A115454 -id:A115454 - cp's OEIS frontend

A106701 a(n) = next-to-most-significant binary digit of n-th composite positive integer.

0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Leroy Quet, Jan 22 2006

The length of each run of zeros and ones: 1,3,6,13,25,53,107,219,445,899,1821,... and 1,3,5,12,26,52,106,218,442,894,1811,2838,..., . - Robert G. Wilson v

			a(2) = 1 because 6 is the second composite and because the next-to-most-significant binary digit (which happens to be the middle binary digit) of 6 = 110 (in binary) is 1.

Cf. A115454, A112416.

f[n_] := IntegerDigits[ FixedPoint[n + PrimePi[ # ] + 1 &, n], 2][[2]]; Array[f, 105] (* Robert G. Wilson v *)

a(n) = floor((c(n) - 2^m)/2^(m-1)), where c(n) is the n-th composite and m = floor(log(c(n))/log(2)).

More terms from Robert G. Wilson v, Jan 24 2006

A175349 a(n) is the smallest positive integer that, when written in binary, contains the binary representations of both the n-th prime and the n-th composite as (possibly overlapping) substrings.

Original entry on oeis.org

4, 6, 40, 39, 43, 108, 113, 79, 368, 466, 500, 149, 361, 344, 377, 53, 59, 988, 542, 2272, 2121, 1103, 2259, 356, 609, 1253, 3304, 3434, 876, 2929, 4078, 387, 393, 2226, 4787, 1687, 630, 2615, 1336, 5561, 2874, 5820, 382, 4033, 12608, 8391, 13506, 14276, 8931, 14662

Offset: 1

Leroy Quet, Apr 19 2010

			The 7th prime is 17, which is 10001 in binary. The 7th composite is 14, which is 1110 in binary. The smallest positive integer that, when written in binary, contains these binary representations as substrings is 113, which is 1110001 in binary. a(7) = 113, therefore.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A004676, A115454, A378479.

nextcomp:= proc(c)
  if isprime(c+1) then c+2 else c+1 fi
end proc:
f:= proc(p,c)
   local Bp, dp, Bc, dc, m, flag1, flag2, x1, x2;
   Bp:= convert(convert(p,binary),string); dp:= length(Bp);
   Bc:= convert(convert(c,binary),string); dc:= length(Bc);
   if dc < dp and StringTools:-Search(Bc, Bp) <> 0 then return p
   elif dp < dc and StringTools:-Search(Bp, Bc) <> 0 then return c
   fi;
   for m from min(dp,dc) to 1 by -1 do
     flag1:= Bp[1..m] = Bc[-m..-1];
     flag2:= Bc[1..m] = Bp[-m..-1];
     if flag1 then x1:= 2^(dp-m) * c + (p mod 2^(dp-m)) fi;
     if flag2 then x2:= 2^(dc-m) * p + (c mod 2^(dc-m)) fi;
     if flag1 and flag2 then return min(x1,x2)
     elif flag1 then return x1
     elif flag2 then return x2
     fi;
   od;
   min(p*2^dc + c, c*2^dp+p)
end proc:
p:= 1: c:= 2: R:= NULL:
for n from 1 to 100 do
  p:= nextprime(p); c:= nextcomp(c);
  R:= R, f(p,c)
od:
R; # Robert Israel, Nov 28 2024

comp[n_] := FixedPoint[n + 1 + PrimePi[#] &, n + 1 + PrimePi[n]]; sub[n_, x_] := MemberQ[Partition[IntegerDigits[n, 2], IntegerLength[x, 2], 1],
IntegerDigits[x, 2]]; a[n_] := Block[{c = comp[n], p = Prime[n], k}, k = Max[p, c]; While[! sub[k,p] || ! sub[k,c], k++]; k]; Array[a, 50] (* Giovanni Resta, Jul 02 2018 *)

a(9)-a(50) from Giovanni Resta, Jul 02 2018

cp's OEIS Frontend

A106701 a(n) = next-to-most-significant binary digit of n-th composite positive integer.

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