A319161 -id:A319161 - cp's OEIS frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 101, 103, 104, 107, 108, 109, 112, 113, 116

Offset: 1

Reinhard Zumkeller, May 06 2007

nonn

This sequence does not contain any number of the form 36n-6 or 36n+6, as such numbers are divisible by 6 but not by 4 or 9. Consequently, this sequence does not contain 24 consecutive integers. The quest for the greatest number of consecutive integers in this sequence has ties to the ABC conjecture (see the MathOverflow link). - Danny Rorabaugh, Sep 23 2015
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions with distinct multiplicities. The enumeration of these partitions by sum is given by A098859. - Gus Wiseman, May 04 2019
Aktaş and Ram Murty (2017) called these terms "special numbers" ("for lack of a better word"). They prove that the number of terms below x is ~ c*x/log(x), where c > 1 is a constant. - Amiram Eldar, Feb 25 2021
Sequence A005940(1+A328592(n)), n >= 1, sorted into ascending order. - Antti Karttunen, Apr 03 2022

			From _Gus Wiseman_, May 04 2019: (Start)
The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  23: {9}
  24: {1,1,1,2}
  25: {3,3}
  27: {2,2,2}
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Kevser Aktaş and M. Ram Murty, On the number of special numbers, Proceedings - Mathematical Sciences, Vol. 127, No. 3 (2017), pp. 423-430; alternative link.
MathOverflow, Consecutive numbers with mutually distinct exponents in their canonical prime factorization
Carlo Sanna, On the number of distinct exponents in the prime factorization of an integer, arXiv:1902.09224 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Prime Factorization

Complement of A130092. A351564 is the characteristic function.
Subsequence of A351294.
Cf. A000961, A006939, A181818, A304686, A319161, A342028, A342029, A342030, A342031, A342032 (subsequences).
Cf. A005940, A048767, A048768, A056239, A098859, A112798, A118914, A181796, A217605, A325326, A325337, A325368, A327498, A327523, A328592, A336423, A336424, A336569, A336570, A336571, A343012, A343013.

filter:= proc(t) local f;
f:= map2(op,2,ifactors(t)[2]);
nops(f) = nops(convert(f,set));
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 30 2015

t[n_] := FactorInteger[n][[All, 2]]; Select[Range[400],  Union[t[#]] == Sort[t[#]] &]  (* Clark Kimberling, Mar 12 2015 *)

isok(n) = {nbf = omega(n); f = factor(n); for (i = 1, nbf, for (j = i+1, nbf, if (f[i, 2] == f[j, 2], return (0)););); return (1);} \\ Michel Marcus, Aug 18 2013

isA130091(n) = issquarefree(factorback(apply(e->prime(e), (factor(n)[, 2])))); \\ Antti Karttunen, Apr 03 2022

a(n) < A130092(n) for n<=150, a(n) > A130092(n) for n>150.

A356862 Numbers with a unique largest prime exponent.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104

Offset: 1

Jens Ahlström, Sep 01 2022

If the prime factorization of k has a unique largest exponent, then k is a term.
Numbers whose multiset of prime factors (with multiplicity) has a unique mode. - Gus Wiseman, May 12 2023
Disjoint union of A246655 and A376250. The asymptotic density of this sequence, 0.3660366524547281232052..., is equal to the density of A376250 since the prime powers have a zero density. - Amiram Eldar, Sep 17 2024

			Prime powers (A246655) are in the sequence, since they have only one prime exponent in their prime factorization, hence a unique largest exponent.
144 is in the sequence, since 144 = 2^4 * 3^2 and there is the unique largest exponent 4.
225 is not in the sequence, since 225 = 3^2 * 5^2 and the largest exponent 2 is not unique, but rather it is the exponent of both the prime factor 3 and of the prime factor 5.

Jens Ahlström, Table of n, a(n) for n = 1..10000
Wikipedia, Mode (statistics).

Subsequence of A319161 (which has additional terms 1, 180, 252, 300, 396, 450, 468, ...).
Cf. A246655, A356838, A356840, A376250.
For factors instead of exponents we have A102750.
For smallest instead of largest we have A359178, counted by A362610.
The complement is A362605, counted by A362607.
The complement for co-mode is A362606, counted by A362609.
Partitions of this type are counted by A362608.
These are the positions of 1's in A362611, for co-modes A362613.
A001221 is the number of prime exponents, sum A001222.
A027746 lists prime factors, A112798 indices, A124010 exponents.
A362614 counts partitions by number of modes, A362615 co-modes.
Cf. A000041, A002865, A051903, A056239, A070003, A247180, A283050, A327473.

Select[Range[2, 100], Count[(e = FactorInteger[#][[;; , 2]]), Max[e]] == 1 &] (* Amiram Eldar, Sep 01 2022 *)

isok(k) = if (k>1, my(f=factor(k), m=vecmax(f[,2]), w=select(x->(f[x,2] == m), [1..#f~])); #w == 1); \\ Michel Marcus, Sep 01 2022

from sympy import factorint
from collections import Counter
def ok(k):
    c = Counter(factorint(k)).most_common(2)
    return not (len(c) > 1 and c[0][1] == c[1][1])
print([k for k in range(2, 105) if ok(k)])

from sympy import factorint
from itertools import count, islice
def A356862_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:len(f:=sorted(factorint(n).values(),reverse=True))==1 or f[0]!=f[1],count(max(startvalue,2)))
A356862_list = list(islice(A356862_gen(),30)) # Chai Wah Wu, Sep 10 2022

A329139 Numbers whose prime signature is an aperiodic word.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A319161 in having 1260 = 2*2 * 3^2 * 5^1 * 7^1. First differs from A325370 in having 420 = 2^2 * 3^1 * 5^1 * 7^1.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A sequence is aperiodic if its cyclic rotations are all different.

			The sequence of terms together with their prime signatures begins:
   1: ()
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   7: (1)
   8: (3)
   9: (2)
  11: (1)
  12: (2,1)
  13: (1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  20: (2,1)
  23: (1)
  24: (3,1)
  25: (2)
  27: (3)

Complement of A329140.
Aperiodic compositions are A000740.
Aperiodic binary words are A027375.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is a necklace are A329138.
Cf. A025487, A097318, A112798, A124010, A178472, A181819, A304678, A329133, A329135, A329136, A329137, A329142.

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],aperQ[Last/@FactorInteger[#]]&]

A319163 Perfect powers whose prime multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 144, 169, 243, 256, 289, 324, 343, 361, 400, 512, 529, 576, 625, 729, 784, 841, 961, 1024, 1331, 1369, 1600, 1681, 1728, 1849, 1936, 2025, 2048, 2187, 2197, 2209, 2304, 2401, 2500, 2704, 2809, 2916, 3125

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

Perfect powers n such that A181819(n) is not a perfect power (i.e. belongs to A007916).

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (11), (111), (22), (1111), (33), (222), (11111), (44), (111111), (2222), (55), (333), (1111111), (221111), (66), (22222), (11111111), (77), (222211), (444), (88), (331111), (111111111), (99), (22111111), (3333), (222222), (441111).

Cf. A001597, A007916, A056239, A072774, A181819, A289509, A296150, A298748, A319151, A319161, A319162, A319165.

Select[Range[1000],And[GCD@@FactorInteger[#][[All,2]]>1,GCD@@Length/@Split[Sort[FactorInteger[#][[All,2]]]]==1]&]

A325370 Numbers whose prime signature has multiplicities covering an initial interval of positive integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A319161 in lacking 420.
The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.
Numbers whose prime signature covers an initial interval are given by A317090.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities have multiplicities covering an initial interval of positive integers. The enumeration of these partitions by sum is given by A325330.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   12: {1,1,2}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}
   27: {2,2,2}
For example, the prime indices of 1890 are {1,2,2,2,3,4}, whose multiplicities give the prime signature {1,1,1,3}, and since this does not cover an initial interval (2 is missing), 1890 is not in the sequence.

Cf. A000009, A055932, A056239, A112798, A118914, A317081, A317089, A317090, A319161, A325326, A325330, A325337, A325369, A325371.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],normQ[Length/@Split[Sort[Last/@FactorInteger[#]]]]&]

A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

Original entry on oeis.org

1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 45, 58, 83, 108, 142, 188, 250, 315, 417, 528, 674, 861, 1094, 1363, 1724, 2152, 2670, 3311, 4105, 5021, 6193, 7561, 9216, 11219, 13614, 16419, 19886, 23920, 28733, 34438, 41272, 49184, 58746, 69823, 82948, 98380, 116567

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

From Gus Wiseman, Jul 11 2023: (Start)
A partition is aperiodic (A000837) if its multiplicities are relatively prime. This sequence counts partitions whose multiplicities are aperiodic.
For example:
- The multiplicities of (5,3) are (1,1), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (3,2,2,1) are (2,1,1), with multiplicities (2,1), which are relatively prime, so it is counted under a(8).
- The multiplicities of (3,3,1,1) are (2,2), with multiplicities (2), with common divisor 2, so it is not counted under a(8).
- The multiplicities of (4,4,4,3,3,3,2,1) are (3,3,1,1), with multiplicities (2,2), which have common divisor 2, so it is not counted under a(24).
(End)

			The a(8) = 16 partitions:
  (8),
  (44),
  (332), (422), (611),
  (2222), (3221), (4211), (5111),
  (22211), (32111), (41111),
  (221111), (311111),
  (2111111),
  (11111111).
Missing from this list are: (53), (62), (71), (431), (521), (3311).

These partitions have ranks A319161.
For distinct instead of relatively prime multiplicities we have A325329.
Cf. A000837, A001597, A007916, A047966, A071625, A098859, A100953, A181819, A182850, A182857, A305563, A319149, A319162, A319164.

Table[Length[Select[IntegerPartitions[n], GCD@@Length/@Split[Sort[Length/@Split[#]]]==1&]],{n,30}]

A325369 Numbers with no two prime exponents appearing the same number of times in the prime signature.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 81, 82, 83, 84, 85, 86

Offset: 1

Gus Wiseman, May 02 2019

nonn

The prime signature (A118914) is the multiset of exponents appearing in a number's prime factorization.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose multiplicities appear with distinct multiplicities. The enumeration of these partitions by sum is given by A325329.

			Most small numbers are in the sequence. However the sequence of non-terms together with their prime indices begins:
  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}
  88: {1,1,1,5}
For example, the prime indices of 1260 are {1,1,2,2,3,4}, whose multiplicities give the prime signature {1,1,2,2}, and since 1 and 2 appear the same number of times, 1260 is not in the sequence.

Cf. A056239, A098859, A112798, A118914, A130091, A317090, A319161, A325326, A325329, A325331, A325337, A325370, A325371.

Select[Range[100],UnsameQ@@Length/@Split[Sort[Last/@FactorInteger[#]]]&]

A319165 Perfect powers whose prime indices are not relatively prime.

Original entry on oeis.org

9, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 441, 529, 625, 729, 841, 961, 1331, 1369, 1521, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3249, 3481, 3721, 3969, 4225, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7569, 7921, 8281, 9261, 9409

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (22), (33), (222), (44), (2222), (55), (333), (66), (22222), (77), (444), (88), (4422), (99), (3333), (222222).

Cf. A001597, A056239, A072774, A181819, A289509, A296150, A298748, A319151, A319161, A319163, A319164.

Select[Range[10000],With[{t=Transpose[FactorInteger[#]]},And[GCD@@PrimePi/@t[[1]]>1,GCD@@t[[2]]>1]]&]

A319180 Perfect powers whose prime indices are relatively prime.

Original entry on oeis.org

4, 8, 16, 32, 36, 64, 100, 128, 144, 196, 216, 225, 256, 324, 400, 484, 512, 576, 676, 784, 900, 1000, 1024, 1089, 1156, 1225, 1296, 1444, 1600, 1728, 1764, 1936, 2025, 2048, 2116, 2304, 2500, 2601, 2704, 2744, 2916, 3025, 3136, 3364, 3375, 3600, 3844, 4096

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (11), (111), (1111), (11111), (2211), (111111), (3311), (1111111), (221111), (4411), (222111), (3322), (11111111), (222211), (331111), (5511), (111111111), (22111111), (6611), (441111), (332211), (333111).

Cf. A001597, A056239, A072774, A289509, A298748, A319161, A319163, A319165, A319179, A319181.

Select[Range[1000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]==1,GCD@@FactorInteger[#][[All,2]]>1]&]

A319181 Numbers that are not perfect powers but whose prime indices have a common divisor > 1.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 21, 23, 29, 31, 37, 39, 41, 43, 47, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 171, 173, 179, 181, 183, 185, 189

Offset: 1

Gus Wiseman, Sep 12 2018

nonn

A prime index of n is a number m such that prime(m) divides n.

			The sequence of integer partitions whose Heinz numbers are in the sequence begins: (2), (3), (4), (5), (6), (7), (8), (4,2), (9), (10), (11), (12), (6,2), (13), (14), (15), (16), (8,2), (17), (18), (4,2,2), (6,3).

Cf. A007916, A056239, A072774, A289509, A298748, A319161, A319163, A319165, A319179, A319180.

Select[Range[1000],And[GCD@@PrimePi/@FactorInteger[#][[All,1]]>1,GCD@@FactorInteger[#][[All,2]]==1]&]

cp's OEIS Frontend

A130091 Numbers having in their canonical prime factorization mutually distinct exponents.

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A356862 Numbers with a unique largest prime exponent.

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A329139 Numbers whose prime signature is an aperiodic word.

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A319163 Perfect powers whose prime multiplicities appear with relatively prime multiplicities.

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A325370 Numbers whose prime signature has multiplicities covering an initial interval of positive integers.

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A319160 Number of integer partitions of n whose multiplicities appear with relatively prime multiplicities.

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A325369 Numbers with no two prime exponents appearing the same number of times in the prime signature.

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A319165 Perfect powers whose prime indices are not relatively prime.

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