A112769 -id:A112769 - cp's OEIS frontend

A304678 Numbers with weakly increasing prime multiplicities.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, May 16 2018

nonn

Complement of A112769.

			12 = 2*2*3 has prime multiplicities (2,1) so is not in the sequence.
36 = 2*2*3*3 has prime multiplicities (2,2) so is in the sequence.
150 = 2*3*5*5 has prime multiplicities (1,1,2) so is in the sequence.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A071365, A112769, A130091, A133808, A133811, A242031, A304465, A304679, A304687.

q:= n-> (l-> (t-> andmap(i-> l[i, 2]<=l[i+1, 2],
        [$1..t-1]))(nops(l)))(sort(ifactors(n)[2])):
select(q, [$1..120])[];  # Alois P. Heinz, Nov 11 2019

Select[Range[200],OrderedQ[FactorInteger[#][[All,2]]]&]
Select[Range[90],Min[Differences[FactorInteger[#][[;;,2]]]]>=0&] (* Harvey P. Dale, Jan 28 2024 *)

isok(n) = my(vm = factor(n)[,2]); vm == vecsort(vm); \\ Michel Marcus, May 17 2018

A304686 Numbers with strictly decreasing prime multiplicities.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 52, 53, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121

Offset: 1

Gus Wiseman, May 16 2018

nonn

			10 = 2*5 has prime multiplicities (1,1) so is not in the sequence.
20 = 2*2*5 has prime multiplicities (2,1) so is in the sequence
90 = 2*3*3*5 has prime multiplicities (1,2,1) so is not in the sequence.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A001597, A005117, A055932, A071365, A112769, A130091, A133808, A133811, A181819, A242031, A304678, A304679, A283050.

Select[Range[200],Greater@@FactorInteger[#][[All,2]]&]

isok(n) = my(vm = factor(n)[,2]); vm == vecsort(vm,,4) && (#vm == #Set(vm)); \\ Michel Marcus, May 17 2018

list(lim)=my(v=List()); forfactored(n=1,lim\1, if(n[2][,2]==vecsort(n[2][,2],,8), listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Oct 28 2021

a(n) ~ n log n. - Charles R Greathouse IV, Oct 28 2021

A332831 Numbers whose unsorted prime signature is neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

90, 126, 198, 234, 270, 300, 306, 342, 350, 378, 414, 522, 525, 540, 550, 558, 588, 594, 600, 630, 650, 666, 702, 738, 756, 774, 810, 825, 846, 850, 918, 950, 954, 975, 980, 990, 1026, 1050, 1062, 1078, 1098, 1134, 1150, 1170, 1176, 1188, 1200, 1206, 1242

Offset: 1

Gus Wiseman, Mar 02 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The sequence of terms together with their prime indices begins:
   90: {1,2,2,3}
  126: {1,2,2,4}
  198: {1,2,2,5}
  234: {1,2,2,6}
  270: {1,2,2,2,3}
  300: {1,1,2,3,3}
  306: {1,2,2,7}
  342: {1,2,2,8}
  350: {1,3,3,4}
  378: {1,2,2,2,4}
  414: {1,2,2,9}
  522: {1,2,2,10}
  525: {2,3,3,4}
  540: {1,1,2,2,2,3}
  550: {1,3,3,5}
  558: {1,2,2,11}
  588: {1,1,2,4,4}
  594: {1,2,2,2,5}
  600: {1,1,1,2,3,3}
  630: {1,2,2,3,4}
For example, the prime signature of 540 is (2,3,1), so 540 is in the sequence.

The version for run-lengths of partitions is A332641.
The version for run-lengths of compositions is A332833.
The version for compositions is A332834.
Prime signature is A124010.
Unimodal compositions are A001523.
Partitions with weakly increasing run-lengths are A100883.
Partitions with weakly increasing or decreasing run-lengths are A332745.
Compositions with weakly increasing or decreasing run-lengths are A332835.
Compositions with weakly increasing run-lengths are A332836.
Cf. A100882, A115981, A328509, A329398, A332281, A332640, A332643, A332746, A332836, A332870.

Select[Range[1000],!Or[LessEqual@@Last/@FactorInteger[#],GreaterEqual@@Last/@FactorInteger[#]]&]

Intersection of A071365 and A112769.

A097320 Numbers with more than one distinct prime factor and, in the ordered (canonical) factorization, the exponent always decreases when read from left to right.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 63, 68, 72, 76, 80, 88, 92, 96, 99, 104, 112, 116, 117, 124, 135, 136, 144, 148, 152, 153, 160, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 244, 248, 261, 268, 272, 275, 279, 284, 288

Offset: 1

Ralf Stephan, Aug 04 2004

The numbers in A304686 that are not prime powers. - Peter Munn, Jun 01 2025

			The ordered (canonical) factorization of 80 is 2^4 * 5^1 and 4 > 1, so 80 is in sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Subsequence of A126706, A097318, A112769, A304686.
Subsequences: A057715, A096156.
Cf. A097319, A230766.

fQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Length[f] > 1 && Max[Differences[f]] < 0]; Select[Range[2, 288], fQ] (* T. D. Noe, Nov 04 2013 *)

for(n=1, 320, F=factor(n); t=0; s=matsize(F)[1]; if(s>1, for(k=1, s-1, if(F[k, 2]<=F[k+1, 2], t=1; break)); if(!t, print1(n", "))))

is(n) = my(f = factor(n)[,2]); #f > 1 && vecsort(f,,12) == f \\ Rick L. Shepherd, Jan 17 2018

from sympy import factorint
def ok(n):
    e = list(factorint(n).values())
    return 1 < len(e) == len(set(e)) and e == sorted(e, reverse=True)
print([k for k in range(289) if ok(k)]) # Michael S. Branicky, Dec 20 2021

If n = Product_{k=1..m} p(k)^e(k), with p(k) > p(k-1) for k > 1, then m > 1, e(1) > e(2) > ... > e(m).

Edited by Peter Munn, Jun 01 2025

A329142 Numbers whose prime signature is not a necklace.

Original entry on oeis.org

1, 12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

After a(1) = 1, first differs from A112769 in lacking 1350.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their prime signatures begins:
   1: ()
  12: (2,1)
  20: (2,1)
  24: (3,1)
  28: (2,1)
  40: (3,1)
  44: (2,1)
  45: (2,1)
  48: (4,1)
  52: (2,1)
  56: (3,1)
  60: (2,1,1)
  63: (2,1)
  68: (2,1)
  72: (3,2)
  76: (2,1)
  80: (4,1)
  84: (2,1,1)
  88: (3,1)
  90: (1,2,1)
  92: (2,1)

Complement of A329138.
Binary necklaces are A000031.
Non-necklace compositions are A329145.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is periodic are A329140.
Cf. A001037, A008965, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329139.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[100],#==1||!neckQ[Last/@FactorInteger[#]]&]

A362620 Numbers whose greatest prime factor is not a mode, meaning it appears fewer times than some other.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212

Offset: 1

Gus Wiseman, May 11 2023

nonn

First differs from A112769 in lacking 300.

			The prime factorization of 90 is 2*3*3*5, with modes {3} and maximum 5, so 90 is in the sequence.

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

Partitions of this type are counted by A240302.
The complement is A362619, counted by A171979.
A027746 lists prime factors, A112798 indices, length A001222, sum A056239.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362605 ranks partitions with a more than one mode, counted by A362607.
A362606 ranks partitions with a more than one co-mode, counted by A362609.
A362611 counts modes in prime factorization, triangle version A362614.
A362613 counts co-modes in prime factorization, triangle version A362615.
A362621 ranks partitions with median equal to maximum, counted by A053263.
Cf. A000040, A002865, A237824, A237984, A327473, A327476, A359908, A362616.

filter:= proc(n) local F;
  F:= sort(ifactors(n)[2], (a,b) -> a[1]Robert Israel, Dec 15 2023

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2,100],FreeQ[Commonest[prifacs[#]],Max[prifacs[#]]]&]

A317258 Heinz numbers of integer partitions that are not totally nonincreasing.

Original entry on oeis.org

18, 50, 54, 75, 90, 98, 108, 126, 147, 150, 162, 180, 198, 234, 242, 245, 250, 252, 270, 294, 300, 306, 324, 338, 342, 350, 363, 375, 378, 396, 414, 450, 468, 486, 490, 500, 507, 522, 525, 540, 550, 558, 578, 588, 594, 600, 605, 612, 630, 648, 650, 666, 684

Offset: 1

Gus Wiseman, Jul 25 2018

nonn

An integer partition is totally nonincreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly decreasing and are themselves a totally nonincreasing integer partition.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of all integer partitions that are not totally nonincreasing begins: (221), (331), (2221), (332), (3221), (441), (22211), (4221), (442), (3321), (22221), (32211), (5221), (6221), (551), (443), (3331), (42211), (32221), (4421), (33211), (7221), (222211), (661), (8221), (4331), (552), (3332), (42221), (52211), (9221), (33221).

Cf. A056239, A071365, A100883, A112769, A181819, A182850, A242031, A296150, A305733, A317256, A317257.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totincQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totincQ[Reverse[Length/@Split[q]]]]];
Select[Range[1000],!totincQ[Reverse[primeMS[#]]]&]

A316597 Heinz numbers of integer partitions that are not totally nondecreasing.

Original entry on oeis.org

12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 150, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208

Offset: 1

Gus Wiseman, Jul 29 2018

nonn

The first term of this sequence that is absent from A112769 is 150.

An integer partition is totally nondecreasing if either it is empty or a singleton or its multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are weakly increasing and, taken in reverse order, are themselves a totally nondecreasing integer partition.

			150 is the Heinz number of (3,3,2,1), with multiplicities (1,1,2), which has multiplicities (2,1), which are decreasing, so 150 does not belong to the sequence.

Complement of A316529.

Cf. A056239, A071365, A112769, A181819, A182850, A242031, A296150, A305733, A316496, A317258.

A328870 Numbers whose lengths of runs of 1's in their reversed binary expansion are not weakly increasing.

Original entry on oeis.org

11, 19, 22, 23, 35, 38, 39, 43, 44, 45, 46, 47, 55, 67, 70, 71, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 103, 107, 110, 111, 131, 134, 135, 139, 140, 141, 142, 143, 147, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 163, 166, 167

Offset: 1

Gus Wiseman, Nov 12 2019

nonn

			The sequence of terms together with their reversed binary expansions begins:
  11: (1101)
  19: (11001)
  22: (01101)
  23: (11101)
  35: (110001)
  38: (011001)
  39: (111001)
  43: (110101)
  44: (001101)
  45: (101101)
  46: (011101)
  47: (111101)
  55: (111011)
  67: (1100001)
  70: (0110001)
  71: (1110001)
  75: (1101001)
  76: (0011001)
  77: (1011001)
  78: (0111001)

Complement of A328869.
The version for prime indices is A112769.
The binary expansion of n has A069010(n) runs of 1's.
Cf. A000120, A003714, A014081, A164707, A245563, A304678, A328592.

Select[Range[100],!LessEqual@@Length/@Split[Join@@Position[Reverse[IntegerDigits[#,2]],1],#2==#1+1&]&]

A332871 Number of compositions of n whose run-lengths are not weakly increasing.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 8, 24, 55, 128, 282, 625, 1336, 2855, 6000, 12551, 26022, 53744, 110361, 225914, 460756, 937413, 1902370, 3853445, 7791647, 15732468, 31725191, 63907437, 128613224, 258626480, 519700800, 1043690354, 2094882574, 4202903667, 8428794336, 16897836060

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

Also compositions whose run-lengths are not weakly decreasing.

			The a(4) = 1 through a(6) = 8 compositions:
  (112)  (113)   (114)
         (221)   (1113)
         (1112)  (1131)
         (1121)  (1221)
                 (2112)
                 (11112)
                 (11121)
                 (11211)
For example, the composition (2,1,1,2) has run-lengths (1,2,1), which are not weakly increasing, so (2,1,1,2) is counted under a(6).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for the compositions themselves (not run-lengths) is A056823.
The version for unsorted prime signature is A112769, with dual A071365.
The case without weakly decreasing run-lengths either is A332833.
The complement is counted by A332836.
Compositions that are not unimodal are A115981.
Compositions with equal run-lengths are A329738.
Compositions whose run-lengths are not unimodal are A332727.
Cf. A001523, A072704, A100883, A181819, A329744, A329766, A332641, A332669, A332726, A332745, A332746, A332834, A332835.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!LessEqual@@Length/@Split[#]&]],{n,0,10}]

a(n) = 2^(n - 1) - A332836(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

cp's OEIS Frontend

A304678 Numbers with weakly increasing prime multiplicities.

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A304686 Numbers with strictly decreasing prime multiplicities.

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A332831 Numbers whose unsorted prime signature is neither weakly increasing nor weakly decreasing.

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A097320 Numbers with more than one distinct prime factor and, in the ordered (canonical) factorization, the exponent always decreases when read from left to right.

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A329142 Numbers whose prime signature is not a necklace.

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A362620 Numbers whose greatest prime factor is not a mode, meaning it appears fewer times than some other.

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A317258 Heinz numbers of integer partitions that are not totally nonincreasing.

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A316597 Heinz numbers of integer partitions that are not totally nondecreasing.