A085309 -id:A085309 - cp's OEIS frontend

A085307 a(1) = 1; for n > 1, concatenate distinct prime factors of n in decreasing order.

1, 2, 3, 2, 5, 32, 7, 2, 3, 52, 11, 32, 13, 72, 53, 2, 17, 32, 19, 52, 73, 112, 23, 32, 5, 132, 3, 72, 29, 532, 31, 2, 113, 172, 75, 32, 37, 192, 133, 52, 41, 732, 43, 112, 53, 232, 47, 32, 7, 52, 173, 132, 53, 32, 115, 72, 193, 292, 59, 532, 61, 312, 73, 2, 135, 1132, 67

Offset: 1

Labos Elemer, Jun 27 2003

n and a(n) have the same parity.

			m = 100 = 2*2*5*5 -> {2,5} -> {5,2} -> 52 = a(100);
a(510510) = 1713117532, while A084317(510510) = 2357111317.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

In A084317 the order of factors is increasing.

Cf. A084317, A084318, A084319, A085308, A085309, A084796, A084797.

with(numtheory):
a:= n-> parse(cat(`if`(n=1, 1, sort([factorset(n)[]], `>`)[]))):
seq(a(n), n=1..100);  # Alois P. Heinz, May 02 2016

f[n_] := FromDigits[ Flatten[ IntegerDigits /@ Reverse[ Flatten[ Table[ # [[1]], {1}] & /@ FactorInteger[n]]]]]; Table[ f[n], {n, 1, 70}]
Table[FromDigits[Flatten[IntegerDigits/@Reverse[FactorInteger[n][[All, 1]]]]],{n,90}] (* Harvey P. Dale, Oct 10 2017 *)

Algorithm:
factorize n;
order prime factors by decreasing size;
concatenate prime factors and interpret the result as a decimal number.

Edited by Robert G. Wilson v, Jul 15 2003

A085308 Iterate function described in A085308 (= reverse concatenation of prime factors); a(n) is either 1# the fixed point[=prime] if it exists at all: 2# a(2k)=1 labels that no convergence with most even initial values, in contrary mostly rapid divergence is the case; 3# a(n)=0 if n=1 or if the iteration results in nontrivial attractor with cycle length larger than one.

Original entry on oeis.org

0, 2, 3, 2, 5, 2, 7, 2, 3, 1, 11, 2, 13, 2, 53, 2, 17, 2, 19, 1, 73, 2, 23, 2, 5, 1, 3, 2, 29, 1, 31, 2, 113, 2, 53, 2, 37, 2, 197, 1, 41, 1, 43, 2, 53, 1, 47, 2, 7, 1, 173, 1, 53, 2, 41113, 2, 193, 1, 59, 1, 61, 1, 73, 2, 53, 1, 67, 1, 233, 1, 2, 73, 1, 53, 1, 197, 1, 79, 1, 3, 1, 83, 1, 53, 1

Offset: 1

Labos Elemer, Jun 27 2003

			n=even: remains even: m = 100 = 2*2*5*5 -> {2,5} -> {5,2} -> 52 = a(100);
n = 2^i*3^j: a(n)=2 since iteration list is {n,32,2}; these
are the known convergent even cases of initial value.
n=143: a(143) = 44864859110711 because the iteration list is
{143, 1311, 23193, 8593, 66113, 388917, 547793, 2273241, 55311373, 989474313, 8914183373, 84859143973, 528059391607, 44864859110711};
a(n) = 0 for n = 213, 323, 639, 713 ending in {713, 3123, 3473, 15123}; terminal orbit of length = 4.
All possible cases occur: fixed point, divergence, terminal cycle.

Cf. A084317-A084319, A085308, A085309.

ffi[x_] := Flatten[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] lf[x_] := Length[FactorInteger[x]] nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] rec[x_] := Fold[nd, 0, Flatten[IntegerDigits[Reverse[ba[x]]], 1]] Table[rec[w], {w, 1, 128}]

Algorithm:
factorize n;
arrange prime factors by decreasing size;
concatenate prime factors and interpret the result as a decimal number.

A084685 a(n) is the least x such that length of fixed-point-list when function-A085307[] was iterated and started at a(n) equals n.

Original entry on oeis.org

2, 4, 6, 14, 22, 115, 55, 105, 155, 145, 341, 501, 489, 143, 437, 301, 395, 665

Offset: 1

Labos Elemer, Jul 16 2003

			Some lists of iterated values started at a(n):
n=1: a(1)=first prime; n=2: a(2)=first true prime factor;
n=3: a(3)=first 2^j.3^i number, that is 6;
n=5: {22,112,72,32,2}
n=12: length=a(12); iv=501; fixed-point=1354674597313; list as follows
{501, 1673, 2397, 47173, 293237, 2571637, 23593109, 273116353, 522211523, 8866073119, 57914987307, 1354674597313}

Cf. A085307, A085308, A085309, A085317-A085323.

cp's OEIS Frontend

A085307 a(1) = 1; for n > 1, concatenate distinct prime factors of n in decreasing order.

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A084685 a(n) is the least x such that length of fixed-point-list when function-A085307[] was iterated and started at a(n) equals n.

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