A177425 -id:A177425 - cp's OEIS frontend

A182854 Integers whose prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number.

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, 245

Offset: 1

Matthew Vandermast, Jan 04 2011

nonn

Numbers that require exactly four 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.

Not the same sequence as A177425, which is a proper subsequence. 1260 is the first nonmember of A177425 that belongs to this sequence; its prime signature is (2,2,1,1).

			The prime signature of 12 (2^2*3^1) is (2,1). Since (2,1) contains at least two distinct numbers, and since each number that appears in (2,1) appears exactly as often as any other number that appears, 12 belongs to this sequence.
12 also requires exactly four iterations under the x -> A181819(x) map to reach a fixed point (namely, 2) .  A181819(12) = 6;  A181819(6) = 4; A181819(4) = 3;  A181819(3) = 2 (and A181819(2) = 2).

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) = 4. See also A182853, A182855.

Subsequence of A059404 and A182852.

aQ[n_] := Length[v = Values @ Counts @ FactorInteger[n][[;;,2]]] > 1 && Length @ Union @ v == 1; Select[Range[250], aQ] (* Amiram Eldar, Aug 08 2019 *)

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

Original entry on oeis.org

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

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[#]&]

A177426 Average of twin prime pairs with multiple and strictly distinct powers.

Original entry on oeis.org

12, 18, 72, 108, 192, 432, 600, 1152, 1620, 1872, 2088, 2268, 2592, 3168, 4050, 4800, 7128, 7488, 9432, 9720, 10008, 13932, 15972, 17208, 18288, 20232, 21492, 21600, 23040, 23832, 25848, 26892, 27108, 27792, 31392, 32832, 33588, 33750, 34128

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 07 2010

nonn

			12 = 2^2*3, 18 = 2*3^2, ... 600 = 2^3*3*5^2, ... 158760 = 2^3*3^4*5*7^2, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001358, A014574, A177425.

f[n_]:=Sort[Last/@FactorInteger[n]];lst={};Do[If[Length[f[n]]>1&&f[n]==Union@f[n]&&PrimeQ[n-1]&&PrimeQ[n+1],AppendTo[lst,n]],{n,0,5*8!}];lst

cp's OEIS Frontend

A182854 Integers whose prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number.

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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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A177426 Average of twin prime pairs with multiple and strictly distinct powers.

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