A087712 -id:A087712 - cp's OEIS frontend

A098282 Iterate the map k -> A087712(k) starting at n; a(n) is the number of steps at which we see a repeated term for the first time; or -1 if the trajectory never repeats.

1, 2, 3, 6, 4, 31, 7, 55, 4, 33, 5, 30, 32, 1, 4, 19, 8, 112, 56, 16, 27, 4, 4, 26, 2, 20, 223, 102, 34, 14, 6, 162, 2, 9, 10, 75, 31, 113, 21, 100, 33, 20, 2, 23, 30, 57, 5, 28, 24, 30, 224, 269, 20, 295, 11, 85, 103, 140, 9, 71, 113, 55, 34, 110, 76, 49, 57

Offset: 1

Eric Angelini, Feb 02 2009

The old entry with this A-number was a duplicate of A030298.
a(52) is currently unknown. - Donovan Johnson
a(52)-a(10000) were found using a conjunction of Mathematica and Kim Walisch's primecount program. The additional values of the prime-counting function can be found in the second a-file. - Matthew House, Dec 23 2016

			1 -> 1; 1 step to see a repeat, so a(1) = 1.
2 -> 1 -> 1; 2 steps to see a repeat.
3 -> 2 -> 1 -> 1; 3 steps to see a repeat.
4 -> 11 -> 5 -> 3 -> 2 -> 1 -> 1; 6 steps to see a repeat.
6 -> 12 -> 112 -> 11114 -> 1733 -> 270 -> 12223 -> 7128 -> 11122225 -> 33991010 -> 13913661 -> 2107998 -> 12222775 -> 33910130 -> 131212367 -> 56113213 -> 6837229 -> 4201627 -> 266366 -> 112430 -> 131359 -> 7981 -> 969 -> 278 -> 134 -> 119 -> 47 -> 15 -> 23 -> 9 -> 22 -> 15; 31 steps to see a repeat.
9 -> 22 -> 15 -> 23 -> 9; 4 steps to see a repeat.
From _David Applegate_ and _N. J. A. Sloane_, Feb 09 2009: (Start)
The trajectories of the numbers 1 through 17, up to and including the first repeat, are as follows. Note that a(n) is one less than the number of terms shown.
[1, 1]
[2, 1, 1]
[3, 2, 1, 1]
[4, 11, 5, 3, 2, 1, 1]
[5, 3, 2, 1, 1]
[6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15]
[7, 4, 11, 5, 3, 2, 1, 1]
[8, 111, 212, 1116, 112211, 52626, 124441, 28192, 11111152, 111165448, 1117261018, 1910112963, 252163429, 42205629, 2914219, 454002, 127605, 231542, 110938, 15631, 44510, 13605, 23155, 3582, 12246, 12637, 1509, 296, 11112, 111290, 131172, 1127117, 76613, 9470, 13161, 21328, 11111114, 14142115, 3625334, 1125035, 348169, 78151, 11369, 1373, 220, 1135, 349, 70, 134, 119, 47, 15, 23, 9, 22, 15]
[9, 22, 15, 23, 9]
[10, 13, 6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15]
[11, 5, 3, 2, 1, 1]
[12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15]
[13, 6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15]
[14, 14]
[15, 23, 9, 22, 15]
[16, 1111, 526, 156, 1126, 1103, 185, 312, 11126, 1734, 1277, 206, 127, 31, 11, 5, 3, 2, 1, 1]
[17, 7, 4, 11, 5, 3, 2, 1, 1]
For n = 18 see A077960.
(End)

Matthew House, Table of n, a(n) for n = 1..10000
Farideh Firoozbakht, Notes on the missing terms in this sequence
Matthew House, Values found using primecount

Cf. A087712, A007097, A077960. See also A145077, A145078, A145079, A144760, A144813, A144814, A144915, A144914.

See A156055 for another version.

void ea (n)
{
mpz u[] ; // factors
mpz tr[]; // sequence
print(n);
while(n > 1)
{
lfactors(u,n); // factorize into u
vmap(u,pi); // replace factors by rank
n = catv(u); // concatenate
print(n);
if(vsearch(tr,n) > 0) break; // loop found
vpush(tr,n); // remember n
}
println('');
}
// Jacques Tramu

import Data.List (genericIndex)
a098282 n = f [n] where
   f xs = if y `elem` xs then length xs else f (y:xs) where
     y = genericIndex (map a087712 [1..]) (head xs - 1)
-- Reinhard Zumkeller, Jul 14 2013

with(numtheory):
f := proc(n) local t1, v, r, x, j;
if (n = 1) then return 1; end if;
t1 := ifactors(n): v := 0;
for x in op(2,t1) do r := pi(x[1]):
for j from 1 to x[2] do
v := v * 10^length(r) + r;
end do; end do; v; end proc;
t := proc(n) local v, l, s; v := n; s := {v}; l := [v]; v := f(v);
while not v in s do s := s union {v}; l := [op(l),v]; v := f(v); end do;
[op(l),v];
end proc; [seq(nops(t(n))-1, n=1..17)];
# David Applegate and N. J. A. Sloane, Feb 09 2009

f[n_] := If[n==1,1,FromDigits@ Flatten[ IntegerDigits@# & /@ (PrimePi@#
& /@ Flatten[ Table[ First@#, {Last@#}] & /@ FactorInteger@n])]];
g[n_] := Length@ NestWhileList[f, n, UnsameQ, All] - 1; Array[g, 39]
(* Robert G. Wilson v, Feb 02 2009; modified slightly by Farideh Firoozbakht, Feb 10 2009 *)

a(8) and a(10) found by Jacques Tramu
Extended through a(39) by Robert G. Wilson v, Feb 02 2009
Terms through a(39) corrected by Farideh Firoozbakht, Feb 10 2009
a(40)-a(51) from Donovan Johnson, Jan 08 2011
More terms from and a(40) corrected by Matthew House, Dec 23 2016

A077960 Trajectory of 18 under iteration of the map k -> A087712(k).

Original entry on oeis.org

18, 122, 118, 117, 226, 130, 136, 1117, 187, 57, 28, 114, 128, 1111111, 52628, 111748, 114663, 212174, 110111, 35131, 81414, 122615, 33341, 4584, 111243, 25475, 33171, 21339, 22351, 41127, 21621, 2920, 111321, 2224811, 223249

Offset: 0

N. J. A. Sloane, Feb 10 2009

a(n) = 1 for n >= 111.

The old entry with this A-number was a duplicate of A077919.

N. J. A. Sloane, Table of n, a(n) for n = 0..120

Cf. A087712, A098282, A144914, A077919.

A144914 Trajectory of 40 under iteration of the map k -> A087712(k).

Original entry on oeis.org

40, 1113, 2416, 111136, 11111936, 1111111115298, 1291622143616, 11111114421614356, 1180446322201364, 114902402126478, 12851299161710, 1326073277802, 124682398018, 113183781186, 124271634111, 21770237322, 1222221326084, 1151561573555, 33189494813, 542767316

Offset: 0

N. J. A. Sloane, Feb 18 2009

Does this trajectory converge?

Yes, for n >= 99 a(n) = 1. - Andrew Howroyd, Feb 06 2018

Andrew Howroyd, Table of n, a(n) for n = 0..100

Cf. A087712, A098282, A077960.

Terms a(8) and beyond corrected by Andrew Howroyd, Feb 06 2018

A144760 Trajectory of 6 under iteration of the map k -> A087712(k).

Original entry on oeis.org

6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15, 23, 9, 22

Offset: 0

N. J. A. Sloane, Feb 18 2009

nonn

31 steps to see a repeat.

Eric Angelini, Prime's Rank (French)
E. Angelini, Prime's Rank [Cached, with permission]

Cf. A087712, A098282.

A144813 Trajectory of 8 under iteration of the map k -> A087712(k).

Original entry on oeis.org

8, 111, 212, 1116, 112211, 52626, 124441, 28192, 11111152, 111165448, 1117261018, 1910112963, 252163429, 42205629, 2914219, 454002, 127605, 231542, 110938, 15631, 44510, 13605, 23155, 3582, 12246, 12637, 1509, 296

Offset: 0

N. J. A. Sloane, Feb 18 2009

55 steps to see a repeat.

Andrew Howroyd, Table of n, a(n) for n = 0..55

Cf. A087712, A098282.

A156055 Define a map f by f(0) = f(1) = 0, otherwise f(k) = A087712(k); then a(n) is the number of steps for the trajectory of n under repeated iteration of f to "terminate".

Original entry on oeis.org

1, 2, 3, 6, 4, 30, 7, 54, 3, 32, 5, 29, 31, 0, 3, 19, 8, 112, 55, 15, 27, 3, 3, 26, 1, 20, 223, 102, 33, 13, 6, 162, 1, 9, 10, 75, 30, 113, 21

Offset: 1

Robert G. Wilson v, Feb 02 2009

Here "terminate" means reaching 0 or a cycle.
From M. F. Hasler, Feb 11 2009: (Start)
"Reaching a cycle" could be better defined: does it mean "reach a value that occurred earlier" or "reach an element belonging to a cycle"?
I think the second is the case, but the value 0 is currently listed at n=14, wouldn't it correspond to x=15 = least element of a nontrivial cycle?
So would the offset be 2 ? or is there a missing term (since the first terms 1,2,3 seem well to correspond to x=1,2,3)? (End)

			a(4) = 6 because 4 -> [{2,2}->{1,1}] ->[{11}->{5}] -> [{5}->{3}] -> [{3}->{2}] -> [{2}->{1}] -> [{1}->{0}].

A variant of A098282, which is the official version of this sequence.

Cf. A087712.

f[n_] := FromDigits@ Flatten[ IntegerDigits@# & /@ (PrimePi@# & /@ Flatten[ Table[ First@#, {Last@#}] & /@ FactorInteger@n])]; g[n_] := Length@ NestWhileList[f, n, UnsameQ, All] - 2; Array[g, 39]

Edited by N. J. A. Sloane, Feb 10 2009

A144814 Trajectory of 10 under iteration of the map k -> A087712(k).

Original entry on oeis.org

10, 13, 6, 12, 112, 11114, 1733, 270, 12223, 7128, 11122225, 33991010, 13913661, 2107998, 12222775, 33910130, 131212367, 56113213, 6837229, 4201627, 266366, 112430, 131359, 7981, 969, 278, 134, 119, 47, 15, 23, 9, 22, 15

Offset: 0

N. J. A. Sloane, Feb 18 2009

nonn

33 steps to see a repeat.

Cf. A087712, A098282.

A144915 Trajectory of 16 under iteration of the map k -> A087712(k).

Original entry on oeis.org

16, 1111, 526, 156, 1126, 1103, 185, 312, 11126, 1734, 1277, 206, 127, 31, 11, 5, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Feb 18 2009

nonn

19 steps to see a repeat.

Cf. A087712, A098282.

A359486 Indices of primes in A087712.

Original entry on oeis.org

3, 4, 5, 10, 11, 15, 17, 20, 31, 34, 41, 45, 46, 59, 60, 67, 69, 75, 80, 82, 83, 85, 90, 93, 102, 109, 119, 127, 136, 153, 155, 157, 170, 179, 191, 205, 206, 207, 211, 221, 230, 236, 241, 246, 249, 253, 254, 272, 276, 277, 283, 295, 309, 314, 322, 327, 328, 331, 332, 334, 345

Offset: 1

Jean-Marc Rebert, Jan 02 2023

nonn

Cf. A027746, A049084, A098282, A112798, A087712.

A145077 Highest point reached in trajectory of n described in A098282, or -1 if no cycle is ever reached.

Original entry on oeis.org

1, 2, 3, 11, 5, 131212367, 11, 1910112963, 23, 131212367, 11, 131212367, 131212367, 14, 23, 11126, 17, 222312455509, 1910112963, 1135, 1112122, 23, 23, 1112122, 33, 11126, 133156118699543, 222312455509, 131212367, 1135, 31, 111151786119, 33, 34, 35, 2455612

Offset: 1

David Applegate and N. J. A. Sloane, Feb 09 2009

a(52) is currently unknown. - Donovan Johnson, Jan 08 2011

a(40) is probably incorrect and should be 11111114421614356. (See corrections to A098282 and A144914.)

Donovan Johnson, Table of n, a(n) for n = 1..51

Cf. A077960, A087712, A098282, A145078, A145079.

More terms from Donovan Johnson, Jan 08 2011

cp's OEIS Frontend

A098282 Iterate the map k -> A087712(k) starting at n; a(n) is the number of steps at which we see a repeated term for the first time; or -1 if the trajectory never repeats.

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A077960 Trajectory of 18 under iteration of the map k -> A087712(k).

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A144914 Trajectory of 40 under iteration of the map k -> A087712(k).

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A144760 Trajectory of 6 under iteration of the map k -> A087712(k).

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A144813 Trajectory of 8 under iteration of the map k -> A087712(k).

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A156055 Define a map f by f(0) = f(1) = 0, otherwise f(k) = A087712(k); then a(n) is the number of steps for the trajectory of n under repeated iteration of f to "terminate".

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A144814 Trajectory of 10 under iteration of the map k -> A087712(k).

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A144915 Trajectory of 16 under iteration of the map k -> A087712(k).

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A359486 Indices of primes in A087712.

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A145077 Highest point reached in trajectory of n described in A098282, or -1 if no cycle is ever reached.

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