A085629 -id:A085629 - cp's OEIS frontend

A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1152, 1296, 1728, 2048, 2592, 3456, 5184, 7776, 8192, 10368, 13824, 15552, 18432, 20736, 31104, 41472, 55296, 62208, 73728, 86400, 108000, 129600, 131072, 165888, 194400, 216000, 221184, 259200, 279936, 324000

Offset: 1

Gus Wiseman, May 17 2022

nonn

All terms are highly powerful (A005934), but that sequence looks only at first appearances that reach a record, and is missing 1152, 2048, 8192, etc.

			The prime exponents of 86400 are (7,3,2), and this is the first case of product 42, so 86400 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wikipedia, Highly powerful number.

These are the positions of first appearances in A005361, counted by A266477.
This is the sorted version of A085629.
The version for shadows instead of exponents is A353397, firsts in A353394.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices, counted by A339095.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime exponents, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
Cf. A070175, A097318, A116608, A162247, A182850, A304678, A325131, A325238, A353399, A353503, A353506, A353507.
Subsequence of A181800.

nn=1000;
d=Table[Times@@Last/@FactorInteger[n],{n,nn}];
Select[Range[nn],!MemberQ[Take[d,#-1],d[[#]]]&]
lps[fct_] := Module[{nf = Length[fct]}, Times @@ (Prime[Range[nf]]^Reverse[fct])]; lps[{1}] = 1; q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, (n == 1 || AllTrue[e, # > 1 &]) && n == Min[lps /@ f[Times @@ e]]]; Select[Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]], q] (* Amiram Eldar, Sep 29 2024, using the function f by T. D. Noe at A162247 *)

A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).

Original entry on oeis.org

1, 4, 8, 16, 32, 64, 128, 144, 216, 288, 432, 864, 1296, 1728, 2592, 3456, 5184, 7776, 10368, 15552, 20736, 31104, 41472, 62208, 86400, 108000, 129600, 194400, 216000, 259200, 324000, 432000, 518400, 648000, 972000, 1296000, 1944000, 2592000

Offset: 1

N. J. A. Sloane

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..609 (terms 1..300 from T. D. Noe)
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991.
G. E. Hardy and M. V. Subbarao, Highly powerful numbers, Congress. Numer., Vol. 37 (1983), pp. 277-307. (Annotated scanned copy)
C. B. Lacampagne and J. L. Selfridge, Large highly powerful numbers are cubeful, Proc. Amer. Math. Soc., Vol. 91, No. 2 (1984), pp. 173-181.
Wikipedia, Highly powerful number.
Index entries for sequences related to powerful numbers

Cf. A036965, A001694, A007532, A005361, A005188, A003321, A014576, A023052, A046074.

Cf. A085629.

a = {1}; b = {1}; f[n_] := Times @@ Last /@ FactorInteger[n]; Do[If[f@ n > Max[b], And[AppendTo[b, f@ n], AppendTo[a, n]]], {n, 1000000}]; a (* Michael De Vlieger, Aug 28 2015 *)
With[{s = Array[Times @@ FactorInteger[#][[All, -1]] &, 3*10^6]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Oct 15 2017 *)
DeleteDuplicates[Table[{n,Times@@FactorInteger[n][[All,2]]},{n,26*10^5}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, May 13 2022 *)

{prdex(n)=local(s,fac); s=1; fac=factor(n); for(k=1,matsize(fac)[1],s=s*fac[k,2]); return(s)} {hp(m)=local(rec); rec=0; for(n=1,m,if(prdex(n)>rec,rec=prdex(n); print1(n",")))}

For n = Product p_i^e_i, let b(n) = Product e_i; then n is highly powerful if b(n) sets a new record.

Hardy and Subbarao give an extensive table.

Corrected and extended by Jason Earls, Jul 10 2003

A181800 Powerful numbers that are the least integer of their prime signature.

Original entry on oeis.org

1, 4, 8, 16, 32, 36, 64, 72, 128, 144, 216, 256, 288, 432, 512, 576, 864, 900, 1024, 1152, 1296, 1728, 1800, 2048, 2304, 2592, 3456, 3600, 4096, 4608, 5184, 5400, 6912, 7200, 7776, 8192, 9216, 10368, 10800, 13824, 14400, 15552, 16384, 18432, 20736, 21600, 27000

Offset: 1

Matthew Vandermast, Dec 02 2010

nonn

Apart from 1, all terms in this sequence are multiples of 4. - Charles R Greathouse IV, Dec 13 2016

Apart from 1, the prime factor exponents of the terms in this sequence represent exactly every factorization of every integer k >= 2. E.g. the two factorizations of 4 are represented by 16 (2^4 -> {4}) and 36 (2^2 * 3^2 -> {2, 2}; 2 * 2 = 4). The smallest such term for each k is subsequence A085629. - Charles L. Hohn, Apr 04 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Charles R Greathouse IV)

Cf. A001694 (powerful numbers), A025487 (least integer of each prime signature).

Cf. A085629.

is(n)=if(n<4,return(n==1)); my(f=factor(n)); vecmin(f[,2])>1 && f[,1]~ == primes(#f~) && f[,2] == vecsort(f[,2],,4) \\ Charles R Greathouse IV, Sep 14 2015

Intersection of A001694 and A025487.

A353507 Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Gus Wiseman, May 19 2022

nonn

Warning: If the prime multiplicities of n are a multiset y, this sequence gives the product of multiplicities in y, not the product of y.
Differs from A351946 at A351946(1260) = 4, a(1260) = 2.
Differs from A327500 at A327500(450) = 3, a(450) = 2.
We set a(1) = 0 so that the positions of first appearances are the primorials A002110.
Also the product of the prime metasignature of n (row n of A238747).

			The prime signature of 13860 is (2,2,1,1,1), with multiplicities (2,3), so a(13860) = 6.

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

Positions of first appearances are A002110.
The prime indices themselves have product A003963, counted by A339095.
The prime signature itself has product A005361, counted by A266477.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A071625 counts distinct prime exponents (third omega).
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A238747 gives prime metasignature, sorted A353742.
A323022 gives fourth omega.
Cf. A085629, A097318, A182850, A304678, A353394, A353399, A353500, A353503.

f:= proc(n) local M,s;
  M:= ifactors(n)[2][..,2];
  mul(numboccur(s,M),s=convert(M,set));
end proc:
f(1):= 0:
map(f, [$1..100]); # Robert Israel, May 19 2023

Table[If[n==1,0,Times@@Length/@Split[Sort[Last/@FactorInteger[n]]]],{n,100}]
Join[{0},Table[Times@@(Length/@Split[FactorInteger[n][[;;,2]]]),{n,2,100}]] (* Harvey P. Dale, Oct 20 2024 *)

from math import prod
from itertools import groupby
from sympy import factorint
def A353507(n): return 0 if n == 1 else prod(len(list(g)) for k, g in groupby(factorint(n).values())) # Chai Wah Wu, May 20 2022

a(n) = A005361(A181819(n)) = A003963(A181819(A181819(n))).

A353742 Sorted prime metasignature of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1

Offset: 1

Gus Wiseman, May 20 2022

The prime metasignature counts the multiplicities of each value in the prime signature of n. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}.

			The prime indices, sorted prime signatures, and sorted prime metasignatures of selected n:
      n = 1: {}             -> {}         -> {}
      n = 2: {1}            -> {1}        -> {1}
      n = 6: {1,2}          -> {1,1}      -> {2}
     n = 12: {1,1,2}        -> {1,2}      -> {1,1}
     n = 30: {1,2,3}        -> {1,1,1}    -> {3}
     n = 60: {1,1,2,3}      -> {1,1,2}    -> {1,2}
    n = 210: {1,2,3,4}      -> {1,1,1,1}  -> {4}
    n = 360: {1,1,1,2,2,3}  -> {1,2,3}    -> {1,1,1}

Row-sums are A001221.
Row-lengths are A071625.
Positions of first appearances are A182863.
This is the sorted version of A238747.
Row-products are A353507.
A001222 counts prime factors with multiplicity.
A003963 gives product of prime indices.
A005361 gives product of prime signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with strict signature, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
Cf. A000040, A000720, A070175, A097318, A182850, A304678, A325238, A353503.

Join@@Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,100}]

A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

Original entry on oeis.org

1, 2, 12, 36, 40, 112, 352, 832, 960, 1296, 2176, 2880, 4864, 5376, 11776, 12544, 16128, 29696, 33792, 34560, 38400, 63488, 64000, 101376, 115200, 143360, 151552, 159744, 335872, 479232, 704512, 835584, 1540096, 1658880, 1802240

Offset: 1

Gus Wiseman, May 17 2022

nonn

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

			The terms together with their prime indices begin:
     1: {}
     2: {1}
    12: {1,1,2}
    36: {1,1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   352: {1,1,1,1,1,5}
   832: {1,1,1,1,1,1,6}
   960: {1,1,1,1,1,1,2,3}
  1296: {1,1,1,1,2,2,2,2}
  2176: {1,1,1,1,1,1,1,7}
  2880: {1,1,1,1,1,1,2,2,3}
  4864: {1,1,1,1,1,1,1,1,8}
  5376: {1,1,1,1,1,1,1,1,2,4}

For shadows instead of exponents we get A003586, counted by A008619.
The LHS (product of prime indices) is A003963, counted by A339095.
The RHS (product of prime exponents) is A005361, counted by A266477.
The version for shadows instead of indices is A353399, counted by A353398.
These partitions are counted by A353506.
A001222 counts prime factors with multiplicity, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353394 gives product of shadows of prime indices, firsts A353397.
Cf. A000720, A008480, A085629, A097318, A109297, A304678, A318871, A320325, A325131, A325755, A353500, A353507.

Select[Range[1000],Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]==Times@@Last/@FactorInteger[#]&]

from itertools import count, islice
from math import prod
from sympy import primepi, factorint
def A353503_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: n == 1 or prod((f:=factorint(n)).values()) == prod(primepi(p)**e for p,e in f.items()), count(max(startvalue,1)))
A353503_list = list(islice(A353503_gen(),20)) # Chai Wah Wu, May 20 2022

A003963(a(n)) = A005361(a(n)).

A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 210, 360, 420, 1260, 2310, 2520, 4620, 13860, 27720, 30030, 60060, 75600, 138600, 180180, 360360, 510510, 831600, 900900, 1021020, 1801800, 3063060, 6126120, 9699690, 10810800, 15315300, 19399380, 30630600, 37837800

Offset: 1

Matthew Vandermast, Jan 14 2011

nonn

Members m of A025487 such that A181819(m) is also a member of A025487.
If prime signatures are considered as partitions, these are the members of A025487 whose prime signature is conjugate to the prime signature of a member of A181818.
Also the least number with each sorted prime metasignature, where a number's metasignature is the sequence of multiplicities of exponents in its prime factorization. For example, 2520 has prime indices {1,1,1,2,2,3,4}, sorted prime signature {1,1,2,3}, and sorted prime metasignature {1,1,2}. - Gus Wiseman, May 21 2022

			The prime signature of 360360 = 2^3*3^2*5*7*11*13 is (3,2,1,1,1,1). 2 appears as many times as 3 in 360360's prime signature, and 1 appears more times than 2. Since 360360 is also a member of A025487, it is a member of this sequence.
From _Gus Wiseman_, May 21 2022: (Start)
The terms together with their sorted prime signatures and sorted prime metasignatures begin:
      1: {}                -> {}            -> {}
      2: {1}               -> {1}           -> {1}
      6: {1,2}             -> {1,1}         -> {2}
     12: {1,1,2}           -> {1,2}         -> {1,1}
     30: {1,2,3}           -> {1,1,1}       -> {3}
     60: {1,1,2,3}         -> {1,1,2}       -> {1,2}
    210: {1,2,3,4}         -> {1,1,1,1}     -> {4}
    360: {1,1,1,2,2,3}     -> {1,2,3}       -> {1,1,1}
    420: {1,1,2,3,4}       -> {1,1,1,2}     -> {1,3}
   1260: {1,1,2,2,3,4}     -> {1,1,2,2}     -> {2,2}
   2310: {1,2,3,4,5}       -> {1,1,1,1,1}   -> {5}
   2520: {1,1,1,2,2,3,4}   -> {1,1,2,3}     -> {1,1,2}
   4620: {1,1,2,3,4,5}     -> {1,1,1,1,2}   -> {1,4}
  13860: {1,1,2,2,3,4,5}   -> {1,1,1,2,2}   -> {2,3}
  27720: {1,1,1,2,2,3,4,5} -> {1,1,1,2,3}   -> {1,1,3}
  30030: {1,2,3,4,5,6}     -> {1,1,1,1,1,1} -> {6}
  60060: {1,1,2,3,4,5,6}   -> {1,1,1,1,1,2} -> {1,5}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1444 terms from Amiram Eldar)
Eric Weisstein's World of Mathematics, Conjugate Partition.

Intersection of A025487 and A179983.
Subsequence of A129912 and A181826.
Includes all members of A182862.
Positions of first appearances in A353742, unordered version A238747.
A001222 counts prime factors with multiplicity, distinct A001221.
A003963 gives product of prime indices.
A005361 gives product of prime signature, firsts A353500 (sorted A085629).
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A130091 lists numbers with distinct prime exponents, counted by A098859.
A181819 gives prime shadow, with an inverse A181821.
A182850 gives frequency depth of prime indices, counted by A225485.
A323014 gives adjusted frequency depth of prime indices, counted by A325280.
Cf. A000040, A055932, A070175, A097318, A304678, A325238, A353507, A353745.

nn=1000;
r=Table[Sort[Length/@Split[Sort[Last/@If[n==1,{},FactorInteger[n]]]]],{n,nn}];
Select[Range[nn],!MemberQ[Take[r,#-1],r[[#]]]&] (* Gus Wiseman, May 21 2022 *)

A353506 Number of integer partitions of n whose parts have the same product as their multiplicities.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 1, 2, 1, 2, 0, 3, 3, 2, 3, 2, 0, 2, 3, 2, 1, 3, 1, 6, 3, 2, 3, 3, 2, 3, 4, 1, 2, 3, 6, 3, 2, 2, 3, 3, 1, 2, 6, 6, 4, 7, 2, 3, 6, 4, 3, 3, 0, 4, 5, 3, 5, 5, 6, 5, 3, 3, 3, 6, 5, 5, 6, 6, 3, 3, 3, 4, 4, 4, 6, 7, 2, 5, 7, 6, 2, 3, 4, 6, 11, 9, 4, 4, 1, 5, 6, 4, 7, 9, 6, 4

Offset: 0

Gus Wiseman, May 17 2022

nonn

			The a(0) = 1 through a(18) = 2 partitions:
  n= 0: ()
  n= 1: (1)
  n= 2:
  n= 3:
  n= 4: (211)
  n= 5:
  n= 6: (3111) (2211)
  n= 7:
  n= 8: (41111)
  n= 9:
  n=10: (511111)
  n=11: (32111111)
  n=12: (6111111) (22221111)
  n=13: (322111111)
  n=14: (71111111) (4211111111)
  n=15:
  n=16: (811111111) (4411111111) (42211111111)
  n=17: (521111111111) (332111111111) (322211111111)
  n=18: (9111111111) (333111111111)
For example, the partition y = (322111111) has multiplicities (1,2,6) with product 12, and the product of parts is also 3*2*2*1*1*1*1*1*1 = 12, so y is counted under a(13).

LHS (product of parts) is ranked by A003963, counted by A339095.
RHS (product of multiplicities) is ranked by A005361, counted by A266477.
For shadows instead of prime exponents we have A008619, ranked by A003586.
Taking sum instead of product of parts gives A266499.
For shadows instead of prime indices we have A353398, ranked by A353399.
These partitions are ranked by A353503.
Taking sum instead of product of multiplicities gives A353698.
A008284 counts partitions by length.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A353507 gives product of multiplicities (of exponents) in prime signature.
Cf. A085629, A114640, A116608, A118914, A124010, A319000, A325702, A353394, A353500.
Cf. A000792, A266480.

Table[Length[Select[IntegerPartitions[n], Times@@#==Times@@Length/@Split[#]&]],{n,0,30}]

a(n) = {my(nb=0); forpart(p=n, my(s=Set(p), v=Vec(p)); if (vecprod(vector(#s, i, #select(x->(x==s[i]), v))) == vecprod(v), nb++);); nb;} \\ Michel Marcus, May 20 2022

a(71)-a(100) from Alois P. Heinz, May 20 2022

A368473 Product of exponents of prime factorization of the exponentially 2^n-numbers (A138302).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2

Offset: 1

Amiram Eldar, Dec 26 2023

The terms of A005361 that are powers of 2 (A000079).

The first position of 2^k, for k = 0, 1, ..., is 1, 4, 15, 126, 1134, ..., which is the position of A085629(2^k) in A138302.

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

Cf. A000079, A005361, A085629, A138302, A271727.
Cf. A366538, A367169.
Similar sequences: A322327, A368472, A368474.

f[n_] := Module[{p = Times @@ FactorInteger[n][[;; , 2]]}, If[p == 2^IntegerExponent[p, 2], p, Nothing]]; Array[f, 150]

lista(kmax) = {my(p); for(k = 1, kmax, p = vecprod(factor(k)[, 2]); if(p >> valuation(p,2) == 1, print1(p, ", ")));}

a(n) = A005361(A138302(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1/d) * Product_{p prime} (1 + Sum_{k>=0} 2^k/p^(2^k)) = 1.504710204899266020302..., where d = A271727 is the asymptotic density of the exponentially 2^n-numbers.

A340232 a(n) is the least number with exactly 2*n bi-unitary divisors.

Original entry on oeis.org

2, 6, 32, 24, 512, 96, 8192, 120, 131072, 1536, 2097152, 480, 33554432, 24576, 536870912, 840, 8589934592, 7776, 137438953472, 7680, 2199023255552, 6291456, 35184372088832, 3360, 562949953421312, 100663296, 9007199254740992, 122880, 144115188075855872, 124416

Offset: 1

Amiram Eldar, Jan 01 2021

nonn

Every integer except 1 has an even number of bi-unitary divisors.

			a(1) = 2 since 2 is the least number with 2*1 = 2 bi-unitary divisors, 1 and 2.
a(2) = 6 since 6 is the least number with 2*2 = 4 bi-unitary divisors, 1, 2, 3 and 6.

Amiram Eldar, Table of n, a(n) for n = 1..62

Subsequence of A025487.
Cf. A222266, A286324, A293185.
Similar sequences: A005179 (all divisors), A038547 (odd divisors), A085629 (coreful divisors), A309181 (nonunitary), A340233 (exponential).

f[p_, e_] := If[OddQ[e], e + 1, e]; d[1] = 1; d[n_] := Times @@ (f @@@ FactorInteger[n]);  max = 10; s = Table[0, {max}]; c = 0; n = 2;  While[c < max, i = d[n]/2; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s

A286324(a(n)) = 2*n and A286324(k) != 2*n for all k < a(n).

cp's OEIS Frontend

A353500 Numbers that are the smallest number with product of prime exponents k for some k. Sorted positions of first appearances in A005361, unsorted version A085629.

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A005934 Highly powerful numbers: numbers with record value of the product of the exponents in prime factorization (A005361).

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A181800 Powerful numbers that are the least integer of their prime signature.

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A353507 Product of multiplicities of the prime exponents (signature) of n; a(1) = 0.

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A353742 Sorted prime metasignature of n.

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A353503 Numbers whose product of prime indices equals their product of prime exponents (prime signature).

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A182863 Members m of A025487 such that, if k appears in m's prime signature, k-1 appears at least as often as k (for any integer k > 1).

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A353506 Number of integer partitions of n whose parts have the same product as their multiplicities.

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