A084317 -id:A084317 - cp's OEIS frontend

A084318 Iterate function described in A084317 if started at initial value n until reaching a fixed point.

0, 2, 3, 2, 5, 23, 7, 2, 3, 5, 11, 23, 13, 3, 1129, 2, 17, 23, 19, 5, 37, 211, 23, 23, 5, 3251, 3, 3, 29, 547, 31, 2, 311, 31397, 1129, 23, 37, 373, 313, 5, 41, 379, 43, 211, 1129, 223, 47, 23, 7, 5, 317, 3251, 53, 23, 773, 3, 1129, 229, 59, 547, 61, 31237, 37, 2, 1129, 2311

Offset: 1

Labos Elemer, Jun 16 2003

Conjecture: fixed point always exists.
Some initial values capriciously provide very large prime fixed-points. This behavior is illustrated in A084319 for initial value n=91.
Unlike the related home primes A037274, the trajectory of numbers in this procedure is not strictly increasing. Of the 8770 numbers < 10000 that have trajectories (that is, that are neither 1 nor prime) 3727 decrease at least once before reaching 30 digits. A sequence with no decreases is twice as likely to not terminate before 30 digits (10.0%) as one that has at least one decrease (4.8%). - Christian N. K. Anderson, May 04 2013

			a(0)=0 since no prime factors to concatenate;
a[p^j]=p for p prime(powers);
n=95=519: fixed-point list is {95,519,3173,19167,36389},
so a(95)=36389, a prime.

Christian N. K. Anderson, Table of n, steps to reach a(n), and a(n) for numbers where a(n) has fewer than 30 digits; NA otherwise. Also includes trajectories, factors separated by |s.

Cf. A084317, A084319.

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] conc[x_] := Fold[nd, 0, Flatten[IntegerDigits[ba[x]], 1]] Table[FixedPoint[conc, w], {w, 1, 90}] Table[conc[w], {w, 1, 128}]

A084319 Orbit of 91 under function described in A084317.

Original entry on oeis.org

91, 713, 2331, 3737, 37101, 383149, 1329473, 10912197, 328312853, 1129846623, 3735159117, 31245053039, 173977184859, 3293176308321, 319269241788861, 371325123869195203, 1278647733810375857, 1665622037676698019, 31742715741254857303, 56627509560552923867

Offset: 0

Labos Elemer, Jun 16 2003

This sequence takes 64 steps to reach a prime (which implies A343156(91)=64). - N. J. A. Sloane, Apr 07 2021

			a(29) = 19797186041838357425713338412621, the 29th iterate.

Eric Angelini, W. Edwin Clark, Hans Havermann, Frank Stevenson, Allan C. Wechsler, and others, Postings to Math Fun mailing list, April 2021.

W. Edwin Clark, Table of n, a(n) for n = 0..63 [The trajectory to the first prime.]

Cf. A084317, A056938, A084318, A343156.

NestList[FromDigits@ Flatten@ IntegerDigits@ Map[First, FactorInteger@ #] &, 91, 17] (* Michael De Vlieger, Mar 25 2017 *)

Example corrected by Rémy Sigrist, Apr 07 2021

A343156 Starting at n, a(n) = number of iterations of the map x -> A084317(x) (concatenate distinct prime factors of x) required to reach a prime, or -1 if no prime is ever reached.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 2, 0, 1, 0, 2, 4, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 4, 1, 2, 0, 2, 0, 1, 1, 5, 3, 1, 0, 2, 1, 2, 0, 2, 0, 1, 4, 1, 0, 1, 1, 2, 1, 4, 0, 1, 2, 2, 2, 1, 0, 2, 0, 3, 1, 1, 3, 1, 0, 5, 3, 1, 0, 1, 0, 2, 4, 2, 2, 2, 0, 2, 1, 1, 0, 2, 3, 2, 3, 1, 0, 2, 64, 1, 1, 2, 4, 1, 0, 2, 1, 2

Offset: 2

N. J. A. Sloane, Apr 07 2021

Judging by the behavior of similar sequences, it is likely that almost all values of a(n) are -1. n = 407 (see A343157) seems to be the first open case.

			10 = 2*5 -> 25 = 5^2 -> 5, prime, taking two steps, so a(10)=2.
a(91) = 64: see A084319.

Eric Angelini, W. Edwin Clark, Hans Havermann, Frank Stevenson, Allan C. Wechsler, and others, Postings to Math Fun mailing list, April 2021.

Hans Havermann, Table of n, a(n) for n = 2..406

Cf. A084317, A037274, A037276, A084319, A195264, A343157.

See A343158 for when k first appears.

A249125 Composite numbers which are a multiple of the concatenation of their prime factors A084317.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 50, 64, 81, 100, 121, 125, 128, 169, 200, 243, 250, 256, 289, 343, 361, 400, 500, 512, 529, 625, 729, 800, 841, 961, 1000, 1024, 1250, 1331, 1369, 1600, 1681, 1849, 2000, 2048, 2187, 2197, 2209, 2401, 2500, 2809, 3125, 3200, 3481, 3721, 4000, 4096, 4489, 4913, 5000, 5041, 5329, 6241, 6250, 6400

Offset: 1

M. F. Hasler, Oct 21 2014

Prime numbers are excluded since they trivially satisfy the condition.
Multiplicity of the prime factors is ignored.
Among the first 10000 terms, the 182 which are not prime powers are of the form 2^h * 5^k. - Giovanni Resta, May 29 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A084317, A069872, A224930, A240265.

Select[Range[6400], CompositeQ[#] &&  Mod[#, FromDigits@ Flatten[ IntegerDigits /@ First /@ FactorInteger@#]] == 0 &] (* Giovanni Resta, May 29 2017 *)

for(n=2,9999,isprime(n)||n%A084317(n)||print1(n","))

A343157 Trajectory of 407 under the map x -> A084317(x).

Original entry on oeis.org

407, 1137, 3379, 31109, 132393, 344131, 1731653, 71143523, 115771019, 7133141039, 18152375353, 723112747673, 1938058565667, 372411163329269, 646991575604859, 3500960117162747, 19920988418382133, 479222853318661919, 3877130279948783893, 71942196909541476259, 7170749184914732550379

Offset: 0

Hans Havermann, Apr 07 2021

It is not known if any a(n) is a prime (see discussion in A343156). - N. J. A. Sloane, Apr 07 2021

Eric Angelini, W. Edwin Clark, Hans Havermann, Frank Stevenson, Allan C. Wechsler, and others, Postings to Math Fun mailing list, April 2021.

Jinyuan Wang, Table of n, a(n) for n = 0..91 (terms 0..84 from Hans Havermann)

Cf. A056938, A084317, A343156.

A084323 Fixed points reached when prime-factor-concatenation function [A084317] is started at n!.

Original entry on oeis.org

0, 2, 23, 23, 547, 547, 2357, 2357, 2357, 2357, 4359293547691, 4359293547691, 325798243129564339, 325798243129564339, 325798243129564339, 325798243129564339, 3947306373286437248759663633906484193454376823

Offset: 1

Labos Elemer, Jun 20 2003

			n=11: 11!=256.81.25.7.11; a(11)=iter[concatenate[{2,3,5,7,11}]] =A084318[39916800]; the list of iteration:
{39916800, 235711, 7151223, 34495309, 41841349, 1116722777, 1958774883, 313113444469, 744730492067, 4359293547691}
at each step the ordered prime factors of previous term are concatenated.

Cf. A001405, A084317, A084318, A084322.

ffi[x_] := Flatten[FactorInteger[x]] ba[x_] := Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}] q[x_] := Apply[Times, Table[Prime[w], {w, 1, x}]] lf[x_] := Length[FactorInteger[x]] nd[x_, y_] := 10*x+y tn[x_] := Fold[nd, 0, x] coc[x_] := Fold[nd, 0, Flatten[IntegerDigits[ba[x]], 1]] Table[FixedPoint[coc, q[w]], {w, 1, 7}]

a(n)=A084318[A000142(n)]=A084318[n! ]

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

Original entry on oeis.org

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

A192137 Numbers m such that their concatenation of prime divisors are palindromic numbers.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 16, 25, 27, 32, 39, 49, 64, 69, 81, 101, 117, 119, 121, 125, 128, 129, 131, 151, 159, 181, 191, 207, 219, 243, 249, 256, 259, 313, 329, 339, 343, 351, 353

Offset: 1

Jaroslav Krizek, Jun 24 2011

The corresponding values of palindromic concatenation in A192138. Superset of A002385 (palindromic primes), A192139 and A192140.

			Concatenation of prime divisors of 39 = 3 * 13 is 313 (palindromic number).

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002113, A002385, A084317, A084092, A192138, A192139, A192140, A192141.

Select[Range[2,500],PalindromeQ[FromDigits[Flatten[IntegerDigits/@ FactorInteger[ #][[All,1]]]]]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 02 2017 *)

A192139 Powers p^m, m >= 0, of palindromic primes p (A002385).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 25, 27, 32, 49, 64, 81, 101, 121, 125, 128, 131, 151, 181, 191, 243, 256, 313, 343, 353, 373, 383, 512, 625, 727, 729, 757, 787, 797, 919, 929, 1024, 1331, 2048, 2187, 2401, 3125, 4096, 6561, 8192, 10201, 10301, 10501, 10601, 11311, 11411, 12421, 12721

Offset: 1

Jaroslav Krizek, Jun 24 2011

Superset of A002385 and A084092. Subset of A192137.

M. F. Hasler, Table of n, a(n) for n = 1..858 (all terms up to 10^7).

Cf. A002113, A002385, A084317, A084092, A192137, A192138, A192140, A192141.

A192139=[1]; forprime(p=1,M=1e7, p==A004086(p) && A192139=setunion(A192139,vector(log(M)\log(p),k,p^k))) \\ M. F. Hasler, May 11 2015

Missing term 625 inserted and more terms added by M. F. Hasler, May 11 2015

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.

cp's OEIS Frontend

A084318 Iterate function described in A084317 if started at initial value n until reaching a fixed point.

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A084319 Orbit of 91 under function described in A084317.

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A343156 Starting at n, a(n) = number of iterations of the map x -> A084317(x) (concatenate distinct prime factors of x) required to reach a prime, or -1 if no prime is ever reached.

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A249125 Composite numbers which are a multiple of the concatenation of their prime factors A084317.

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A343157 Trajectory of 407 under the map x -> A084317(x).

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A084323 Fixed points reached when prime-factor-concatenation function [A084317] is started at n!.

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

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A192137 Numbers m such that their concatenation of prime divisors are palindromic numbers.

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