A055945 -id:A055945 - cp's OEIS frontend

A331632 Distinct values of A055945 in order of their appearance as n grows.

0, 1, 3, 7, 5, -2, 9, 2, 15, -6, 21, 6, 31, 17, -14, 27, -4, 13, -18, 35, 4, -10, 45, 14, 41, 10, 49, 18, 63, 33, -30, 51, -12, -42, 75, 12, 93, 30, 81, 105, 42, 127, 65, -62, 99, -28, 37, -90, 119, -8, 57, -70, 91, -36, 29, -98, 135, 8, 73, -54, 107, -20, -82

Offset: 1

Rémy Sigrist, Jan 23 2020

			A055945 starts: 0, 0, 1, 0, 3, 0, 3, 0, 7, 0, 5, -2, 9, 2, 7, 0, 15, 0, ...
We keep:        0,    1,    3,          7,    5, -2, 9, 2,       15,    ...

Rémy Sigrist, Table of n, a(n) for n = 1..8032
Rémy Sigrist, Colored scatterplot of the first 74383 terms (where the color is function of the 2-adic valuation of the least k such that a(n) = A055945(k))

Cf. A030101, A055945, A331633.

DeleteDuplicates[Array[# - IntegerReverse[#, 2] &, 200]] (* Paolo Xausa, Apr 28 2025 *)

s=[]; for (n=0, 151, v=n-fromdigits(Vecrev(binary(n)),2); if (!setsearch(s,v), print1 (v ", "); s=setunion(s, [v])))

A265326 n-th prime minus its binary reversal.

Original entry on oeis.org

1, 0, 0, 0, -2, 2, 0, -6, -6, 6, 0, -4, 4, -10, -14, 10, 4, 14, -30, -42, 0, -42, -18, 12, 30, 18, -12, 0, 18, 42, 0, -62, -8, -70, -20, -82, -28, -34, -62, -8, -26, 8, -62, 62, 34, -28, 8, -28, 28, 62, 82, -8, 98, 28, 0, -186, -84, -210, -60

Offset: 1

Max Barrentine, Dec 07 2015

a(n) = 0 iff A000040(n) is in A016041. - Altug Alkan, Dec 07 2015

The graph consists of a succession of parallelograms. The parallelograms end when there is a long run of mostly positive terms followed by a long run of mostly negative terms. The places where the successive parallelograms end are the primes just before a power of 2: 3, 7, 13, 31, 61, 127, 251, 509, 1021, 2039, 4093, 8191, 16381, 32749, ..., which are terms with indices 2, 4, 6, 11, 18, 31, 54, 97, 172, 309, 564, 1028, 1900, 3512, 6542, 12251, 23000, 43390, 82025, ... (see A014234 and A007053). - N. J. A. Sloane, May 29 2016

			n=5: prime(5) = 11_10 = 1011_2, reversing gives 1101_2 = 13_10, so a(5) = 11-13 = -2.

N. J. A. Sloane, Table of n, a(n) for n = 1..23000, May 29 2016 [First 10000 terms from _Robert Israel_]
Rémy Sigrist, Colored scatterplot of the first 82025 terms (corresponding to the prime numbers < 2^20) (where the color is function of A000040(n) mod 8)
N. J. A. Sloane, Table of n, a(n) for n = 1..78498
N. J. A. Sloane and Brady Haran, Amazing Graphs, Numberphile video (2019)

Cf. A055945, A068396, A098957, A007053, A014234.

revdigs:= proc(n) local L, j;
  L:= convert(n,base,2);
  add(L[-j]*2^(j-1),j=1..nops(L))
end proc:
map(t -> t - revdigs(t),  select(isprime, [2,seq(i,i=3..1000,2)])); # Robert Israel, Dec 08 2015

Table[# - FromDigits[Reverse@ IntegerDigits[#, 2], 2] &@ Prime@ n, {n, 60}] (* Michael De Vlieger, Dec 09 2015 *)

a098957(n) = my(v=binary(prime(n)), s); forstep(i=#v, 1, -1, s+=s+v[i]); s
a(n) = prime(n) - a098957(n); \\ Altug Alkan, Dec 07 2015

a(n) = A000040(n) - A098957(n).

a(n) = A055945(A000040(n)). - Michel Marcus, Dec 08 2015

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.

Original entry on oeis.org

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

A349240 a(n) = n - (reversal of digits in the Zeckendorf representation of n).

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 0, 3, 7, 0, 4, 7, 0, 12, 0, 6, 10, -2, 14, 2, 8, 20, 0, 9, 15, -5, 20, 0, 9, 25, 5, 14, 20, 0, 33, 0, 14, 23, -10, 30, -3, 11, 36, 3, 17, 26, -7, 43, 10, 24, 33, 0, 40, 7, 21, 54, 0, 22, 36, -18, 46, -8, 14, 54, 0, 22, 36, -18, 62, 8, 30, 44

Offset: 0

Kevin Ryde, Nov 11 2021

Kevin Ryde, Table of n, a(n) for n = 0..10000
Kevin Ryde, PARI/GP Code

Cf. A189920 (Zeckendorf digits), A349238 (reverse), A349239 (reverse and add).
Cf. A094202 (indices of 0's).
Other bases: A055945 (binary), A056965 (decimal).

PARI
```
\\ See links.
```

# Using functions NumToFib and RevFibToNum from A349238.
n, a = 0, 0
print(a - a, end = ", ")
while n < 71:
    n += 1
    print(n - RevFibToNum(NumToFib(n)), end = ", ") # A.H.M. Smeets, Nov 14 2021

a(n) = n - A349238(n).

a(n) = 2*n - A349239(n).

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

cp's OEIS Frontend

A331632 Distinct values of A055945 in order of their appearance as n grows.

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A265326 n-th prime minus its binary reversal.

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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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A349240 a(n) = n - (reversal of digits in the Zeckendorf representation of n).

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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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