Dylan Hamilton - cp's OEIS frontend

A259656 Let f(x) be the absolute value of the difference between x and its base-2 reversal. a(n) is the number of times f(x) must be applied starting with n for the result to be 0.

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 4, 3, 2, 3, 4, 3, 4, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 3, 4, 3, 2, 3

Offset: 1

Dylan Hamilton, Jul 02 2015

First differences appear to always be odd.

More precisely, a(n) is even if n is even and a(n) is odd when n is odd. This is an immediate consequence of the parities in A055945 (which represents f apart from the sign) and the fact that we count iterations of f until the result is even. - Jörgen Backelin, Nov 04 2015

Cf. A055945.

A259656 := proc(n)
    local f,a ;
    f := n ;
    a := 0 ;
    while f <> 0 do
        f := abs(A055945(f)) ;
        a := a+1 ;
    end do:
    a;
end proc: # R. J. Mathar, Nov 04 2015

A259658 Let f(x) be the absolute value of the difference between x and its base-2 reversal. Let g(x) be the number of times f(x) must be applied to x for the result to be 0. a(n) is the smallest value of x for which g(x) is n.

Original entry on oeis.org

0, 1, 2, 11, 38, 271, 544, 2093, 2624, 8607, 17984, 35343, 35904, 70671, 71744, 141327, 143424, 282639, 286784, 565263, 573504, 1130511, 1146944, 2261007, 2293824, 4521999, 4587584, 9043983, 9175104, 18087951, 18350144, 36175887, 36700224, 72351759, 73400384

Offset: 0

Dylan Hamilton, Jul 02 2015

f(x) = abs(A055945(x)).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A030101, A055945.

I:=[0,1,2,11,38,271,544,2093,2624, 8607,17984,35343, 35904,70671]; [n le 14 select I[n] else 3*Self(n-2)-2*Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jul 10 2015

CoefficientList[Series[x (18144 x^12 + 12800 x^11 - 13708 x^10 -11200 x^9 - 2870 x^8 - 1068 x^7 - 1302 x^6 - 434 x^5 - 240 x^4 - 32 x^3 - 8 x^2 - 2 x - 1)/((1 - x) (x + 1) (2 x^2 - 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 10 2015 *)
LinearRecurrence[{0,3,0,-2},{0,1,2,11,38,271,544,2093,2624,8607,17984,35343,35904,70671},50] (* Harvey P. Dale, Nov 23 2022 *)

G.f.: -x*(18144*x^12 +12800*x^11 -13708*x^10 -11200*x^9 -2870*x^8 -1068*x^7 -1302*x^6 -434*x^5 -240*x^4 -32*x^3 -8*x^2 -2*x-1)/ ((x-1) *(x+1) *(2*x^2-1)). - Alois P. Heinz, Jul 02 2015

a(0), a(19)-a(34) from Alois P. Heinz, Jul 02 2015

A211518 a(n) is the sum of all distinct integers that can be produced by reversing the digits of n in any base b >= 2.

Original entry on oeis.org

1, 3, 4, 5, 13, 21, 37, 40, 71, 82, 150, 140, 232, 252, 327, 352, 520, 497, 711, 729, 881, 1027, 1325, 1214, 1567, 1700, 1904, 2016, 2388, 2523, 2997, 3178, 3583, 3758, 4406, 4244, 5138, 5379, 6055, 5948, 6988, 7027, 8150, 8240, 8971, 9303, 10476, 10441, 11808, 12088, 13139, 13571, 15009, 15047, 16473, 16620, 18263, 19020

Offset: 1

Dylan Hamilton, Jun 26 2012

			If n=3, we can get 3 from base 10 (or any other base except 3) and 1 from reversing the base-3 expansion 10, so a(3) = 3+1 = 4.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

rev[x_,b_]:=FromDigits[Reverse[IntegerDigits[x,b]],b];Total/@Union/@Table[Table[rev[x,b],{b,2,x+1}],{x,Startpoint,Endpoint}]

rev(n,B)=my(m=n%B);n\=B;while(n>0,m=m*B+n%B;n\=B);m
a(n)=if(n<3,2*n-1,my(v=vecsort(vector(n-1,k,rev(n,k+1)),,8));sum(i=1,#v,v[i])) \\ Charles R Greathouse IV, Aug 05 2012

A175947 A175945(n)-A175946(n).

Original entry on oeis.org

1, 0, 3, -2, 1, 2, 7, -6, -1, 0, 3, 0, 5, 6, 15, -14, -5, -4, 1, -2, 3, 2, 7, -4, 3, 4, 11, 4, 13, 14, 31, -30, -13, -12, -3, -10, -1, -2, 5, -6, 1, 0, 9, 0, 7, 6, 15, -12, -1, 0, 9, 2, 11, 10, 23, 0, 11, 12, 27, 12, 29, 30, 63, -62, -29, -28, -11, -26, -9, -10, 1, -22, -7, -8, 5, -8

Offset: 1

Dylan Hamilton, Oct 28 2010

takelist[l_, t_] := Module[{lent, term},Set[lent, Length[t]]; Table[l[[t[[y]]]], {y, 1, lent}]]
frombinrep[x_] := FromDigits[Flatten[Table[Table[If[OddQ[n], 1, 0], {d, 1, x[[n]]}], {n, 1, Length[x]}]], 2]
binrep[x_] := repcount[IntegerDigits[x, 2]]
onebinrep[x_]:=Module[{b},b=binrep[x];takelist[b,Range[1,Length[b],2]]]
zerobinrep[x_]:=Module[{b},b=binrep[x];takelist[b,Range[2,Length[b],2]]]
Table[frombinrep[onebinrep[n]], {n,START,END}]-Table[frombinrep[zerobinrep[n]], {n,START,END}]

A175949 Numbers obtained by concatenation of the binary representation of A175946(n) and A175945(n).

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 7, 14, 13, 10, 11, 12, 9, 8, 15, 30, 29, 26, 27, 18, 21, 20, 23, 28, 25, 22, 19, 24, 17, 16, 31, 62, 61, 58, 59, 50, 53, 52, 55, 34, 37, 42, 43, 36, 41, 40, 47, 60, 57, 54, 51, 38, 45, 44, 39, 56, 49, 46, 35, 48, 33, 32, 63, 126, 125, 122, 123, 114, 117, 116

Offset: 1

Dylan Hamilton, Oct 28 2010

The operation as in A175948, but the run-length encoding of zeros (A175946) is placed left from the run-length encoding of ones (A175945).

			n=9 is 1001 in binary. Run lengths of 0's are 2 (one run of length 2) and of 1's are 11 (two runs each of length 1). The concatenation of these lengths is 211, which is interpreted as 2 one's, 1 zero, 1 one, binary 1101, and recoded decimal as a(9)=8+4+1 =13.

takelist[l_, t_] := Module[{lent, term},Set[lent, Length[t]]; Table[l[[t[[y]]]], {y, 1, lent}]]
frombinrep[x_] := FromDigits[Flatten[Table[Table[If[OddQ[n], 1, 0], {d, 1, x[[n]]}], {n, 1, Length[x]}]], 2]
binrep[x_] := repcount[IntegerDigits[x, 2]]
onebinrep[x_]:=Module[{b},b=binrep[x];takelist[b,Range[1,Length[b],2]]]
zerobinrep[x_]:=Module[{b},b=binrep[x];takelist[b,Range[2,Length[b],2]]]
Table[frombinrep[Flatten[{zerobinrep[n], onebinrep[n]}]], {n,START,END}]

A175950 A175948(n)-A175949(n).

Original entry on oeis.org

0, 0, 0, -2, 0, 2, 0, -6, -2, 0, -2, 0, 4, 6, 0, -14, -6, -4, -8, 2, 0, -2, -6, -4, 2, 4, 6, 4, 12, 14, 0, -30, -14, -12, -20, -6, -12, -14, -20, 6, 6, 0, 2, 0, -4, -6, -14, -12, -2, 0, 0, 14, 8, 6, 10, 0, 10, 12, 22, 12, 28, 30, 0, -62, -30, -28, -44, -22, -36, -38, -48, -10, -18

Offset: 1

Dylan Hamilton, Oct 28 2010

A difference between two ways of encoding-decoding run lengths of 0's and 1's in the binary representation of n.

takelist[l_, t_] := Module[{lent, term},Set[lent, Length[t]]; Table[l[[t[[y]]]], {y, 1, lent}]]
frombinrep[x_] := FromDigits[Flatten[Table[Table[If[OddQ[n], 1, 0], {d, 1, x[[n]]}], {n, 1, Length[x]}]], 2]
binrep[x_] := repcount[IntegerDigits[x, 2]]
onebinrep[x_]:=Module[{b},b=binrep[x];takelist[b,Range[1,Length[b],2]]]
zerobinrep[x_]:=Module[{b},b=binrep[x];takelist[b,Range[2,Length[b],2]]]
Table[frombinrep[Flatten[{onebinrep[n], zerobinrep[n]}]], {n,START,END}]-Table[frombinrep[Flatten[{zerobinrep[n], onebinrep[n]}]], {n,START,END}]

A175928 Number of bases in which the concatenation of the consecutive increasing integers 1...n is greater than the concatenation of consecutive decreasing integers n...1 in the same base.

Original entry on oeis.org

1, 1, 3, 3, 4, 3, 4, 5, 5, 5, 5, 5, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 7, 7, 7, 7

Offset: 2

Dylan Hamilton, Oct 20 2010

Table[Count[ Table[Negative[FromDigits[Flatten[IntegerDigits[Reverse[Range[x]], ba]], \ ba] - FromDigits[Flatten[IntegerDigits[Range[x], ba]], ba]], {ba, 2, x + 1}], True], {x, 1, 200}]

A175994 Decimal expansion of the definite integral of y=x^(1/x) for x=0 to e, the only maximum of this graph.

Original entry on oeis.org

2, 6, 6, 1, 8, 2, 5, 7, 0, 5, 3, 8, 0, 4, 1, 7, 8, 2, 8, 4, 9, 7, 0, 3, 9, 3, 3, 7, 6, 5, 1, 3, 9, 5, 8, 3, 0, 2, 1, 4, 9, 7, 0, 8, 2, 0, 9, 8, 3, 3, 0, 3, 5, 4, 8, 2, 1, 4, 6, 7, 8, 4, 8, 5, 0, 9, 1, 4, 7, 0, 2, 1, 0, 6, 5, 7, 1, 7, 5, 1, 6, 6, 2, 4, 6, 8, 2, 8, 2, 9, 3, 5, 6, 2, 4, 3, 5, 1, 4, 0

Offset: 1

Dylan Hamilton, Nov 05 2010

			2.6618257053804178284970393376513958302149708209833035482146784850914702106571...

J. Sondow and D. Marques, Algebraic and transcendental solutions of some exponential equations, Annales Mathematicae et Informaticae 37 (2010) 151-164; see Figure 5.

Cf. A073229 (decimal expansion of e^(1/e)).

RealDigits[ NIntegrate[ x^(1/x), {x, 0, E}, WorkingPrecision -> 105]][[1]] (* Jean-François Alcover, Nov 07 2012 *)

A175998 Decimal expansion of (e-1)*(e^(1/e)-1) - int(x^(1/x)-1, x=1..e).

Original entry on oeis.org

1, 7, 4, 0, 1, 7, 6, 2, 9, 0, 5, 2, 8, 8, 3, 0, 0, 3, 2, 3, 8, 4, 1, 1, 4, 4, 6, 2, 2, 6, 1, 6, 9, 0, 6, 1, 5, 3, 2, 7, 3, 0, 5, 3, 8, 9, 3, 0, 0, 6, 3, 4, 8, 1, 6, 8, 9, 1, 0, 5, 2, 6, 0, 2, 5, 4, 0, 3, 1, 1, 9, 0, 7, 9, 4, 2, 1, 0, 8, 6, 4, 6, 4, 3, 4, 5, 9, 4, 2, 6, 3, 0, 4, 5, 2, 9, 7, 2, 5, 0

Offset: 0

Dylan Hamilton, Nov 05 2010

Area contained by y=x^(1/x), x=e, and y=1.

Difference between the numbers defined in A175996 and A175997.

			0.174017629052883003238411446226169...

Cf. A073229, A175996, A175997.

e=exp(1); (e-1)*(e^(1/e)-1)-intnum(x=1,e,x^(1/x)-1)  \\ - M. F. Hasler, Nov 27 2012

A176000 Decimal expansion of 1 - A175999.

Original entry on oeis.org

6, 4, 6, 5, 0, 3, 1, 9, 9, 2, 9, 8, 5, 7, 7, 9, 4, 4, 5, 3, 4, 1, 6, 3, 6, 2, 9, 7, 9, 3, 3, 0, 1, 7, 5, 4, 9, 0, 9, 7, 4, 3, 1, 9, 9, 1, 9, 1, 2, 2, 6, 0, 0, 6, 1, 9, 2, 1, 9, 2, 0, 7, 5, 3, 9, 2, 1, 9, 9, 8, 1, 5, 4, 0, 2, 9, 9, 7, 4, 6, 6, 0, 9, 5, 9, 5, 9, 7, 0, 9, 3, 5, 7, 2, 3, 4, 9, 0, 8, 0

Offset: 0

Dylan Hamilton, Nov 05 2010

Area contained by y=x^(1/x),x=1, and y=0

Cf. A073229, A175999.

cp's OEIS Frontend

User: Dylan Hamilton

A259656 Let f(x) be the absolute value of the difference between x and its base-2 reversal. a(n) is the number of times f(x) must be applied starting with n for the result to be 0.

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A259658 Let f(x) be the absolute value of the difference between x and its base-2 reversal. Let g(x) be the number of times f(x) must be applied to x for the result to be 0. a(n) is the smallest value of x for which g(x) is n.

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A211518 a(n) is the sum of all distinct integers that can be produced by reversing the digits of n in any base b >= 2.

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PARI

A175947 A175945(n)-A175946(n).

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Mathematica

A175949 Numbers obtained by concatenation of the binary representation of A175946(n) and A175945(n).

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A175950 A175948(n)-A175949(n).

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Mathematica

A175928 Number of bases in which the concatenation of the consecutive increasing integers 1...n is greater than the concatenation of consecutive decreasing integers n...1 in the same base.

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Mathematica

A175994 Decimal expansion of the definite integral of y=x^(1/x) for x=0 to e, the only maximum of this graph.

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Mathematica

A175998 Decimal expansion of (e-1)*(e^(1/e)-1) - int(x^(1/x)-1, x=1..e).

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PARI

A176000 Decimal expansion of 1 - A175999.

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