A066120 -id:A066120 - cp's OEIS frontend

A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

1, 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310

Offset: 1

David W. Wilson

All numbers of the form 2^k1*3^k2*...*p_n^k_n, where k1 >= k2 >= ... >= k_n, sorted.
A111059 is a subsequence. - Reinhard Zumkeller, Jul 05 2010
Choie et al. (2007) call these "Hardy-Ramanujan integers". - Jean-François Alcover, Aug 14 2014
The exponents k1, k2, ... can be read off Abramowitz & Stegun p. 831, column labeled "pi".
For all such sequences b for which it holds that b(n) = b(A046523(n)), the sequence which gives the indices of records in b is a subsequence of this sequence. For example, A002182 which gives the indices of records for A000005, A002110 which gives them for A001221 and A000079 which gives them for A001222. - Antti Karttunen, Jan 18 2019
The prime signature corresponding to a(n) is given in row n of A124832. - M. F. Hasler, Jul 17 2019

			The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..10001 (first 291 terms from Will Nicholes)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Kevin Broughan, Equivalents of the Riemann Hypothesis, Vol. 1: Arithmetic Equivalents, Cambridge University Press, 2017. See section 8.2, "Hardy-Ramanujan Numbers".
YoungJu Choie, Nicolas Lichiardopol, Pieter Moree and Patrick Solé, On Robin's criterion for the Riemann hypothesis, Journal de théorie des nombres de Bordeaux, Vol. 19, No. 2 (2007), pp. 357-372. See section 5, p. 367.
Asaf Cohen Antonir and Asaf Shapira, An Elementary Proof of a Theorem of Hardy and Ramanujan (2022). arXiv:2207.09410 [math.NT]
Michael De Vlieger, Relations of A025487 to A002110, A002182, and A002201.
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, pp. 9-10.
G. H. Hardy and S. Ramanujan, Asymptotic formulae for the distribution of integers of various types, Proc. London Math. Soc, Ser. 2, Vol. 16 (1917), pp. 112-132. Also published in the book Collected Papers of Srinivasa Ramanujan, Chelsea, 1962, pages 245-261.
Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
L. B. Richmond, Asymptotic results for partitions (I) and the distribution of certain integers, Journal of Number Theory, Vol. 8, No. 4 (1976), pp. 372-389. See page 388.

Subsequence of A055932, A191743, and A324583.
Cf. A025488, A051282, A036041, A051466, A061394, A124832, A161360, A166469, A181815, A181817, A283980, A306802, A322584, A322585 (characteristic function), A329897, A329898, A329899, A329900, A329904, A330683.
Cf. A085089, A101296 (left inverses).
Equals range of values taken by A046523.
Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).
Subsequences of this sequence include: A000079, A000142, A000400, A001013, A001813, A002110, A002182, A005179, A006939, A025527, A056836, A061742, A064350, A066120, A087980, A097212, A097213, A111059, A119840, A119845, A126098, A129912, A140999, A166338, A166470, A166472, A166473, A166475, A167448, A168262, A168263, A168264, A179215, A181555, A181804, A181806, A181809, A181818, A181822, A181823, A181824, A181825, A181826, A181827, A182763, A182862, A182863, A212170, A220264, A220423, A250269, A250270, A260633, A266047, A284456, A300357, A304938, A329894, A330687; also A037019 and A330681 (when sorted), possibly also A289132.
Rearrangements of this sequence include A036035, A059901, A063008, A077569, A085988, A086141, A087443, A108951, A181821, A181822, A322827, A329886, A329887.
Cf. also array A124832 (row n = prime signature of a(n)) and A304886, A307056.

import Data.Set (singleton, fromList, deleteFindMin, union)
a025487 n = a025487_list !! (n-1)
a025487_list = 1 : h [b] (singleton b) bs where
   (_ : b : bs) = a002110_list
   h cs s xs'@(x:xs)
     | m <= x    = m : h (m:cs) (s' `union` fromList (map (* m) cs)) xs'
     | otherwise = x : h (x:cs) (s  `union` fromList (map (* x) (x:cs))) xs
     where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Apr 06 2013

isA025487 := proc(n)
    local pset,omega ;
    pset := sort(convert(numtheory[factorset](n),list)) ;
    omega := nops(pset) ;
    if op(-1,pset) <> ithprime(omega) then
        return false;
    end if;
    for i from 1 to omega-1 do
        if padic[ordp](n,ithprime(i)) < padic[ordp](n,ithprime(i+1)) then
            return false;
        end if;
    end do:
    true ;
end proc:
A025487 := proc(n)
    option remember ;
    local a;
    if n = 1 then
        1 ;
    else
        for a from procname(n-1)+1 do
            if isA025487(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A025487(n),n=1..100) ; # R. J. Mathar, May 25 2017

PrimeExponents[n_] := Last /@ FactorInteger[n]; lpe = {}; ln = {1}; Do[pe = Sort@PrimeExponents@n; If[ FreeQ[lpe, pe], AppendTo[lpe, pe]; AppendTo[ln, n]], {n, 2, 2350}]; ln (* Robert G. Wilson v, Aug 14 2004 *)
(* Second program: generate all terms m <= A002110(n): *)
f[n_] := {{1}}~Join~
  Block[{lim = Product[Prime@ i, {i, n}],
   ww = NestList[Append[#, 1] &, {1}, n - 1], dec},
   dec[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]];
   Map[Block[{w = #, k = 1},
      Sort@ Prepend[If[Length@ # == 0, #, #[[1]]],
        Product[Prime@ i, {i, Length@ w}] ] &@ Reap[
         Do[
          If[# < lim,
             Sow[#]; k = 1,
             If[k >= Length@ w, Break[], k++]] &@ dec@ Set[w,
             If[k == 1,
               MapAt[# + 1 &, w, k],
               PadLeft[#, Length@ w, First@ #] &@
                 Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1] ]],
           {i, Infinity}] ][[-1]]
] &, ww]]; Sort[Join @@ f@ 13] (* Michael De Vlieger, May 19 2018 *)

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))) \\ Charles R Greathouse IV, Jun 10 2011

factfollow(n)={local(fm, np, n2);
  fm=factor(n); np=matsize(fm)[1];
  if(np==0,return([2]));
  n2=n*nextprime(fm[np,1]+1);
  if(np==1||fm[np,2]Franklin T. Adams-Watters, Dec 01 2011 */

is(n) = {if(n==1, return(1)); my(f = factor(n));  f[#f~, 1] == prime(#f~) && vecsort(f[, 2],,4) == f[, 2]} \\ David A. Corneth, Feb 14 2019

upto(Nmax)=vecsort(concat(vector(logint(Nmax,2),n,select(t->t<=Nmax,if(n>1,[factorback(primes(#p),Vecrev(p)) || p<-partitions(n)],[1,2]))))) \\ M. F. Hasler, Jul 17 2019

\\ For fast generation of large number of terms, use this program:
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
A025487list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t); 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++); vecsort(Vec(lista)); }; \\ Returns a list of terms up to the term 2^e.
v025487 = A025487list(101);
A025487(n) = v025487[n];
for(n=1,#v025487,print1(A025487(n), ", ")); \\ Antti Karttunen, Dec 24 2019

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
N = 2310
nmax = 2^floor(log(N,2))
sorted([j for j in (prod(sharp_primorial(t[0])^t[1] for k, t in enumerate(factor(n))) for n in (1..nmax)) if j <= N])
# Giuseppe Coppoletta, Jan 26 2015

What can be said about the asymptotic behavior of this sequence? - Franklin T. Adams-Watters, Jan 06 2010
Hardy & Ramanujan prove that there are exp((2 Pi + o(1))/sqrt(3) * sqrt(log x/log log x)) members of this sequence up to x. - Charles R Greathouse IV, Dec 05 2012
From Antti Karttunen, Jan 18 & Dec 24 2019: (Start)
A085089(a(n)) = n.
A101296(a(n)) = n [which is the first occurrence of n in A101296, and thus also a record.]
A001221(a(n)) = A061395(a(n)) = A061394(n).
A007814(a(n)) = A051903(a(n)) = A051282(n).
a(A101296(n)) = A046523(n).
a(A306802(n)) = A002182(n).
a(n) = A108951(A181815(n)) = A329900(A181817(n)).
If A181815(n) is odd, a(n) = A283980(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).
(End)
Sum_{n>=1} 1/a(n) = Product_{n>=1} 1/(1 - 1/A002110(n)) = A161360. - Amiram Eldar, Oct 20 2020

Offset corrected by Matthew Vandermast, Oct 19 2008

Minor correction by Charles R Greathouse IV, Sep 03 2010

A181818 Products of superprimorials (A006939).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 24, 32, 48, 64, 96, 128, 144, 192, 256, 288, 360, 384, 512, 576, 720, 768, 1024, 1152, 1440, 1536, 1728, 2048, 2304, 2880, 3072, 3456, 4096, 4320, 4608, 5760, 6144, 6912, 8192, 8640, 9216, 11520, 12288, 13824, 16384, 17280, 18432, 20736, 23040, 24576, 27648, 32768

Offset: 1

Matthew Vandermast, Nov 30 2010

nonn

Sorted list of positive integers with a factorization Product p(i)^e(i) such that (e(1) - e(2)) >= (e(2) - e(3)) >= ... >= (e(k-1) - e(k)) >= e(k), with k = A001221(n), and p(k) = A006530(n) = A000040(k), i.e., the prime factors p(1) .. p(k) must be consecutive primes from 2 onward. - Comment clarified by Antti Karttunen, Apr 28 2022
Subsequence of A025487. A025487(n) belongs to this sequence iff A181815(n) is 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 A182863. - Matthew Vandermast, May 20 2012

			2, 12, and 360 are all superprimorials (i.e., members of A006939). Therefore, 2*2*12*360 = 17280 is included in the sequence.
From _Gus Wiseman_, Aug 12 2020 (Start):
The sequence of factorizations (which are unique) begins:
    1 = empty product
    2 = 2
    4 = 2*2
    8 = 2*2*2
   12 = 12
   16 = 2*2*2*2
   24 = 2*12
   32 = 2*2*2*2*2
   48 = 2*2*12
   64 = 2*2*2*2*2*2
   96 = 2*2*2*12
  128 = 2*2*2*2*2*2*2
  144 = 12*12
  192 = 2*2*2*2*12
  256 = 2*2*2*2*2*2*2*2
(End)

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

A181817 rearranged in numerical order. Also includes all members of A000079, A001021, A006939, A009968, A009992, A066120, A166475, A167448, A181813, A181814, A181816, A182763.
Subsequence of A025487, A055932, A087980, A130091, A181824.
A001013 is the version for factorials.
A336426 is the complement.
A336496 is the version for superfactorials.
A001055 counts factorizations.
A006939 lists superprimorials or Chernoff numbers.
A317829 counts factorizations of superprimorials.
Cf. A022915, A076954, A304686, A325368, A336419, A336420, A336421, A353518 (characteristic function).

Select[Range[100],PrimePi[First/@If[#==1,{}, FactorInteger[#]]]==Range[ PrimeNu[#]]&&LessEqual@@Differences[ Append[Last/@FactorInteger[#],0]]&] (* Gus Wiseman, Aug 12 2020 *)

firstdiffs0forward(vec) = { my(v=vector(#vec)); for(n=1,#v,v[n] = vec[n]-if(#v==n,0,vec[1+n])); (v); };
A353518(n) = if(1==n,1,my(f=factor(n), len=#f~); if(primepi(f[len,1])!=len, return(0), my(diffs=firstdiffs0forward(f[,2])); for(i=1,#diffs-1,if(diffs[i+1]>diffs[i],return(0))); (1)));
isA181818(n) = A353518(n); \\ Antti Karttunen, Apr 28 2022

A166469 Number of divisors of n which are not multiples of consecutive primes.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 4, 2, 4, 2, 4, 3, 5, 2, 4, 2, 6, 4, 4, 2, 5, 3, 4, 4, 6, 2, 5, 2, 6, 4, 4, 3, 5, 2, 4, 4, 8, 2, 6, 2, 6, 4, 4, 2, 6, 3, 6, 4, 6, 2, 5, 4, 8, 4, 4, 2, 7, 2, 4, 6, 7, 4, 6, 2, 6, 4, 6, 2, 6, 2, 4, 4, 6, 3, 6, 2, 10, 5, 4, 2, 8, 4, 4, 4, 8, 2, 6, 4, 6, 4, 4, 4, 7, 2, 6, 6, 9, 2, 6, 2, 8, 5

Offset: 1

Matthew Vandermast, Nov 05 2009

nonn

Links various subsequences of A025487 with an unusual number of important sequences, including the Fibonacci, Lucas, and other generalized Fibonacci sequences (see cross-references).

If a number is a product of any number of consecutive primes, the number of its divisors which are not multiples of n consecutive primes is always a Fibonacci n-step number. See also A073485, A167447.

			Since 3 of 30's 8 divisors (6, 15, and 30) are multiples of 2 or more consecutive primes, a(30) = 8 - 3 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Eric Weisstein's World of Mathematics, Fibonacci n-Step Number

Cf. A000040, A006094, A104210, A296210.
A(A002110(n)) = A000045(n+2); A(A097250(n)) = A000032(n+1). For more relationships involving Fibonacci and Lucas numbers, see A166470-A166473, comment on A081341.
A(A061742(n)) = A001045(n+2); A(A006939(n)) = A000085(n+1); A(A212170(n)) = A000142(n+1). A(A066120(n)) = A166474(n+1).

Array[DivisorSum[#, 1 &, FreeQ[Differences@ PrimePi@ FactorInteger[#][[All, 1]], 1] &] &, 105] (* Michael De Vlieger, Dec 16 2017 *)

A296210(n) = { if(1==n,return(0)); my(ps=factor(n)[,1], pis=vector(length(ps),i,primepi(ps[i])), diffsminusones = vector(length(pis)-1,i,(pis[i+1]-pis[i])-1)); !factorback(diffsminusones); };
A166469(n) = sumdiv(n,d,!A296210(d)); \\ Antti Karttunen, Dec 15 2017

a) If n has no prime gaps in its factorization (cf. A073491), then, if the canonical factorization of n into prime powers is the product of p_i^(e_i), a(n) is the sum of all products of one or more nonadjacent exponents, plus 1. For example, if A001221(n) = 3, a(n) = e_1*e_3 + e_1 + e_2 + e_3 + 1. If A001221(n) = k, the total number of terms always equals A000045(k+2).
The answer can also be computed in k steps, by finding the answers for the products of the first i powers, for i = 1 to i = k. Let the result of the i-th step be called r(i). r(1) = e_1 + 1; r(2) = e_1 + e_2 +1; for i > 2, r(i) = r(i-1) + e_i * r(i-2).
b) If n has prime gaps in its factorization, express it as a product of the minimum number of A073491's members possible. Then apply either of the above methods to each of those members, and multiply the results to get a(n). a(n) = A000005(n) iff n has no pair of consecutive primes as divisors.
a(n) = Sum_{d|n} (1-A296210(d)). - Antti Karttunen, Dec 15 2017

Edited by Matthew Vandermast, May 24 2012

A181555 a(n) = A002110(n)^n.

Original entry on oeis.org

1, 2, 36, 27000, 1944810000, 65774855015100000, 733384949590939374729000000, 9037114296609938214167920266348510000000, 78354300210436852307898467208663359164858967744100000000

Offset: 0

Matthew Vandermast, Oct 31 2010

For n>0, a(n)= first counting number whose prime signature consists of n repeated n times (cf. A002024). Subsequence of A025487.

			a(4) = 1944810000 = 210^4 = 2^4 * 3^4 * 5^4 * 7^4.

Amiram Eldar, Table of n, a(n) for n = 0..26

A061742(n) = A002110(n)^2. See also A006939, A066120, A166475, A167448.
A000005(a(n)) = A000169(n). The divisors of a(n) appear as the first A000169(n) terms of A178479, with A178479(A000169(n)) = a(n).
A071207(n, k) gives the number of divisors of n with (n-k) distinct prime factors, A181567(n, k) gives the number of divisors of n with k prime factors counted with multiplicity.
Cf. A001221, A001222, A079474, A146290, A146292.

a[0] = 1; a[n_] := Product[Prime[i], {i, 1, n}]^n; Array[a, 9, 0] (* Amiram Eldar, Aug 08 2019 *)

a(n) = A079474(2n,n). - Alois P. Heinz, Aug 22 2019

A212165 Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 96, 97, 98, 99, 100, 101, 103, 104

Offset: 1

Matthew Vandermast, May 22 2012

Union of A212164 and A212166.  Includes numerous subsequences that are subsequences of neither A212164 nor A212166.
Includes all factorials except A000142(3) = 6.
Observation: all terms in DATA section are also the first 65 numbers n whose difference between the arithmetic derivative of n and the sum of the divisors of n is nonnegative. - Omar E. Pol, Dec 19 2012

			10 = 2^1*5^1 has 2 distinct prime factors, hence, 2 positive exponents in its prime factorization (although 1s are often left implicit).  2 is larger than the maximal exponent in 10's prime factorization, which is 1. Therefore, 10 does not belong to the sequence. But 20 = 2^2*5^1 and 40 = 2^3*5^1 belong, since the largest exponents in their prime factorizations are 2 and 3 respectively.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Primefan, The First 2500 Integers Factored (first of 5 pages).

Complement of A212168.
See also A212167.
Subsequences (none of which are subsequences of A212164 or A212166) include A000079, A001021, A066120, A087980, A130091, A141586, A166475, A181818, A181823, A181824, A182763, A212169. Also includes all terms in A181813 and A181814.
Cf. A001221, A051903, A188654, A225230.

import Data.List (findIndices)
a212165 n = a212165_list !! (n-1)
a212165_list = map (+ 1) $ findIndices (<= 0) a225230_list
-- Reinhard Zumkeller, May 03 2013

okQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Max[f] >= Length[f]]; Select[Range[1000], okQ] (* T. D. Noe, May 24 2012 *)

is(k) = {my(e = factor(k)[, 2]); !(#e) || vecmax(e) >= #e;} \\ Amiram Eldar, Sep 08 2024

A225230(a(n)) <= 0. - Reinhard Zumkeller, May 03 2013

A166475 4th level primorials: product of first n superduperprimorials.

Original entry on oeis.org

1, 2, 48, 414720, 270888468480000, 30900096179361042923520000000000, 1848494880770448654906901042987600267878400000000000000000000

Offset: 0

Matthew Vandermast, Nov 05 2009

Next term has 110 digits.
a(n) = first counting number with n distinct positive tetrahedral exponents in its prime factorization (cf. A000292).
Note: a(n) is not the first counting number with n distinct square exponents in its prime factorization, as previously stated.  That sequence is A212170. - Matthew Vandermast, May 23 2012

			a(3) = 414720 = 2^10*3^4*5^1 has 3 positive tetrahedral exponents in its prime factorization (cf. A000292).  It is the smallest number with this property.

Dario Alpern, Factorization using the Elliptic Curve Method

Subsequence of A025487.

Cf. A002110, A006939, A066120 for first, second and third level primorials.

a(n) = Product_{k=1..n} prime(k)^((n-k+1)^2).

Offset corrected by Matthew Vandermast, Nov 07 2009
Edited by Matthew Vandermast, Nov 10 2009, May 23 2012
Name changed by Arkadiusz Wesolowski, Feb 21 2014

A066119 Multi-level primorials: triangle with a(n,k)=a(n-1,k-1)*a(n-1,k) but with a(n,1)=p(n) and a(n,n)=2.

Original entry on oeis.org

2, 3, 2, 5, 6, 2, 7, 30, 12, 2, 11, 210, 360, 24, 2, 13, 2310, 75600, 8640, 48, 2, 17, 30030, 174636000, 653184000, 414720, 96, 2, 19, 510510, 5244319080000, 114069441024000000, 270888468480000, 39813120, 192, 2, 23, 9699690

Offset: 1

Henry Bottomley, Dec 05 2001

			a(4,3)=a(3,2)*a(3,3)=6*2=12. Rows start 2; 3,2; 5,6,2; 7,30,12,2; ...

Columns include A000040, A002110, A006939 and A066120. Right hand side includes A007395 and A007283. Cf. A066121.

A166474 a(1)=1; a(2)=2; for n>2, a(n) = a(n-1) + A000217(n-1)*a(n-2).

Original entry on oeis.org

1, 2, 5, 17, 67, 322, 1729, 10745, 72989, 556514, 4570909, 41300833, 397831735, 4156207538, 45928539713, 544673444273, 6790954845241, 90125991819010, 1251379270355221, 18375317715967121, 281164964490563531, 4525863356878968482

Offset: 1

Matthew Vandermast, Nov 05 2009

nonn

Equals the eigensequence of an infinite lower triangular matrix with 1's in the main diagonal and the triangular series in the subdiagonal.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Cf. A066120, A166469, A351046.

a166474 n = a166474_list !! (n-1)
a166474_list = 1 : 2 : zipWith (+)
   (tail a166474_list) (zipWith (*) a166474_list $ drop 2 a000217_list)
-- Reinhard Zumkeller, Feb 27 2012

[n le 2 select n else Self(n-1) + Binomial(n,2)*Self(n-2): n in [1..41]]; // G. C. Greubel, Aug 02 2024

Rest[CoefficientList[Series[-2*E^(Sqrt[2]*ArcTanh[x/Sqrt[2]])/(x^2-2), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 19 2012 *)

@CachedFunction
def A166474(n):
    if n<3: return n
    else: return A166474(n-1) + binomial(n,2)*A166474(n-2)
[A166474(n) for n in range(1,41)] # G. C. Greubel, Aug 02 2024

a(n+1) = A166469(A066120(n)).
E.g.f.: -2*exp(sqrt(2)*arctanh(x/sqrt(2)))/(x^2-2) = ((sqrt(2) + x)^2/(2 - x^2))^(1/sqrt(2))*2/(2 - x^2). - Vaclav Kotesovec, Oct 19 2012
a(n) ~ n!*2^(1/sqrt(2)-n/2)*n^(1/sqrt(2))/(2*Gamma(1+1/sqrt(2))). - Vaclav Kotesovec, Oct 19 2012

More terms from Sean A. Irvine, Jun 16 2011

cp's OEIS Frontend

A203015 Primes p such that p + 1 or p - 1 is in A066120.

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A025487 Least integer of each prime signature A124832; also products of primorial numbers A002110.

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A181818 Products of superprimorials (A006939).

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A166469 Number of divisors of n which are not multiples of consecutive primes.

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A181555 a(n) = A002110(n)^n.

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A212165 Numbers k such that the maximum exponent in its prime factorization is not less than the number of positive exponents (A051903(k) >= A001221(k)).

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