A025487 -id:A025487 - cp's OEIS frontend

A181824 Members of A025487 such that A025487(n) <= A181822(n).

1, 2, 4, 8, 12, 16, 24, 32, 36, 48, 64, 72, 96, 120, 128, 144, 192, 216, 240, 256, 288, 360, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1080, 1152, 1296, 1440, 1536, 1680, 1728, 1920, 2048, 2160, 2304, 2592, 2880, 3072, 3360, 3456, 3600, 3840, 4096

Offset: 1

Matthew Vandermast, Dec 08 2010

nonn

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

Union of A181823 and A181825.

Cf. A025487, A181822, A181826, A181827.

A181825 Members of A025487 whose prime signature is self-conjugate (as a partition).

Original entry on oeis.org

1, 2, 12, 36, 120, 360, 1680, 5040, 5400, 27000, 36960, 75600, 110880, 378000, 960960, 1587600, 1663200, 2882880, 7938000, 8316000, 32672640, 34927200, 43243200, 98017920, 174636000, 216216000, 277830000, 908107200, 1152597600, 1241560320, 1470268800, 1944810000

Offset: 1

Matthew Vandermast, Dec 08 2010

A025487(n) is included iff A025487(n) = A181822(n).

Closed under the binary operations of GCD and LCM, since a self-conjugate partition of Omega(a(n)) (which the prime signature of these numbers is) is the concatenation of self-conjugate hooks of decreasing size while moving downward and to the right in the Ferrers diagram, and the GCD (or LCM) of two terms a(i) and a(j) is obtained by taking the smaller (or larger, respectively) of the corresponding hooks. For example, GCD(a(8),a(11)) = GCD(5040,36960) = 1680 = a(7), and LCM(a(8),a(11)) = 110880 = a(13). The two binary operations make the set {a(n)} into a lattice order. - Richard Peterson, May 29 2020

			A025487(11) = 36 = 2^2*3^2 has a prime signature of (2,2), which is a self-conjugate partition; hence, 36 is included in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..11879 (first 578 terms from Amiram Eldar, terms <= 10^70)
David A. Corneth, PARI-program
Eric Weisstein's World of Mathematics, Self-Conjugate Partition

Cf. A025487, A181822, A303557.
Includes subsequences A006939 and A181555.
See also A181823, A181824, A181826, A181827.

\\ See Corneth link \\ David A. Corneth, Jun 03 2020

a(18)-a(32) from Amiram Eldar, Jan 19 2019

A181826 Members of A025487 such that A025487(n) >= A181822(n).

Original entry on oeis.org

1, 2, 6, 12, 30, 36, 60, 120, 180, 210, 360, 420, 840, 900, 1260, 1680, 1800, 2310, 2520, 4620, 5040, 5400, 6300, 7560, 9240, 12600, 13860, 18480, 25200, 27000, 27720, 30030, 36960, 37800, 44100, 55440, 60060, 69300, 75600, 83160, 88200, 110880, 120120

Offset: 1

Matthew Vandermast, Dec 08 2010

nonn

Includes all members of A003418, A051451 and A129912.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition

Cf. A003418, A025487, A051451, A129912, A181822, A181823, A181824, A181825, A181827.

Union of A181825 and A181827.

A036041 Number of prime divisors, counted with multiplicity, of prime signature A025487(n); equals size of associated partition.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 3, 5, 4, 5, 4, 6, 5, 6, 5, 7, 6, 5, 7, 4, 6, 6, 8, 7, 6, 8, 5, 7, 7, 9, 8, 7, 9, 6, 8, 6, 8, 10, 7, 9, 6, 8, 8, 10, 7, 9, 7, 9, 11, 8, 10, 5, 7, 9, 9, 11, 8, 10, 8, 10, 12, 9, 11, 6, 8, 10, 8, 10, 12, 7, 9, 9, 11, 9, 8, 11, 10, 13, 10, 12, 7, 9, 11, 9, 11, 13, 8, 10, 10, 12

Offset: 1

Alford Arnold

			a(3) = 2 since A025487(3) = 4 = 2*2; a(5) = 3 since A025487(5) = 8 = 2*2*2; ...

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Cf. A025487, A000041, A035796, A036042, A001222.

a(n) = A001222(A025487(n)) = A001222(A181822(n)).

More terms from Henry Bottomley, Apr 30 2001
Edited to accommodate change in A025487's offset by Matthew Vandermast, Nov 08 2008
Definition corrected by Álvar Ibeas, Nov 01 2014

A051282 2-adic valuation of A025487: largest k such that 2^k divides A025487(n), where A025487 gives products of primorials.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 4, 3, 1, 5, 2, 4, 2, 6, 3, 5, 3, 7, 4, 2, 6, 1, 3, 4, 8, 5, 3, 7, 2, 4, 5, 9, 6, 4, 8, 3, 5, 2, 6, 10, 3, 7, 2, 4, 5, 9, 4, 6, 3, 7, 11, 4, 8, 1, 3, 5, 6, 10, 5, 7, 4, 8, 12, 5, 9, 2, 4, 6, 3, 7, 11, 2, 4, 6, 8, 5, 3, 9, 5, 13, 6, 10, 3, 5, 7, 4, 8, 12, 3, 5, 7, 9, 2, 6, 4, 10, 6, 14, 7, 11, 4, 6, 8, 5, 9, 13, 4, 6, 8, 3, 10, 3, 7, 1, 5, 11, 7, 4

Offset: 1

Alford Arnold

a(n) can be used for resorting A025487 and sequences indexed by A025487, e.g., A050322, A050323, A050324 and A050325.

a(n) is the number of primorial numbers (A002110) larger than 1 in the representation of A025487(n) as a product of primorial numbers. - Amiram Eldar, Jun 03 2023

			a(8) = 3 because A025487(8) = 24 and 2^3 divides 24.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001055, A002033, A007814, A025487, A045778, A050320, A051903.

Cf. A050322, A050323, A050324, A050325.

a051282 = a007814 . a025487  -- Reinhard Zumkeller, Apr 06 2013

max = 40000; A025487 = {1}; lpe = {}; Do[ pe = Sort[ FactorInteger[n][[All, 2]]]; If[FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[A025487, n]], {n, 2, max}]; a[n_] := FactorInteger[ A025487[[n]] ][[1, 2]]; a[1] = 0; Table[a[n], {n, 1, Length[A025487]}] (* Jean-François Alcover, Jun 14 2012, after Robert G. Wilson v *)

isA025487(n)=my(k=valuation(n, 2), t); n>>=k; forprime(p=3, default(primelimit), t=valuation(n, p); if(t>k, return(0), k=t); if(k, n/=p^k, return(n==1)))
[valuation(n,2) | n <- [1..1000], isA025487(n)]
\\ Or, for older versions:
apply(n->valuation(n,2), select(isA025487, [1..1000])) \\ Charles R Greathouse IV, Nov 07 2014

a(n) = A007814(A025487(n)) = A051903(A025487(n)). - Matthew Vandermast, Jul 03 2012

More terms from Naohiro Nomoto, Mar 11 2001

A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 1, 4, 1, 4, 3, 1, 3, 3, 1, 1, 5, 1, 4, 4, 1, 5, 4, 1, 4, 5, 2, 1, 6, 1, 5, 6, 1, 6, 5, 1, 5, 7, 3, 1, 7, 1, 6, 8, 1, 5, 8, 4, 1, 7, 6, 1, 4, 6, 4, 1, 1, 6, 9, 1, 6, 9, 4, 1, 8, 1, 7, 10, 1, 6, 11, 6, 1, 8, 7, 1, 5, 9, 7, 2, 1, 7, 12, 1, 7, 11, 5, 1, 9, 1, 8, 12, 1, 7, 14

Offset: 1

Matthew Vandermast, Nov 11 2008

The formula used in obtaining the A025487(n)th row (see below) also gives the number of divisors of the k-th power of A025487(n).
Every row that appears in A146289 appears exactly once in the table. Rows appear in order of first appearance in A146289.
T(n,0)=1.

			Rows begin:
  1;
  1,1;
  1,2;
  1,2,1;
  1,3;
  1,3,2;
  1,4;
  1,4,3;...
36's 9 divisors include 1 divisor with 0 distinct prime factors (1); 4 with 1 (2, 3, 4 and 9); and 4 with 2 (6, 12, 18 and 36). Since 36 = A025487(11), the 11th row of the table therefore reads (1, 4, 4). These are the positive coefficients of the polynomial equation 1 + 4k + 4k^2 = (1 + 2k)(1 + 2k), derived from the prime factorization of 36 (namely, 2^2*3^2).

Anonymous?, Polynomial calculator
Eric Weisstein's World of Mathematics, Distinct Prime Factors
G. Xiao, WIMS server, Factoris (both expands and factors polynomials)

For the number of distinct prime factors of n, see A001221.
Row sums equal A146288(n). T(n, 1)=A036041(n) for n>1. T(n, A061394(n))=A052306(n).
Row A098719(n) of this table is identical to row n of A007318.
Cf. A146289. Also cf. A146291, A146292.

If A025487(n)'s canonical factorization into prime powers is Product p^e(p), then T(n, m) is the coefficient of k^m in the polynomial expansion of Product_p (1 + ek).

A329898 a(n) is the position of 2*A025487(n) in A025487.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 74, 75, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Antti Karttunen, Dec 24 2019

nonn

Numbers k for which A007814(A025487(k)) > A007949(A025487(k)), i.e., numbers k for which the 2-adic valuation of A025487(k) is larger than its 3-adic valuation.

Numbers k for which A181815(k) is even.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A329897 (complement), A330683 (and its permutation).
Cf. A007814, A007949, A025487, A329904 (a left inverse), A329906.
Positions of even terms in A181815, zeros in A330682.

(* First, load the function f at A025487, then: *)
With[{s = Union@ Flatten@ f@ 6}, Map[If[2 # > Max@ s, Nothing, FirstPosition[s, 2 #][[1]] ] &, s]] (* Michael De Vlieger, Jan 11 2020 *)

upto_e = 64; \\ 64 -> 43608 terms.
A283980(n) = {my(f=factor(n)); prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(p==2, 6, nextprime(p+1))^e)}; \\ From A283980
A329898list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, v025487); while(lista[i] != u, if(2*lista[i] <= u, listput(lista,2*lista[i]); t =
A283980(lista[i]); if(t <= u, listput(lista,t))); i++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,oo,if(!(t=vecsearch(v025487,2*(v025487[i]))),return(Vec(lista)), listput(lista,t))); };
v329898 = A329898list(upto_e);
A329898(n) = v329898[n];

For all n >= 1, A329904(a(n)) = n.

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 *)

A050324 Number of ordered factorizations indexed by prime signatures: A074206(A025487).

Original entry on oeis.org

1, 1, 2, 3, 4, 8, 8, 20, 13, 16, 26, 48, 44, 32, 76, 112, 132, 64, 208, 176, 256, 75, 252, 368, 128, 544, 604, 576, 308, 768, 976, 256, 1376, 1888, 1280, 1076, 2208, 818, 2496, 512, 2316, 3392, 1460, 2568, 5536, 2816, 3408, 6080, 3172, 6208, 1024, 7968

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

This sequence can help to find terms for A163272, as has been done by Giovanni Resta. A074206(n) is computed only from the prime signature of n. If A074206(k) has the same prime signature as k then A074206(k) is in A163272. - David A. Corneth, Jul 16 2018

The number of ordered prime factorizations of n is A074206(n), not really A002033(n) = A074206(n-1). This has induced confusion in A002033 so it might be worth mentioning the distinction to be made. - M. F. Hasler, Oct 12 2018

David A. Corneth, Table of n, a(n) for n = 1..12651 (first 300 terms by R. J. Mathar)

Cf. A025487, A034776, A074206, A163272.

A050324 := proc(n)
    A002033(A025487(n)-1) ;
end proc: # R. J. Mathar, May 25 2017
A050324 := A074206 @ A025487; # M. F. Hasler, Oct 12 2018

A050324(n)=A074206(A025487(n)) \\ M. F. Hasler, Oct 12 2018

Edited to accommodate change in A025487's offset by Matthew Vandermast, Nov 27 2009

A181823 Members of A025487 such that A025487(n) < A181822(n).

Original entry on oeis.org

4, 8, 16, 24, 32, 48, 64, 72, 96, 128, 144, 192, 216, 240, 256, 288, 384, 432, 480, 512, 576, 720, 768, 864, 960, 1024, 1080, 1152, 1296, 1440, 1536, 1728, 1920, 2048, 2160, 2304, 2592, 2880, 3072, 3360, 3456, 3600, 3840, 4096, 4320, 4608, 5184, 5760, 6144, 6480

Offset: 1

Matthew Vandermast, Dec 08 2010

nonn

			A025487(5) = 8 and A181822(5) = 30 have the prime signatures (3) and (1,1,1) respectively. 8 is the smaller member of the pair and is therefore included in this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Conjugate Partition.

Cf. A025487, A181822, A181824, A181825, A181826, A181827.

cp's OEIS Frontend

A181824 Members of A025487 such that A025487(n) <= A181822(n).

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A181825 Members of A025487 whose prime signature is self-conjugate (as a partition).

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A181826 Members of A025487 such that A025487(n) >= A181822(n).

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A036041 Number of prime divisors, counted with multiplicity, of prime signature A025487(n); equals size of associated partition.

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A051282 2-adic valuation of A025487: largest k such that 2^k divides A025487(n), where A025487 gives products of primorials.

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Haskell

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PARI

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A146290 Triangle T(n,m) read by rows (n >= 1, 0 <= m <= A061394(n)), giving the number of divisors of A025487(n) with m distinct prime factors.

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A329898 a(n) is the position of 2*A025487(n) in A025487.

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Mathematica

PARI

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

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