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.

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A326741 Numbers which converge to 8 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

8, 23, 27, 33, 34, 81, 92, 108, 118, 128, 138, 148, 158, 168, 178, 188, 198, 208, 214, 222, 231, 248, 254, 262, 271, 287, 308, 319, 323, 329, 331, 333, 334, 341, 408, 412, 428, 432, 447, 459, 508, 608, 623, 632, 708, 748, 794, 808, 811, 822, 908, 913, 919, 921
Offset: 1

Views

Author

Martin Renner, Jul 22 2019

Keywords

Examples

			33 -> 3^3 = 27 -> 2^7 = 128 -> 1^2*8 = 8.

Crossrefs

Cf. A133500, A133501, A135384, A135385, A309385, A326733, A326734, A326735, A326736, A326737, A326738, A326739, A326740, A326742.

A326742 Numbers which converge to 9 under repeated application of the powertrain map of A133500.

Original entry on oeis.org

9, 25, 32, 52, 91, 109, 119, 129, 139, 149, 159, 169, 179, 189, 199, 209, 228, 234, 242, 251, 279, 295, 309, 313, 321, 337, 377, 409, 418, 422, 509, 515, 521, 539, 544, 609, 709, 809, 814, 835, 909, 911, 965, 1025, 1032, 1052, 1091, 1125, 1132, 1152, 1191
Offset: 1

Views

Author

Martin Renner, Jul 22 2019

Keywords

Examples

			25 -> 2^5 = 32 -> 3^2 = 9.

Crossrefs

Cf. A133500, A133501, A135384, A135385, A309385, A326733, A326734, A326735, A326736, A326737, A326738, A326739, A326740, A326741.

Programs

  • Python
    def powertrain(n):
    p, s = 1, str(n)
    if len(s)%2 == 1: s += '1'
    for b, e in zip(s[0::2], s[1::2]): p *= int(b)**int(e)
    return p
def aupto(limit, target=0):
    alst = []
    for n in range(1, limit+1):
        m, ptm = n, powertrain(n)
        while m != ptm: m, ptm = ptm, powertrain(ptm)
        if m == target: alst.append(n)
    return alst
print(aupto(1191, target=9)) # Michael S. Branicky, Sep 25 2021

A135381 a(n) = high point in trajectory of n under repeated application of powertrain map (see A133500).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 32, 531441, 128, 256, 512, 30, 31, 32, 128, 81, 5832000, 729, 30840979456, 191102976, 102372436321763328, 40, 41, 42, 531441, 256, 1024, 531441, 531441, 5832000, 8470728, 50, 51, 32, 125, 625
Offset: 0

Views

Author

J. H. Conway and N. J. A. Sloane, Dec 10 2007

Keywords

Examples

			The trajectory of 39 is 39 -> 19683 -> 5038848 -> 214990848 -> 17179869184 -> 1735247072139264 -> 19999187712 -> 102372436321763328 -> 8813365017182208 -> 0, so a(39) = 102372436321763328.

Links

Crossrefs

Cf. A133500, A133501. For records see A135382, A135383.

Programs

  • Maple
    maxtraj := proc(n) local h,p,M,t1,t2,i; M:=100; t1:=n; h:=n; for i from 1 to M do t2:=powertrain(t1); if t2 = t1 then RETURN(n,h); fi; if t2 > t1 then h:=t2; fi; t1:=t2; od; RETURN(n,-1); end;

A133503 Numbers for which iteration of the powertrain map of A133500 takes a record number of steps to converge.

Original entry on oeis.org

0, 10, 24, 26, 39, 3573, 26899, 68697, 497699, 3559595, 555959597395
Offset: 1

Views

Author

J. H. Conway and N. J. A. Sloane, Dec 03 2007, Dec 04 2007, Dec 18 2007

Keywords

Comments

Where records occur in A133501.
This sequence is almost certainly finite.
The number 31395559595973 takes 16 steps to converge and may be the next term. It may also be the last term.
The next term is > 10^7 (and <= 31395559595973).

Examples

			The smallest number that takes 13 steps to converge is 497699, for which the trajectory is 497699 -> 11948427342082473984 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The smallest number that takes 15 steps to converge is 3559595 -> for which the trajectory is 3559595 -> 4634857177734375 -> 23122964691361341376561152 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The number 31395559595973 takes 16 steps to converge and so the next term is >= 16. The trajectory is 31395559595973 -> 471570692025125026702880859375 -> 34755118508614725279865110528 -> 23122964691361341376561152000000 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The smallest number that takes 16 steps to converge is 555959597395, for which the trajectory starts 555959597395 -> 471570692025125026702880859375 and then continues as above. - _Michael S. Branicky_, Jan 24 2022

Crossrefs

See A133508 for the corresponding numbers of steps. Cf. A133500, A133501.
See also A003001.

Extensions

a(11) from Michael S. Branicky, Jan 24 2022

A133508 Record numbers of steps associated with the terms of A133503.

Original entry on oeis.org

0, 1, 2, 5, 9, 10, 11, 12, 13, 15, 16
Offset: 1

Views

Author

J. H. Conway and N. J. A. Sloane, Dec 04 2007, Dec 18 2007

Keywords

Comments

This sequence is almost certainly finite.

Examples

			The smallest number that takes 13 steps to converge is 497699, for which the trajectory is 497699 -> 11948427342082473984 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The smallest number that takes 15 steps to converge is 3559595 -> for which the trajectory is 3559595 -> 4634857177734375 -> 23122964691361341376561152 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The number 31395559595973 takes 16 steps to converge and so the next term is >= 16.
The trajectory of 31395559595973 is 31395559595973 -> 471570692025125026702880859375 -> 34755118508614725279865110528 -> 23122964691361341376561152000000 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.

Crossrefs

Cf. A133500, A133501, A133503.

Extensions

a(11) from Michael S. Branicky, Jan 24 2022

A226135 Let abcd... be the decimal expansion of n. Number of iterations of the map n -> f(n) needed to reach a number < 10, where f(n) = a^b + c^d + ... which ends in an exponent or a base according as the number of digits is even or odd.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 5, 2, 21, 2, 1, 1, 1, 3, 2, 3, 6, 8, 19, 6, 1, 1, 2, 5, 21, 3, 4, 12, 17, 4, 1, 1, 3, 2, 3, 5, 4, 15, 4, 3, 1, 1, 7, 2, 4, 14, 16, 4, 16, 4, 1, 1, 5, 6, 3, 2, 5, 11, 13, 15, 1, 1, 5
Offset: 0

Views

Author

Michel Lagneau, May 27 2013

Keywords

Comments

Inspired by the sequence A133501 (Number of steps for "powertrain" operation to converge when started at n). It is conjectured that the trajectory for each number converges to a single number < 10.
The conjecture is true, since f(x) < x trivially holds for x > 10^10 and I have verified that for every 10 <= x <= 10^10 there is a k such that f^(k)(x) < x, where f^(k) denotes f applied k times. Both the conventions 0^0 = 1 and 0^0 = 0 have been taken into account. - Giovanni Resta, May 28 2013

Examples

			a(62) = 7 because:
62 -> 6^2 = 36;
36 -> 3^6 = 729;
729 -> 7^2 + 9^1 = 58;
58 -> 5^8 = 390625;
390625 -> 3^9 + 0^6 + 2^5 = 19715;
19715 -> 1^9 + 7^1 + 5^1 = 13;
13 -> 1^3 = 1;
62 -> 36 -> 729 -> 58 -> 390625 -> 19715 -> 13 -> 1 with 7 iterations.

Links

Crossrefs

Cf. A000312, A031348, A031349, A045503, A133500, A225974.

Programs

  • Maple
    A133501:= proc(n)
     local a, i, n1, n2, t1, t2;
     n1:=abs(n); n2:=sign(n); t1:=convert(n1, base, 10); t2:=nops(t1); a:=0;
        for i from 0 to floor(t2/2)-1 do
         a := a+t1[t2-2*i]^t1[t2-2*i-1];
       od:
       if t2 mod 2 = 1 then
       a:=a+t1[1]; fi; RETURN(n2*a); end;
A226135:= proc(n)
    local traj , c;
    traj := n ;
    c := [n] ;
    while true do
       traj := A133501(traj) ;
       if member(traj, c) then
       return nops(c)-1 ;
       end if;
       c := [op(c), traj] ;
    end do:
end proc:
seq(A226135(n), n=0..100) ;
# second Maple program:
f:= n-> `if`(n<10, n, `if`(is(length(n), odd), f(10*n+1),
               iquo(irem(n, 100, 'r'), 10, 'h')^h+f(r))):
a:= n-> `if`(n<10, 0, 1+a(f(n))):
seq(a(n), n=0..100);  # Alois P. Heinz, May 27 2013
Previous Showing 11-16 of 16 results.

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