A098282 -id:A098282 - cp's OEIS frontend

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

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

A145078 Minimal value of cycle of trajectory of n described in A098282, or -1 if no cycle is ever reached.

Original entry on oeis.org

1, 1, 1, 1, 1, 9, 1, 9, 9, 9, 1, 9, 9, 14, 9, 1, 1, 1, 9, 9, 1, 9, 9, 1, 25, 1, 1, 1, 9, 9, 1, 1, 25, 1, 1, 1, 9, 1, 1, 9, 9, 1, 14, 1, 9, 9, 9, 9, 1, 9, 1

Offset: 1

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

a(52) is currently unknown. - Donovan Johnson

a(40) is probably incorrect and should be 1. (See corrections to A098282 and A144914.) - Andrew Howroyd, Feb 06 2018

Cf. A098282, A087712, A145077, A145079, A077960.

Comment edited by John W. Layman, Nov 03 2010
a(1)-a(18) confirmed and a(19)-a(39) added by John W. Layman, Nov 03 2010
a(40) and a(51) from Donovan Johnson, Jan 08 2011

A145079 Length of cycle of trajectory of n described in A098282, or -1 if no cycle is ever reached.

Original entry on oeis.org

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

Offset: 1

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

a(52) is currently unknown. - Donovan Johnson

a(40) is probably incorrect and should be 1. (See corrections to A098282 and A144914.)

Cf. A098282, A087712, A145077, A145078, A077960.

a(19)-a(39) from John W. Layman, Mar 12 2010

a(40)-a(51) from Donovan Johnson, Jan 08 2011

A087712 a(1) = 1; if n = k-th prime, a(n) = k; otherwise write all prime factors of n in nondecreasing order, replace each prime with its rank, and concatenate the ranks.

Original entry on oeis.org

1, 1, 2, 11, 3, 12, 4, 111, 22, 13, 5, 112, 6, 14, 23, 1111, 7, 122, 8, 113, 24, 15, 9, 1112, 33, 16, 222, 114, 10, 123, 11, 11111, 25, 17, 34, 1122, 12, 18, 26, 1113, 13, 124, 14, 115, 223, 19, 15, 11112, 44, 133, 27, 116, 16, 1222, 35, 1114, 28, 110, 17, 1123, 18

Offset: 1

Eric Angelini, Feb 02 2009

Concatenations of consecutive entries of A112798. - R. J. Mathar, Feb 09 2009

The old entry with this A-number was a duplicate of A082467.

			n = 2 = first prime, a(2) = 1.
n = 3 = second prime, a(3) = 2.
n = 4 = 2*2 -> 1,1 -> 11, so a(4) = 11.
n = 6 = 2*3 -> 1,2 -> 12, so a(6) = 12.
n = 12 = 2*2*3 -> 1,1,2 -> 112, so a(12) = 112.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

See A098282 for lengths of trajectories. Cf. A077960, A156055.

Cf. A027746, A049084.

a087712 1 = 1
a087712 n = read $ concatMap (show . a049084) $ a027746_row n :: Integer
-- Reinhard Zumkeller, Oct 03 2012

# Maple program from R. J. Mathar, Feb 08 2009: (Start)
cat2 := proc(a,b) a*10^(max(1,ilog10(b)+1))+b ; end:
A049084 := proc(p) if isprime(p) then numtheory[pi](p) ; else 0 ; fi; end:
A087712 := proc(n) local pf,a,p,ex ; if isprime(n) then A049084(n) ; elif n = 1 then 1 ; else pf := ifactors(n)[2] ; a := 0 ; for p in pf do for ex from 1 to op(2,p) do a := cat2(a, A049084(op(1,p)) ) ; od: od: fi; end:
seq(A087712(n),n=1..140); # (End)
# (Maple program from David Applegate and N. J. A. Sloane, Feb 09 2009)
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;

f[n_] := If[n == 1, 1, FromDigits@ Flatten[ IntegerDigits@# & /@ (PrimePi@# & /@ Flatten[ Table[ First@#, {Last@#}] & /@ FactorInteger@ n])]]; Array[f, 61] (* Robert G. Wilson v, Jun 06 2011 *)

from sympy import factorint, primepi
def a(n):
    if n == 1: return 1
    return int("".join(str(primepi(p))*e for p, e in factorint(n).items()))
print([a(n) for n in range(1, 62)]) # Michael S. Branicky, Oct 01 2024

More terms from R. J. Mathar (Feb 08 2009) and independently from David Applegate and N. J. A. Sloane, Feb 09 2009

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.

cp's OEIS Frontend

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

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A145078 Minimal value of cycle of trajectory of n described in A098282, or -1 if no cycle is ever reached.

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A145079 Length of cycle of trajectory of n described in A098282, or -1 if no cycle is ever reached.

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A087712 a(1) = 1; if n = k-th prime, a(n) = k; otherwise write all prime factors of n in nondecreasing order, replace each prime with its rank, and concatenate the ranks.

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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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