A071365 -id:A071365 - cp's OEIS frontend

A096011 Arrange numbers n such that A071364(n) <> A046523(n) by prime signature, cf. A071365.

18, 54, 50, 90, 250, 75, 108, 126, 375, 98, 150, 500, 198, 686, 147, 162, 294, 1125, 234, 1029, 242, 270, 1250, 490, 1372, 306, 1715, 245, 300, 378, 1875, 726, 3087, 342, 2662, 338, 324, 588, 594, 4802, 735, 5324, 350, 3993, 363, 450, 2500, 980, 702

Offset: 0

Alford Arnold, Jul 20 2004

In the example, the first column is sequence A056808; the first row is A095990.

			The array begins
18 50 75 98 147 ...
54 250 375 686
90 126 198
108 500
150
...

Extended by Ray Chandler, Jul 31 2004

A304678 Numbers with weakly increasing prime multiplicities.

Original entry on oeis.org

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

A071364 Smallest number with same sequence of exponents in canonical prime factorization as n.

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 6, 2, 12, 2, 6, 6, 16, 2, 18, 2, 12, 6, 6, 2, 24, 4, 6, 8, 12, 2, 30, 2, 32, 6, 6, 6, 36, 2, 6, 6, 24, 2, 30, 2, 12, 12, 6, 2, 48, 4, 18, 6, 12, 2, 54, 6, 24, 6, 6, 2, 60, 2, 6, 12, 64, 6, 30, 2, 12, 6, 30, 2, 72, 2, 6, 18, 12, 6, 30, 2, 48, 16, 6, 2, 60, 6, 6, 6, 24

Offset: 1

Reinhard Zumkeller, May 21 2002

nonn

A046523(a(n))=A046523(n); A046523(n)<=a(n)<=n; A001221(a(n))=A001221(n), A001222(a(n))=A001222(n); A020639(a(n))=2, A006530(a(n))=A000040(A001221(n))<=A006530(n); A000005(a(n))=A000005(n);
a(a(n))=a(n); a(n)=2^k iff n=p^k, p prime, k>0 (A000961); if n>1 is not a prime power, then a(n) mod 6 = 0; range of values = A055932, as distinct prime factors of a(n) are consecutive: a(n)=n iff n=A055932(k) for some k;
a(A003586(n))=A003586(n).

			a(105875) = a(5*5*5*7*11*11) = 2*2*2*3*5*5 = 600.

Daniel Forgues, Table of n, a(n) for n=1..100000

Cf. A071365, A071366.
Cf. A000040.
Cf. A085079, A089247.
The range is A055932.
The reversed version is A331580.
Unsorted prime signature is A124010.
Numbers whose prime signature is aperiodic are A329139.
Cf. A056239, A112798, A233249, A333217, A333219, A334032, A334033.

a071364 = product . zipWith (^) a000040_list . a124010_row
-- Reinhard Zumkeller, Feb 19 2012

Table[ e = Last /@ FactorInteger[n]; Product[Prime[i]^e[[i]], {i, Length[e]}], {n, 88}] (* Ray Chandler, Sep 23 2005 *)

a(n) = f = factor(n); for (i=1, #f~, f[i,1] = prime(i)); factorback(f); \\ Michel Marcus, Jun 13 2014

from math import prod
from sympy import prime, factorint
def A071364(n): return prod(prime(i+1)**p[1] for i,p in enumerate(sorted(factorint(n).items()))) # Chai Wah Wu, Sep 16 2022

In prime factorization of n, replace least prime by 2, next least by 3, etc.

a(n) = product(A000040(k)^A124010(k): k=1..A001221(n)). - Reinhard Zumkeller, Apr 27 2013

Extended by Ray Chandler, Sep 23 2005

A242031 Numbers n such that prime factorization n = p_1^k_1p_2^k_2...*p_r^k_r satisfies k_1 >= k_2 >= ... >= k_r.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72

Offset: 1

Jean-François Alcover, Aug 14 2014

nonn

Complement sequence begins 18, 50, 54, 75, 90, 98, ... (A071365).

			12 = 2^2*3^1 is in the sequence, but 18 = 2^1*3^2 is not.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

Cf. A071365, A304686 (strictly decreasing).

filter:= proc(n)
local F;
F:= ifactors(n)[2];
F:= sort(F,(s,t) -> s[1]>t[1]);
ListTools:-Sorted(map(t -> t[2],F));
end:
select(filter, [$1..100]); # Robert Israel, Aug 18 2014

Select[Range[100], GreaterEqual @@ (FactorInteger[#][[All, 2]]) &]

s=[]; for(n=1, 10^3, m=factor(n)[,2]; if(vecsort(m,,4)==m, s=concat(s, n))); s \\ Jens Kruse Andersen, Aug 18 2014

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.

A095990 Numbers with ordered prime signature (1,2).

Original entry on oeis.org

18, 50, 75, 98, 147, 242, 245, 338, 363, 507, 578, 605, 722, 845, 847, 867, 1058, 1083, 1183, 1445, 1587, 1682, 1805, 1859, 1922, 2023, 2523, 2527, 2645, 2738, 2883, 3179, 3362, 3698, 3703, 3757, 3971, 4107, 4205, 4418, 4693, 4805, 5043, 5547, 5618, 5819

Offset: 1

Alford Arnold, Jul 18 2004

Numbers of the form p*q^2 where p and q are primes with p < q.

A054753 contains natural numbers with ordered prime signatures (2,1) and (1,2).

			18 = 2*3*3, 50 = 2*5*5, 75 = 3*5*5, 98 = 2*7*7, 147 = 3*7*7, ...

Enrique Pérez Herrero, Table of n, a(n) for n = 1..3000
OEIS Wiki, Ordered prime signatures

Subsequence of A071365.

Cf. A054753, A095904, A096156.

Take[ Sort[ Flatten[ Table[ Prime[p]*Prime[q]^2, {q, 2, 16}, {p, q - 1}]]], 46] (* Robert G. Wilson v, Jul 23 2004 *)

list(lim)=my(v=List());forprime(q=3, sqrtint(lim\2), forprime(p=2, min(lim\q^2,q-1), listput(v,p*q^2))); Set(v) \\ Charles R Greathouse IV, Feb 26 2014

Edited and extended by Robert G. Wilson v, Jul 23 2004

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.

A071366 Numbers n such that A046523(n) <> A071364(n) and A071364(n) <> n.

Original entry on oeis.org

50, 75, 98, 126, 147, 198, 234, 242, 245, 250, 294, 306, 338, 342, 350, 363, 375, 378, 414, 490, 500, 507, 522, 525, 550, 558, 578, 588, 594, 605, 650, 666, 686, 702, 722, 726, 735, 738, 756, 774, 825, 845, 846, 847, 850, 867, 882, 918, 950, 954, 975, 980

Offset: 1

Reinhard Zumkeller, May 21 2002

nonn

Also A046523(n) <> n, as A046523(n) <= A071364(n) <= n.

			n=50=2*5*5: A071364(50)=2*3*3=18, A046523(50)=2*2*3=12;
n=500=2*2*5*5*5: A071364(500)=2*2*3*3*3=108, A046523(500)=2*2*2*3*3=72.

Cf. A071365.

Select[Range[1000], (e = Last /@ FactorInteger[ # ]) != Sort[e, Greater] && Product[Prime[i]^e[[i]], {i, Length[e]}] != # &] (* Ray Chandler, Sep 23 2005 *)

Extended by Ray Chandler, Sep 23 2005

cp's OEIS Frontend

A096011 Arrange numbers n such that A071364(n) <> A046523(n) by prime signature, cf. A071365.

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A304678 Numbers with weakly increasing prime multiplicities.

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A071364 Smallest number with same sequence of exponents in canonical prime factorization as n.

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A242031 Numbers n such that prime factorization n = p_1^k_1*p_2^k_2*...*p_r^k_r satisfies k_1 >= k_2 >= ... >= k_r.

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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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A095990 Numbers with ordered prime signature (1,2).

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A242031 Numbers n such that prime factorization n = p_1^k_1p_2^k_2...*p_r^k_r satisfies k_1 >= k_2 >= ... >= k_r.