A065447 -id:A065447 - cp's OEIS frontend

A262806 a(n) = A065447(n) / A002275(floor((n+1)/2)).

1, 100, 9101, 91010000, 901900901001, 901900901001000000, 901089109010900090010001, 90108910901090009001000100000000, 9010080101801008010090100900009000100001, 90100801018010080100901009000090001000010000000000, 900999901099900099901099900999901000901000900000900001000001

Offset: 1

Max Alekseyev, Oct 03 2015

			a(5) = 100111000011111 / 111 = 901900901001.

a(n) = A065447(n) / A002275(floor((n+1)/2)).

a(2*n) = a(2*n-1) * 10^(2*n).

A087552 a(1) = 1, then the smallest prime divisor of A065447(n) not included earlier.

Original entry on oeis.org

1, 2, 11, 5, 3, 29, 101, 113, 41, 271, 7, 13, 239, 613, 73, 137, 37, 8291, 9091, 157637, 313, 21649, 9901, 2733970560857, 53, 79, 229, 4649, 31, 13001, 17, 19, 6529, 664193, 6781, 52579, 1111111111111111111

Offset: 1

Amarnath Murthy, Sep 13 2003

Conjecture: Every prime is a member and this is a rearrangement of the noncomposite numbers.

Proof of conjecture: primes 2=a(1) and 5=a(3) are terms, while any other prime divides infinitely many numbers of the form A002275([(n+1)/2]) = (10^[(n+1)/2]-1)/9, which in turn divide A065447(n). Thus every prime will sooner or later appear as a(n). - Max Alekseyev, Jul 03 2019

			a(6) = 29; smallest prime divisor of 100111000011111000000 not included earlier is 29. The prime divisors are 2, 3, 5, 29, 37, 97 and 106872959.

Cf. A002275, A065447.

More terms from David Wasserman, Jun 06 2005

Offset corrected by Max Alekseyev, Jul 03 2019

A065760 Concatenation of increasing number of alternating digits in base 2, starting with 1.

Original entry on oeis.org

1, 4, 39, 624, 19999, 1279936, 163831935, 41940975360, 21473779384831, 21989150090066944, 45033779384457103359, 184458360358736295358464, 1511082888058767731576545279, 24757582037954850514150117851136, 811256448219704541647671061746057215

Offset: 1

Lior Manor, Nov 18 2001

			a(5) = 19999 is formed by appending 1 five times (11111) to a(4) in base 2: 100111000011111.

Harry J. Smith, Table of n, a(n) for n = 1..50

Decimal version of A065447.

Cf. A065761.

a:= proc(n) option remember; `if`(n=1, 1,
      (t-> (a(n-1)+t)*2^n-t)(irem(n,2)))
    end:
seq(a(n), n=1..17);  # Alois P. Heinz, Mar 08 2024

With[{nn=20}, Table[FromDigits[Flatten[Take[Table[Table[If[EvenQ[n],0,1], {n}], {n,nn}], j]], 2], {j, nn}]] (* Harvey P. Dale, Sep 09 2012 *)

baseI(x, b)= { local(d, e=0, f=1); while (x>0, d=x-10*(x\10); x\=10; e+=d*f; f*=b); return(e) } { c=1; for (n=1, 50, if (n==1, a=1; b=1, c=c*10 + 1; if (n%2, d=c, d=0); b=b*10^n + d; a=baseI(b, 2)); write("b065760.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 30 2009

a(n) = a(n-1) * 2^n if n is even, (a(n-1) + 1) * 2^n - 1 if n is odd. - Franklin T. Adams-Watters, Sep 18 2012

A065761 Concatenation of increasing number of alternating digits in base 2, starting with 0.

Original entry on oeis.org

0, 3, 24, 399, 12768, 817215, 104603520, 26778501375, 13710592704000, 14039646928897023, 28753196910381103104, 117773094544920998318079, 964797190511992818221703168, 15807237169348490333744384720895

Offset: 1

Lior Manor, Nov 18 2001

The first 5 terms in base 2 are 0, (0)11, (0)11000, (0)110001111, (0)11000111100000.

			a(5) = 12768 is formed by appending 0 five times (00000) to a(4) in base 2: (0)11000111100000.

Harry J. Smith, Table of n, a(n) for n = 1..50

Cf. A065760, A065447.

baseI(x, b)= { local(d, e=0, f=1); while (x>0, d=x-10*(x\10); x\=10; e+=d*f; f*=b); return(e) } { c=1; for (n=1, 50, if (n==1, a=0; b=0, c=c*10 + 1; if (n%2, d=0, d=c); b=b*10^n + d; a=baseI(b, 2)); write("b065761.txt", n, " ",a) ) } \\ Harry J. Smith, Oct 30 2009

a(1) = 0; a(n+1) = append n+1 0's or 1's (alternately) to a(n).

A371032 a(n) is the integer whose decimal digits are 0's or 1's in alternating runs of lengths n, n-1, n-2, ..., 3, 2, 1.

Original entry on oeis.org

1, 110, 111001, 1111000110, 111110000111001, 111111000001111000110, 1111111000000111110000111001, 111111110000000111111000001111000110, 111111111000000001111111000000111110000111001, 1111111111000000000111111110000000111111000001111000110

Offset: 1

Clark Kimberling, Mar 09 2024

			a(1) = 1 has runlength 1; a(2) = 110 has runlengths 2,1; a(3) = 111001 has runlengths 3,2,1.

Cf. A000217 (binary lengths), A007088, A065447, A371033 (decimal version).

f:= proc(n) option remember; (10^(n*(n+1)/2)-1)/9 - procname(n-1) end proc:
f(1):= 1:
map(f, [$1..30]); # Robert Israel, Jul 09 2024

Flatten[Table[Flatten[Map[ConstantArray[Mod[#, 2], n + 1 - #] &, Range[n]]], {n, 10}]]   (* Peter J. C. Moses, Mar 08 2024 *)

def A371032(n):
    c = 0
    for i in range(n):
        c = (m:=10**(n-i))*c
        if i&1^1:
            c += (m-1)//9
    return c # Chai Wah Wu, Mar 18 2024

a(n) = A007088(A371033(n)). - Michel Marcus, Jul 09 2024

a(n) = (10^(n*(n+1)/2) - 1)/9 - a(n-1). - Robert Israel, Jul 09 2024

New name from Michel Marcus, Jul 09 2024

cp's OEIS Frontend

A262806 a(n) = A065447(n) / A002275(floor((n+1)/2)).

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A087552 a(1) = 1, then the smallest prime divisor of A065447(n) not included earlier.

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A065760 Concatenation of increasing number of alternating digits in base 2, starting with 1.

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Maple

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A065761 Concatenation of increasing number of alternating digits in base 2, starting with 0.

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PARI

Formula

A371032 a(n) is the integer whose decimal digits are 0's or 1's in alternating runs of lengths n, n-1, n-2, ..., 3, 2, 1.

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