A070175 -id:A070175 - 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 *)

A050322 Number of factorizations indexed by prime signatures: A001055(A025487).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 5, 7, 9, 12, 11, 11, 16, 19, 21, 15, 29, 26, 30, 15, 31, 38, 22, 47, 52, 45, 36, 57, 64, 30, 77, 98, 67, 74, 97, 66, 105, 42, 109, 118, 92, 109, 171, 97, 141, 162, 137, 165, 56, 212, 181, 52, 198, 189, 289, 139, 250, 257, 269, 254, 77, 382, 267

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

For A025487(m) = 2^k = A000079(k), we have a(m) = A000041(k).

Is a(k) = A000110(k) for A025487(m) = A002110(k)?

			From _Gus Wiseman_, Jan 13 2020: (Start)
The a(1) = 1 through a(11) = 9 factorizations:
  {}  2  4    6    8      12     16       24       30     32         36
         2*2  2*3  2*4    2*6    2*8      3*8      5*6    4*8        4*9
                   2*2*2  3*4    4*4      4*6      2*15   2*16       6*6
                          2*2*3  2*2*4    2*12     3*10   2*2*8      2*18
                                 2*2*2*2  2*2*6    2*3*5  2*4*4      3*12
                                          2*3*4           2*2*2*4    2*2*9
                                          2*2*2*3         2*2*2*2*2  2*3*6
                                                                     3*3*4
                                                                     2*2*3*3
(End)

R. J. Mathar and Michael De Vlieger, Table of n, a(n) for n = 1..5000 (First 300 terms from _R. J. Mathar_)
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

Cf. A000041, A000079, A000110, A001055, A002110, A025487.
The version indexed by unsorted prime signature is A331049.
The version indexed by prime shadow (A181819, A181821) is A318284.
This sequence has range A045782 (same as A001055).
Cf. A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

A050322 := proc(n)
    A001055(A025487(n)) ;
end proc: # R. J. Mathar, May 25 2017

c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{d, i}, d = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/d[[i]], d[[i]]], {i, 1, Length[d]}]]; Map[c[#, #] &, Union@ Table[Times @@ MapIndexed[If[n == 1, 1, Prime[First@ #2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, Product[Prime@ i, {i, 6}]}]] (* Michael De Vlieger, Jul 10 2017, after Dean Hickerson at A001055 *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Length/@facs/@First/@GatherBy[Range[1000],If[#==1,{},Sort[Last/@FactorInteger[#]]]&] (* Gus Wiseman, Jan 13 2020 *)

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 *)

A328957 Numbers k such that sigma_0(k) != omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 41, 42, 43, 47, 49, 53, 59, 61, 64, 66, 67, 70, 71, 72, 73, 78, 79, 81, 83, 89, 97, 100, 101, 102, 103, 105, 107, 108, 109, 110, 113, 114, 120, 121, 125, 127, 128, 130, 131, 137

Offset: 1

Gus Wiseman, Nov 01 2019

nonn

			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}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}
   31: {11}
   32: {1,1,1,1,1}

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

Nonzeros of A328958.
The complement is A328956.
Prime signature is A124010.
Omega-sequence is A323023.
omega(n) * Omega(n) is A113901(n).
(Omega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - Omega(n) * omega(n) is A328958(n).
sigma_0(n) - 2 - (Omega(n) - 1) * omega(n) is A328959(n).
Cf. A060687, A070175, A090858, A112798, A303555, A320632, A328960, A328961, A328962, A328963, A328964, A328965.

Select[Range[100],DivisorSigma[0,#]!=PrimeOmega[#]*PrimeNu[#]&]

is(k) = {my(f = factor(k)); numdiv(f) != omega(f) * bigomega(f);} \\ Amiram Eldar, Jul 28 2024

A000005(a(n)) != A001222(a(n)) * A001221(a(n)).

A355382 Number of divisors d of n such that bigomega(d) = omega(n).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 02 2022

nonn

The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.
If positive integers are regarded as arrows from the number of prime factors to the number of distinct prime factors, this sequence counts divisible composable pairs. Is there a nice choice of a composition operation making this into an associative category?

			The set of divisors of 180 satisfying the condition is {12, 18, 20, 30, 45}, so a(180) = 5.

The version with multiplicity is A181591.
For partitions we have A355383, with multiplicity A339006.
The version for compositions is A355384.
Positions of first appearances are A355386.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 count prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
Cf. A000712, A022811, A056239, A071625, A118914, A133494, A181819, A182850, A323014, A323022, A323023, A355385, A355388.

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,100}]

A098348 Triangular array read by rows: a(n, k) = number of ordered factorizations of a "hook-type" number with n total prime factors and k distinct prime factors. "Hook-type" means that only one prime can have multiplicity > 1.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 20, 44, 75, 16, 48, 132, 308, 541, 32, 112, 368, 1076, 2612, 4683, 64, 256, 976, 3408, 10404, 25988, 47293, 128, 576, 2496, 10096, 36848, 116180, 296564, 545835, 256, 1280, 6208, 28480, 120400, 454608, 1469892, 3816548

Offset: 1

Alford Arnold, Sep 04 2004

The first three columns are A000079, A001792 and A098385.
The first two diagonals are A000670 and A005649.
A070175 gives the smallest representative of each hook-type prime signature, so this sequence is a rearrangement of A074206(A070175).

			a(4, 2) = 20 because 24=2*2*2*3 has 20 ordered factorizations and so does any other number with the same prime signature.

Cf. A050324, A070175, A070826, A074206, A095705. A098349 gives the row sums. A098384.

a(n, k) = 1 + (Sum_{i=1..k-1} binomial(k-1, i)*a(i, i)) + (Sum_{j=1..k} Sum_{i=j..j+n-k-1} binomial(k-1, j-1)*a(i, j)) + (Sum_{j=1..k-1} binomial(k-1,j-1)*a(j+n-k, j)). - David Wasserman, Feb 21 2008
a(n, k) = A074206(2^(n+1-k)*A070826(k)). - David Wasserman, Feb 21 2008
The following conjectural formula for the triangle entries agrees with the values listed above: T(n,k) = Sum_{j = 0..n-k} 2^(n-k-j)*binomial(n-k,j)*a(k,j), where a(k,j) = 2^j*Sum_{i = j+1..k+1} binomial(i,j+1)*(i-1)!*Stirling2(k+1,i). See A098384 for related conjectures. - Peter Bala, Apr 20 2012

Edited and extended by David Wasserman, Feb 21 2008

A355386 Position of first appearance of n in A355382, where A355382(m) = number of divisors d of m such that bigomega(d) = omega(m); or a(n) = -1 if n does not appear in A355382.

Original entry on oeis.org

1, 12, 36, 120, 180, 360, 840, 1260, 5400, 27000, 2520, 5040, 6300, 7560, 15120, 12600, 25200

Offset: 1

Gus Wiseman, Jul 02 2022

The first position of -1 appears to be 18, pointed out by Amiram Eldar.
The terms are not always increasing.
The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.

			The terms together with their prime indices begin:
      1: {}
     12: {1,1,2}
     36: {1,1,2,2}
    120: {1,1,1,2,3}
    180: {1,1,2,2,3}
    360: {1,1,1,2,2,3}
    840: {1,1,1,2,3,4}
   1260: {1,1,2,2,3,4}
   5400: {1,1,1,2,2,2,3,3}
  27000: {1,1,1,2,2,2,3,3,3}
   2520: {1,1,1,2,2,3,4}
   5040: {1,1,1,1,2,2,3,4}
   6300: {1,1,2,2,3,3,4}
   7560: {1,1,1,2,2,2,3,4}
  15120: {1,1,1,1,2,2,2,3,4}
The terms together with their divisors satisfying the condition begin:
      1:   1
     12:   4,   6
     36:   4,   6,   9
    120:   8,  12,  20,  30
    180:  12,  18,  20,  30,  45
    360:   8,  12,  18,  20,  30,  45
    840:  24,  40,  56,  60,  84, 140, 210
   1260:  36,  60,  84,  90, 126, 140, 210, 315
   5400:   8,  12,  18,  20,  27,  30,  45,  50,  75
  27000:   8,  12,  18,  20,  27,  30,  45,  50,  75, 125
   2520:  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   5040:  16,  24,  36,  40,  56,  60,  84,  90, 126, 140, 210, 315
   6300:  36,  60,  84,  90, 100, 126, 140, 150, 210, 225, 315, 350, 525

These are the positions of first appearances in A355382, which is the version of A181591 without multiplicity.
A000005 counts divisors.
A001221 counts prime indices without multiplicity.
A001222 counts prime indices with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
A355383 counts cmpsbl. pairs of partitions with containment, comps. A355384.
Cf. A000712, A022811, A056239, A071625, A181819, A319910, A339006.

tf=Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,1000}];
Table[Position[tf,n][[1,1]],{n,Select[Union[tf],SubsetQ[tf,Range[#]]&]}]

A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

Original entry on oeis.org

1, 2, 12, 120, 1680, 36960, 960960, 32672640, 1241560320, 57111774720, 3312482933760, 205373941893120, 15197671700090880, 1246209079407452160, 107173980829040885760, 10074354197929843261440, 1067881544980563385712640, 126010022307706479514091520, 15373222721540190500719165440

Offset: 0

Ilya Gutkovskiy, Apr 26 2018

nonn

For n > 0, a(n) is the smallest number m having exactly n distinct prime divisors and exactly 2*n - 1 prime divisors counted with multiplicity.

			a(1) = 2^1;
a(2) = 2^2*3;
a(3) = 2^3*3*5;
a(4) = 2^4*3*5*7;
a(5) = 2^5*3*5*7*11, etc.

Central terms of triangle A303555 (for n > 0).

Cf. A000079, A002110, A003680, A005179, A011782, A038547, A061283, A070175, A088860, A102476.

Join[{1}, Table[2^(n - 1) Product[Prime[j], {j, n}], {n, 18}]]

a(n) = A011782(n)*A002110(n).

A323350 Nonprime numbers > 1 whose number of prime factors counted with multiplicity is a perfect square.

Original entry on oeis.org

16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 330, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376

Offset: 1

Gus Wiseman, Jan 15 2019

nonn

First differs from A014613 in having 512.

			360 = 2*2*2*3*3*5 has 6 prime factors, and 6 is not a perfect square, so 360 does not belong to the sequence.
2160 = 2*2*2*2*3*3*3*5 has 8 prime factors, and 8 is not a perfect square, so 2160 does not belong to the sequence.
10800 = 2*2*2*2*3*3*3*5*5 has 9 prime factors, and 9 is a perfect square, so 10800 belongs to the sequence.

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

Cf. A000005, A001222, A063989, A067028, A067340, A070175, A071625, A118914, A167175, A323305.

filter:= proc(n) local t;
  t:= numtheory:-bigomega(n);
  t > 1 and issqr(t)
end proc:
select(filter, [$4..1000]); # Robert Israel, Jan 15 2019

Select[Range[100],#>1&&!PrimeQ[#]&&IntegerQ[Sqrt[PrimeOmega[#]]]&]

isok(n) = (n>1) && !isprime(n) && issquare(bigomega(n)); \\ Michel Marcus, Jan 15 2019

A331049 Number of factorizations of A055932(n), the least representative of the n'th distinct unsorted prime signature, into factors > 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 4, 7, 5, 7, 9, 12, 7, 11, 11, 16, 11, 19, 16, 21, 15, 29, 11, 12, 26, 30, 15, 31, 38, 22, 21, 47, 26, 29, 52, 45, 36, 57, 26, 64, 19, 30, 52, 77, 52, 36, 57, 98, 21, 67, 38, 74, 97, 66, 105, 47, 42, 36, 109, 118, 98, 92, 109, 52, 171, 30

Offset: 1

Gus Wiseman, Jan 10 2020

nonn

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. Factorizations are counted by A001055.

The unsorted prime signature of A055932(n) is given by row n of A124829.

			The a(1) = 1 through a(11) = 7 factorizations:
  {}  2  4    6    8      12     16       18     24       30     32
         2*2  2*3  2*4    2*6    2*8      2*9    3*8      5*6    4*8
                   2*2*2  3*4    4*4      3*6    4*6      2*15   2*16
                          2*2*3  2*2*4    2*3*3  2*12     3*10   2*2*8
                                 2*2*2*2         2*2*6    2*3*5  2*4*4
                                                 2*3*4           2*2*2*4
                                                 2*2*2*3         2*2*2*2*2

The sorted-signature version is A050322.
This sequence has range A045782.
Factorizations are A001055.
Cf. A025487, A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Length@*facs/@First/@GatherBy[Range[1500],If[#==1,{},Last/@FactorInteger[#]]&]

a(n) = A001055(A055932(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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A050322 Number of factorizations indexed by prime signatures: A001055(A025487).

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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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A328957 Numbers k such that sigma_0(k) != omega(k) * Omega(k), where sigma_0 = A000005, omega = A001221, Omega = A001222.

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A355382 Number of divisors d of n such that bigomega(d) = omega(n).

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A098348 Triangular array read by rows: a(n, k) = number of ordered factorizations of a "hook-type" number with n total prime factors and k distinct prime factors. "Hook-type" means that only one prime can have multiplicity > 1.

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A355386 Position of first appearance of n in A355382, where A355382(m) = number of divisors d of m such that bigomega(d) = omega(m); or a(n) = -1 if n does not appear in A355382.

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Mathematica

A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

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Mathematica