefficient predictors in both bases 2 and 3 (for definition, see paper in link).

Next term is > 5*10^300, if it exists.

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

Cf. A089633, A188341.

aQ[n_]:= DigitCount[n, 2, 0] < 2 && DigitCount[n, 3, 0] == 0 && DigitCount[n, 3, 1] < 2; Select[Range[1000], aQ] (* Amiram Eldar, Dec 10 2018 *)

isok2(n) = my(b=binary(n)); #select(x->(x==0), b) <= 1; \\ A089633
isok3(n) = my(d=digits(n, 3)); (#select(x->(x==0), d) == 0) && (#select(x->(x==1), d) <= 1); \\ A188341
isok(n) = isok2(n) && isok3(n); \\ Michel Marcus, Dec 10 2018

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.

A188529 Numbers which contain only the digit 3 in their base-4 representation, with at most one exception. If the exception is the most-significant digit, it must be the digit 1 or 2, otherwise the exception must be the digit 2.

Original entry on oeis.org

1, 2, 3, 7, 11, 14, 15, 31, 47, 59, 62, 63, 127, 191, 239, 251, 254, 255, 511, 767, 959, 1007, 1019, 1022, 1023, 2047, 3071, 3839, 4031, 4079, 4091, 4094, 4095, 8191, 12287, 15359, 16127, 16319, 16367, 16379, 16382, 16383, 32767, 49151, 61439, 64511, 65279, 65471, 65519

Offset: 1

Vladimir Shevelev, Apr 03 2011

The sequence lists the positive binomial coefficient predictors in base 4. For definition, see paper in link.

A subsequence of A089633.

			(767)_10 = (23333)_4 contains only digits 3, with the exception the leading digit which is 2. Therefore 767 is in the sequence.
(4091)_10 = (333323)_4 contains only digits 3, with the exception a digit 2.

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

Cf. A024036, A089633, A188341, A188499.

isA188529 := proc(n) local dgs ,n3,p,d; dgs := convert(n,base,4); n3 := 0 ; for p from 1 to nops(dgs) do d := op(p,dgs) ; if d <> 3 then n3 := n3+1 ; if n3 >=2 then return false; end if; if p < nops(dgs) and d <> 2 then return false; end if; if p = nops(dgs) and d = 0 then return false; end if; end if; end do: return true; end proc:
A188529 := proc(n) option remember; local a; if n = 1 then 1; else for a from procname(n-1)+1 do if isA188529(a) then return a; end if; end do; end if; end proc:
seq(A188529(n),n=1..50) ; # R. J. Mathar, Apr 03 2011

A188531 Numbers which contain only the digit 4 in their base-5 representation, with at most one exception. If the exception is the most-significant digit, it must be the digit 1, 2, or 3, otherwise the exception must be the digit 3.

Original entry on oeis.org

1, 2, 3, 4, 9, 14, 19, 23, 24, 49, 74, 99, 119, 123, 124, 249, 374, 499, 599, 619, 623, 624, 1249, 1874, 2499, 2999, 3099, 3119, 3123, 3124, 6249, 9374, 12499, 14999, 15499, 15599, 15619, 15623, 15624, 31249, 46874, 62499, 74999, 77499, 77999, 78099, 78119

Offset: 1

Vladimir Shevelev, Apr 03 2011

The sequence lists the positive binomial coefficient predictors in base 5. For definition, see paper in link.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
V. Shevelev, Binomial Coefficient Predictors, Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.8.

Cf. A089633, A188341, A188499, A188529.

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, 5, 'm') od;
      l:= array([l, 0]);
      for i while l[i]=4 do od;
      if l[i]<3 then l[i]:= l[i]+1
                else l[i]:= 4;
                     if i>1 then l[i-1]:= 3 fi
      fi;
      add(l[i] *5^(i-1), i=1..t)
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 03 2011

aQ[n_] := 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 *)

listb(nd) = {for (i = 1, nd, my(v = vector(nd, k, 4), kstart = if (i==1, 1, 3)); for (k=kstart,3, v[i] = k; print1(fromdigits(v, 5), ", "););); print1(fromdigits(vector(nd, k, 4), 5), ", ");}
lista(nnd) = {for (nd=1, nnd, listb(nd););} \\ Michel Marcus, Dec 14 2018

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

A188533 Intersection of A089633 and A188531.

Original entry on oeis.org

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

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.

cp's OEIS Frontend

A188499 Intersection of A089633 and A188341.

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A188535 Intersection of A188341 and A188532.

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A188529 Numbers which contain only the digit 3 in their base-4 representation, with at most one exception. If the exception is the most-significant digit, it must be the digit 1 or 2, otherwise the exception must be the digit 2.

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A188531 Numbers which contain only the digit 4 in their base-5 representation, with at most one exception. If the exception is the most-significant digit, it must be the digit 1, 2, or 3, otherwise the exception must be the digit 3.

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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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A188533 Intersection of A089633 and A188531.

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A188534 Intersection of A089633 and A188532.

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