A085307 -id:A085307 - cp's OEIS frontend

A085309 Initial values providing nontrivial cyclic attractor when function defined in A085307 is iterated.

213, 323, 639, 713

Offset: 1

Labos Elemer, Jun 27 2003

			n=213 gives {213,713,3123,3473,15123,713},
n=323 gives {323,1917,713,3123,3473,15123,713},
n=639 gives {639,713,3123,3473,15123,713}.

Cf. A084317-A084319, A085307, A085308, A085309.

Algorithm: 1# factorize n; 2# arrange prime-factors by decreasing size; 3# concatenate prime factors and interpret the result as decimal number.Iterate 1#, 2#, 3#.

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.

A084796 Replace n with concatenation of its prime factors in decreasing order.

Original entry on oeis.org

1, 2, 3, 22, 5, 32, 7, 222, 33, 52, 11, 322, 13, 72, 53, 2222, 17, 332, 19, 522, 73, 112, 23, 3222, 55, 132, 333, 722, 29, 532, 31, 22222, 113, 172, 75, 3322, 37, 192, 133, 5222, 41, 732, 43, 1122, 533, 232, 47, 32222, 77, 552, 173, 1322, 53, 3332, 115, 7222, 193, 292

Offset: 1

N. J. A. Sloane, Jul 19 2003

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

Cf. A032726, A085307, A084797.

with(numtheory):
a:= n-> parse(cat(`if`(n=1, 1,
    sort([seq(i[1]$i[2], i=ifactors(n)[2])], `>`)[]))):
seq(a(n), n=1..100);  # Alois P. Heinz, May 02 2016

Table[FromDigits[Flatten[Table[#[[1]],#[[2]]]&/@ Reverse[ FactorInteger[ n]]]],{n,60}] (* Harvey P. Dale, Aug 29 2016 *)

More terms from Christopher N. Swanson (cswanson(AT)ashland.edu), Jul 22 2003

A302170 Irregular triangle T(n,k) read by rows: first row is 1, n-th row (n > 1) lists distinct prime factors of n in decreasing order.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 2, 7, 2, 3, 5, 2, 11, 3, 2, 13, 7, 2, 5, 3, 2, 17, 3, 2, 19, 5, 2, 7, 3, 11, 2, 23, 3, 2, 5, 13, 2, 3, 7, 2, 29, 5, 3, 2, 31, 2, 11, 3, 17, 2, 7, 5, 3, 2, 37, 19, 2, 13, 3, 5, 2, 41, 7, 3, 2, 43, 11, 2, 5, 3, 23, 2, 47, 3, 2, 7, 5, 2, 17, 3, 13, 2, 53, 3, 2, 11, 5, 7, 2, 19, 3, 29, 2, 59, 5, 3, 2, 61, 31, 2

Offset: 1

Ilya Gutkovskiy, Apr 02 2018

			The irregular triangle begins:
1:  {1}
2:  {2}
3:  {3}
4:  {2}
5:  {5}
6:  {3, 2}
7:  {7}
8:  {2}
9:  {3}
10: {5, 2}
11: {11}
12: {3, 2}

Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A001221 (row lengths), A006530, A008472 (row sums), A020639, A027746, A027748 (another version), A027750, A056538, A085307, A238689.

a302170 n k = a302170_tabl !! (n-1) !! (k-1)
a302170_tabl = map a302170_row [1..]
a302170_row = reverse . a027748_row
-- Brian Chess, Sep 19 2022

Flatten[Table[Reverse[FactorInteger[n][[All, 1]]], {n, 1, 62}]]

T(n,1) = A006530(n).

T(n,A001221(n)) = A020639(n).

A251363 Numbers n such that n is the concatenation of distinct prime factors of phi(n), in decreasing order.

Original entry on oeis.org

237532, 832332, 82953292, 423238803752

Offset: 1

Jahangeer Kholdi, Dec 03 2014

Numbers n such that n = A085307(A000010(n)). - Michel Marcus, Dec 06 2014

			237532 is in the sequence since phi(237532)=23*7*5*3^2*2^4,
832332 is in the sequence since phi(832332)=83*23*3^2*2^4, and
82953292 is in the sequence since phi(82953292)=829*53*29*2^5.

Cf. A000010, A000040, A085307, A251360, A251361, A251362.

a251363[n_Integer] :=
Rest@ Select[Range[n], # == FromDigits[Flatten@ IntegerDigits[
Sort[First@ Transpose@ FactorInteger[EulerPhi[#]], Greater]]] &]; a251363[10^6] (* Michael De Vlieger, Dec 03 2014 *)

a(4) from Max Alekseyev, Feb 10 2025

A084686 Take n-th prime p(n), rewrite it with digits in decreasing order to get b(n), then a(n)=(b(n)-p(n))/9.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 6, 8, 1, 7, 0, 4, 0, 0, 3, 0, 4, 0, 1, 0, 0, 2, 0, 1, 0, 1, 23, 67, 89, 22, 66, 20, 66, 88, 88, 40, 66, 52, 66, 62, 88, 70, 80, 82, 86, 88, 0, 11, 55, 77, 11, 77, 20, 30, 55, 41, 77, 50, 55, 60, 61, 71, 47, 0, 2, 46, 0, 44, 44, 66, 20, 66, 44, 40, 66, 50, 66, 64, 1, 59, 58

Offset: 1

Zak Seidov, Jun 30 2003

			a(7)=6 because p(7)=17, b(n)=71 and (71-17)/9=6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A040164, A085307.

f:= proc(n) local p,L,i;
  p:= ithprime(n);
  L:= sort(convert(p,base,10));
  (add(10^(i-1)*L[i],i=1..nops(L))-p)/9
end proc:
map(f, [$1..100]); # Robert Israel, Nov 26 2019

Table[ -Prime[n]-FromDigits[Sort[ -IntegerDigits[Prime[n]]]], {n, 1, 100}]

a(n) = A283001(A000040(n)). - Robert Israel, Nov 26 2019

A379137 Numbers k such that a nonzero proper substring of the concatenation, in decreasing order, of the prime factors of k (without multiplicity) is divisible by k.

Original entry on oeis.org

66, 95, 132, 995, 9995, 18733, 85713, 93115, 131131, 197591, 316406, 380627, 632812, 999995, 2897105, 4285713, 7231913, 8691315, 58730137, 99999995, 169035711, 507107133, 3005755566, 4870313015, 6011511132, 9023163631, 9091190911

Offset: 1

Jean-Marc Rebert, Dec 15 2024

507107133, 4870313015, and all numbers of the form 5*A055558(k), k>=1, are terms (cf. A378950). - Michael S. Branicky, Dec 16 2024

			66 is a term as 66 = 2 * 3 * 11 -> 1132 contains the substring 132, which is equal to 2 * 66 and is divisible by 66.

Cf. A055558, A085307, A378950.

a(19)-a(20) from Michael S. Branicky, Dec 16 2024

a(21)-a(22) from Jean-Marc Rebert, Dec 16 2024

cp's OEIS Frontend

A085309 Initial values providing nontrivial cyclic attractor when function defined in A085307 is iterated.

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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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A084796 Replace n with concatenation of its prime factors in decreasing order.

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Maple

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A302170 Irregular triangle T(n,k) read by rows: first row is 1, n-th row (n > 1) lists distinct prime factors of n in decreasing order.

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Haskell

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A251363 Numbers n such that n is the concatenation of distinct prime factors of phi(n), in decreasing order.

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Mathematica

Extensions

A084686 Take n-th prime p(n), rewrite it with digits in decreasing order to get b(n), then a(n)=(b(n)-p(n))/9.

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Programs

Maple

Mathematica

Formula

A379137 Numbers k such that a nonzero proper substring of the concatenation, in decreasing order, of the prime factors of k (without multiplicity) is divisible by k.

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Extensions