cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A359208 Maximum value reached when starting from n during iteration of the map x->A359194(x) (binary complement of 3n), or -1 if infinite.

Original entry on oeis.org

0, 1, 2, 300, 300, 5, 300, 10, 10, 300, 10, 300, 328536, 300, 21, 300, 300, 328536, 300, 300, 300, 21, 72, 328536, 300, 328536, 661, 328536, 123130640068522377168864228132316865867184046004226894, 40, 300, 328536, 328536
Offset: 0

Views

Author

Joshua Searle, Dec 20 2022

Keywords

Comments

It is unknown whether any terms are -1. The next term a(33) is equal to a(28), a 54-digit number. a(425720) is >= 2.09 * 10^114778, unresolved after 10^10 iterations.
a(425720) = 7.14... * 10^179246. - Joshua Searle, Jan 10 2023

Examples

			a(3) = 300 because the largest term in the iterated sequence: (3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0) is 300.
		

Crossrefs

Programs

  • Mathematica
    f[n_] := BitXor[3 n, 2^IntegerPart[Log2[3 n] + 1] - 1]; Table[Max@ NestWhileList[f, n, # != 0 &], {n, 0, 32}] (* Michael De Vlieger, Dec 21 2022 *)
  • PARI
    f(n) = if(n, bitneg(n, exponent(n)+1), 1); \\ A035327
    a(n) = my(x=n, m=n); while (m, m=f(3*m); if (m>x, x=m)); x; \\ Michel Marcus, Dec 21 2022
  • Python
    def f(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length())-1)
    def a(n):
        i, fi, m = 0, n, n
        while fi != 0: i, fi, m = i+1, f(fi), max(m, fi)
        return m
    print([a(n) for n in range(33)]) # Michael S. Branicky, Dec 20 2022
    

A359218 Let S(n) be the sequence obtained through the mapping of x->A359194(x) starting with n and stopping when 0 is reached, -1 if 0 is never reached. a(n) = m if appears in S(k), k < n, otherwise -1.

Original entry on oeis.org

0, 0, 1, 0, 3, 0, 6, 1, 7, 4, 10, 9, 10, 13, 0, 15, 16, 12, 18, 6, 3, 21, 22, 12, 24, 25, 3, 27, 21, 7, 30, 31, 31, 28, 34, 22, 19, 37, 13, 39, 40, 4, 1, 43, 123, 58, 46, 4, 187, 49, 27, 102, 52, 96, 42, 55, 87, 57, 58, 21, 30, 61, 48, 63, 64, 60, 66, 54, 51, 69
Offset: 0

Views

Author

Michael De Vlieger, Dec 21 2022

Keywords

Comments

By convention, a(0) = 0 since n = 0.
Regarding A359215(n), this is the value m that had appeared in S(k), k < n.

Examples

			a(1) = 0 since S(1) = {1, 0}, but m = 0 appeared in S(0).
a(2) = 1 since S(2) = {2, 1, ...}, but m = 1 appeared in S(1).
a(3) = 0 since S(3) = {3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0}, but m = 0 appeared in S(0).
a(4) = 3 since S(4) = {4, 3, ...} but 3 appears in S(3), etc.
a(5) = 0 since S(5) = {5, 0}, but 0 appears in S(0).
a(6) = 6 since 6 appears in F(3).
a(7) = 1 since S(7) = {7, 10, 1, ...} but 1 appears in S(1).
a(8) = 7 since S(8) = {8, 7, ...} but 7 appears in S(7)
a(9) = 4 since S(9) = {9, 4, ...} but 4 appears in S(4).
a(10) = 10 since 10 appears in S(7).
a(11) = 9 since S(11) = {11, 30, 37, 16, 15, 18, 9, ...} but 9 appears in S(9).
a(12) = 10 since S(12) = {12, 27, ..., 39, 10, ...} but 10 appears in S(7), etc.
		

Crossrefs

Programs

  • Mathematica
    c[] = -1; c[0] = 0; f[n] := BitXor[3 n, 2^IntegerPart[Log2[3 n] + 1] - 1]; Table[(Map[If[c[#1] == -1, Set[c[#1], #2]] & @@ # &, Partition[#, 2, 1]]; Last@ #) &@ NestWhileList[f, n, c[#] == -1 &], {n, 0, 120}]

A359219 Starting numbers that require more iterations of the map x->A359194(x) (binary complement of 3n) to reach 0 than any smaller number.

Original entry on oeis.org

0, 1, 2, 3, 4, 9, 11, 12, 17, 23, 28, 33, 74, 86, 180, 227, 350, 821, 3822, 4187, 5561, 6380, 6398, 22174, 22246, 26494, 34859, 49827, 70772, 103721, 104282, 204953, 213884, 225095, 407354, 425720
Offset: 1

Views

Author

Joshua Searle, Dec 21 2022

Keywords

Comments

425720 after 10^10 iterations has not yet reached 0 and in general it is unknown whether every starting number does reach 0.
A359207(425720) = 87037147316. - Martin Ehrenstein, Jan 02 2023

Examples

			3 is a term because it requires 11 iterations to reach 0, which is more than any starting number less than 3.
0: (0) -- 0 terms
1: (1, 0) -- 1 term
2: (2, 1, 0) -- 2 terms
3: (3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0) -- 11 terms.
		

Crossrefs

Programs

  • Python
    from itertools import count, islice
    def f(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length())-1)
    def iters(n):
        i, fi = 0, n
        while fi != 0: i, fi = i+1, f(fi)
        return i
    def agen(): # generator of terms
        record = -1
        for m in count(0):
            v = iters(m)
            if v > record: yield m; record = v
    print(list(islice(agen(), 18))) # Michael S. Branicky, Dec 21 2022

Extensions

a(27)-a(36) from Tom Duff (SeqFan mailing list, Dec 19 2022)

A359220 Number of steps to reach 0 from A359219(n) where A359219 are the starting numbers that require more iterations in the map x->A359194(x) than any smaller number.

Original entry on oeis.org

0, 1, 2, 11, 12, 13, 19, 80, 81, 83, 7572, 7573, 7574, 7578, 7580, 664475, 664882, 3180929, 3180930, 3180931, 3181981, 3181988, 3182002, 3182226, 120796790, 556068798, 556068799, 556068871, 556068872, 572086553, 572086610, 1246707529, 1246707552, 1246707555, 1246707602
Offset: 1

Views

Author

Joshua Searle, Dec 21 2022

Keywords

Comments

It is unknown whether all starting numbers reach 0; the next term, a(36) depends on whether 425720 ever reaches 0 (see A359207). It remains nonzero after 10^10 iterations.
A359207(425720) = 87037147316. Calculated by Tom Duff (12/16/22) - Joshua Searle, Jan 10 2023
a(4) - a(7) only differ by a small fraction of their starting terms. The same is true for the terms in the intervals a(8) - a(10), a(11) - a(15), a(16) - a(17) and a(18) - a(24). It may also be true for a(26) - a(29), a(30) - a(31) and a(32) - a(35).

Examples

			a(4) is the step count from the starting number A359219(4) = 3: (3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0) -- 11 steps, hence a(4) = 11.
		

Crossrefs

Programs

  • Python
    from itertools import count, islice
    def f(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length())-1)
    def iters(n):
        i, fi = 0, n
        while fi != 0: i, fi = i+1, f(fi)
        return i
    def agen(): # generator of terms
        record = -1
        for m in count(0):
            v = iters(m)
            if v > record: yield v; record = v
    print(list(islice(agen(), 18))) # Michael S. Branicky, Dec 21 2022

Extensions

a(27) and beyond from Tom Duff (SeqFan mailing list, Dec 19 2022)

A359221 Starting numbers which reach a new record high value when iterating the map x->A359194(x) (binary complement of 3n).

Original entry on oeis.org

0, 1, 2, 3, 12, 28, 227, 821, 22246, 26494, 204953, 425720
Offset: 1

Views

Author

Joshua Searle, Dec 22 2022

Keywords

Comments

It is unknown whether all starting numbers reach 0.
103721 is not a term of this sequence despite having a trajectory of record length, because its maximum of 2.42...*10^14081 is lower than the previous record holder.

Examples

			Let S(x) = iteration sequence of A359194 starting with x; then
S(0) = (0), maximum = 0;
S(1) = (1, 0), maximum = 1;
S(2) = (2, 1, 0), maximum = 2;
S(3) = (3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0), maximum = 300;
Since S(3) contains a higher maximum than any lower positive starting integer, 3 is a term of this sequence.
		

Crossrefs

Programs

  • Python
    from itertools import count, islice
    def f(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length())-1)
    def itersmax(n):
        i, fi, m = 0, n, n
        while fi != 0: i, fi, m = i+1, f(fi), max(m, fi)
        return i, m
    def agen(): # generator of terms
        record = -1
        for m in count(0):
            v, mx = itersmax(m)
            if mx > record:
                yield m # use mx to obtain values
                record = mx
    print(list(islice(agen(), 8))) # Michael S. Branicky, Dec 22 2022

A359222 Number of steps to reach 0 from A359221(n) (Starting numbers that reach a new record high value during iteration by the map x->A359194(x)).

Original entry on oeis.org

0, 1, 2, 11, 80, 7572, 664475, 3180929, 120796790, 556068798, 1246707529, 87037147316
Offset: 1

Views

Author

Joshua Searle, Dec 29 2022

Keywords

Comments

a(12) found by Tom Duff (26 Dec 2022).
It is unknown whether all starting numbers reach 0.

Examples

			a(4) is the step count from the starting number A359221(4) = 3: (3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0) -- 11 steps, hence a(4) = 11.
		

Crossrefs

Programs

  • Python
    from itertools import count, islice
    def f(n): return 1 if n == 0 else (m:=3*n)^((1 << m.bit_length())-1)
    def itersmax(n):
        i, fi, m = 0, n, n
        while fi != 0: i, fi, m = i+1, f(fi), max(m, fi)
        return i, m
    def agen(): # generator of terms
        record = -1
        for m in count(0):
            v, mx = itersmax(m)
            if mx > record:
                yield v # use m to obtain starting numbers
                record = mx
    print(list(islice(agen(), 8))) # Michael S. Branicky, Dec 29 2022
Showing 1-6 of 6 results.