A182854 -id:A182854 - cp's OEIS frontend

A182850 a(n) = number of iterations that n requires to reach a fixed point under the x -> A181819(x) map.

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

Offset: 1

Matthew Vandermast, Jan 04 2011

nonn

The fixed points of the x -> A181819(x) map are 1 and 2. Note that the x -> A000005(x) map has the same fixed points, and that A000005(n) = A181819(n) iff n is cubefree (cf. A004709). Under the x -> A181819(x) map, it seems significantly easier to generalize about which kinds of integers take a given number of iterations to reach a fixed point than under the x -> A000005(x) map.

Also the number of steps in the reduction of the multiset of prime factors of n wherein one repeatedly takes the multiset of multiplicities. For example, the a(90) = 5 steps are {2,3,3,5} -> {1,1,2} -> {1,2} -> {1,1} -> {2} -> {1}. - Gus Wiseman, May 13 2018

			A181819(6) = 4; A181819(4) = 3; A181819(3) = 2; A181819(2) = 2. Therefore, a(6) = 3, a(4) = 2, a(3)= 1, and a(2) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

A182857 gives values of n where a(n) increases to a record.

Cf. A000961, A001222, A003434, A005117, A007916, A036459, A112798, A130091, A181819, A182851-A182858, A238748, A304455, A304464, A304465.

a182850 n = length $ takeWhile (`notElem` [1,2]) $ iterate a181819 n
-- Reinhard Zumkeller, Mar 26 2012

Table[If[n<=2,0,Length[FixedPointList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]]]]-1],{n,100}] (* Gus Wiseman, May 13 2018 *)

;; With memoization-macro definec.
(definec (A182850 n) (if (<= n 2) 0 (+ 1 (A182850 (A181819 n))))) ;; Antti Karttunen, Feb 05 2016

For n > 2, a(n) = a(A181819(n)) + 1.
a(n) = 0 iff n equals 1 or 2.
a(n) = 1 iff n is an odd prime (A000040(n) for n>1).
a(n) = 2 iff n is a composite perfect prime power (A025475(n) for n>1).
a(n) = 3 iff n is a squarefree composite integer or a power of a squarefree composite integer (cf. A182853).
a(n) = 4 iff n's prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number (cf. A182854).

A182853 Squarefree composite integers and powers of squarefree composite integers.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Matthew Vandermast, Jan 04 2011

Numbers that require exactly three iterations to reach a fixed point under the x -> A181819(x) map. In each case, 2 is the fixed point that is reached. (1 is the other fixed point of the x -> A181819(x) map.) Cf. A182850.

Numbers such that A001221(n) > 1 and A071625(n) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

Numbers n such that A182850(n) = 3. See also A182854, A182855.
Subsequence of A072774 and A182851.
Cf. A120944.

isoka(n) = (omega(n) > 1) && issquarefree(n); \\ A120944
isok(n) = isoka(n) || (ispower(n,,&k) && isoka(k)); \\ Michel Marcus, Jun 24 2017

from math import isqrt
from sympy import mobius, primepi, integer_nthroot
def A182853(n):
    def g(x): return int(sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))-primepi(x))
    def f(x): return n-2+x+(y:=x.bit_length())-sum(g(integer_nthroot(x,k)[0]) for k in range(1,y))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 19 2024

(define A182853 (MATCHING-POS 1 1 (lambda (n) (= 3 (A182850 n))))) ;; After the alternative definition of the sequence given by the original author. Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 05 2016

A182855 Numbers that require exactly five iterations to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

Offset: 1

Matthew Vandermast, Jan 04 2011

nonn

In each case, 2 is the fixed point that is reached (1 is the other fixed point of the x -> A181819(x) map).
Includes all integers whose prime signature a) contains two or more distinct numbers, and b) contains no number that occurs the same number of times as any other number. The first member of this sequence that does not fit that description is 75675600, whose prime signature is (4,3,2,2,1,1).
A full characterization is: Numbers whose prime signature (1) has not all equal multiplicities but (2) the numbers of distinct parts appearing with each distinct multiplicity are all equal. For example, the prime signature of 2520 is {1,1,2,3}, which satisfies (1) but fails (2), as the numbers of distinct parts appearing with each distinct multiplicity are 1 (with multiplicity 2, the part being 1) and 2 (with multiplicity 1, the parts being 2 and 3). Hence the sequence does not contain 2520. - Gus Wiseman, Jan 02 2019

			1. 180 requires exactly five iterations under the x -> A181819(x) map to reach a fixed point (namely, 2).  A181819(180) = 18;  A181819(18) = 6; A181819(6) = 4; A181819(4) = 3;  A181819(3) = 2 (and A181819(2) = 2).
2. The prime signature of 180 (2^2*3^2*5) is (2,2,1).
a. Two distinct numbers appear in (2,2,1) (namely, 1 and 2).
b. Neither 1 nor 2 appears in (2,2,1) the same number of times as any other number that appears there.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

Numbers n such that A182850(n) = 5. See also A182853, A182854.
Subsequence of A059404 and A182851.  Includes A085987 and A179642 as subsequences.
Cf. A001221, A001222, A006939, A062770, A071625, A118914, A181819, A181821, A182857, A323014, A323022, A323023.

Select[Range[1000],With[{sig=Sort[Last/@FactorInteger[#]]},And[!SameQ@@Length/@Split[sig],SameQ@@Length/@Union/@GatherBy[sig,Length[Position[sig,#]]&]]]&] (* Gus Wiseman, Jan 02 2019 *)

A382858 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all equal.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 4, 0, 1, 6, 1, 0, 1, 24, 1, 12, 1, 2, 1, 0, 1, 36, 4, 0, 36, 0, 1, 10, 1, 120, 0, 0, 1, 84, 1, 0, 0, 24, 1, 3, 1, 0, 38, 0, 1, 240, 6, 18, 0, 0, 1, 246, 0, 6, 0, 0, 1, 96, 1, 0, 30, 720, 1, 0, 1, 0, 0, 14, 1, 660, 1, 0, 74, 0, 1, 0, 1

Offset: 1

Gus Wiseman, Apr 09 2025

nonn

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The a(9) = 4 permutations are:
  (1,1,2,2)
  (1,2,1,2)
  (2,1,2,1)
  (2,2,1,1)

The anti-run case is A335125.
These permutations for factorials are counted by A335407, distinct A382774.
For distinct instead of equal run-lengths we have A382773.
For prime indices we have A382857 (firsts A382878), distinct A382771 (firsts A382772).
Positions of 0 are A382914, signature restriction of A382915.
A003963 gives product of prime indices.
A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
A047966 counts partitions with equal multiplicities, distinct A098859.
A056239 adds up prime indices, row sums of A112798.
A304442 counts partitions with equal run-sums, ranks A353833.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A382913 ranks Look-and-Say partitions by signature, complement A382912.
Cf. A000720, A000961, A001221, A001222, A003242, A048767, A181821, A182854, A238130, A305936, A351202, A382879.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[Select[Permutations[nrmptn[n]],SameQ@@Length/@Split[#]&]],{n,100}]

a(n) = A382857(A181821(n)) = A382857(A304660(n)).

A383113 Numbers whose prime indices have more than one permutation with all distinct run-lengths.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 216, 224, 232, 236, 242

Offset: 1

Gus Wiseman, Apr 20 2025

nonn

First differs from A177425, A182854, A367589 in having 216.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 360 are {1,1,1,2,2,3}, with six permutations with all distinct run-lengths:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)
so 360 is in the sequence.
The terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  80: {1,1,1,1,3}

For exactly one permutation we have A000961, counted by A000005.
For no choices we have A351293, counted by A351295, conjugate A381433, equal A382879.
For at least one choice we have A351294, conjugate A381432, counted by A239455.
These are positions of terms > 1 in A382771, firsts A382772, equal A382878.
For equal run-lengths we have A383089, positions of terms > 1 in A382857.
Partitions of this type are counted by A383111.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct run-lengths (ordered A242882), ranks A130091.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A000720, A001221, A001222, A047966, A048767, A351013, A351202, A381435, A382876, A383090, A383091, A383092, A383112.

Select[Range[100], Length[Select[Permutations[PrimePi/@Join @@ ConstantArray@@@FactorInteger[#]], UnsameQ@@Length/@Split[#]&]]>1&]

The complement is A000961 \/ A351293, counted by A000005 + A351295.

A367589 Numbers with exactly two distinct prime factors, both appearing with different exponents.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 242, 244

Offset: 1

Gus Wiseman, Dec 01 2023

nonn

First differs from A177425 in lacking 360.
First differs from A182854 in lacking 360.
These are the Heinz numbers of the partitions counted by A182473.

			The terms together with their prime indices begin:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  28: {1,1,4}
  40: {1,1,1,3}
  44: {1,1,5}
  45: {2,2,3}
  48: {1,1,1,1,2}
  50: {1,3,3}
  52: {1,1,6}
  54: {1,2,2,2}
  56: {1,1,1,4}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}

The case of any multiplicities is A007774, counts A002133.
These partitions are counted by A182473.
The case of equal exponents is A367590, counts A367588.
A000041 counts integer partitions, strict A000009.
A091602 counts partitions by greatest multiplicity, least A243978.
A098859 counts partitions with distinct multiplicities, ranks A130091.
A116608 counts partitions by number of distinct parts.
Cf. A071625, A072233, A072774, A109297, A367580.

Select[Range[100], PrimeNu[#]==2&&UnsameQ@@Last/@FactorInteger[#]&]

cp's OEIS Frontend

A182850 a(n) = number of iterations that n requires to reach a fixed point under the x -> A181819(x) map.

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A182853 Squarefree composite integers and powers of squarefree composite integers.

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A182855 Numbers that require exactly five iterations to reach a fixed point under the x -> A181819(x) map.

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A382858 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all equal.

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A383113 Numbers whose prime indices have more than one permutation with all distinct run-lengths.

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A367589 Numbers with exactly two distinct prime factors, both appearing with different exponents.

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