A322827 -id:A322827 - cp's OEIS frontend

A324383 a(n) is the minimal number of primorials that add to A322827(n).

1, 1, 1, 2, 2, 1, 2, 2, 2, 6, 1, 6, 4, 2, 4, 4, 8, 6, 6, 10, 8, 1, 10, 22, 4, 6, 2, 12, 8, 4, 4, 2, 8, 16, 6, 4, 24, 6, 8, 14, 26, 18, 1, 26, 20, 6, 18, 30, 6, 12, 2, 14, 16, 2, 10, 16, 8, 6, 4, 8, 6, 2, 4, 4, 12, 14, 14, 18, 18, 12, 16, 32, 42, 28, 6, 22, 32, 24, 24, 42, 46, 32, 18, 20, 30, 1, 24, 54, 38, 26, 14, 44, 34, 8

Offset: 0

Antti Karttunen, Feb 27 2019

nonn

a(n) is odd if and only if n is one of the terms of A000975: 1, 2, 5, 10, 21, 42, 85, ..., in which case A322827(n) will be one of primorials (A002110), and a(n) = 1. This happens because A276150 is even for all multiples of four, and a product of two or more primorials > 1 is always a multiple of 4. Note that the same property does not hold in factorial system: 36 = 3!*3!, but A034968(36) = 3 as 36 = 4!+3!+3!.

Cf. A000975 (positions of ones), A002110, A003188, A025487, A276150, A322827, A324342, A324382.

Cf. also A324386, A324387 (permutations of this sequence).

A276150(n) = { my(s=0,m); forprime(p=2, , if(!n, return(s)); m = n%p; s += m; n = (n-m)/p); };
A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));
A324383(n) = A276150(A322827(n));

a(n) = A276150(A322827(n)).

a(n) = A324386(A003188(n)).

A323503 a(n) = A322827(2n) / A322827(n).

Original entry on oeis.org

1, 3, 6, 3, 6, 7, 6, 3, 6, 7, 210, 7, 6, 7, 6, 3, 6, 7, 210, 7, 210, 13, 210, 7, 6, 7, 210, 7, 6, 7, 6, 3, 6, 7, 210, 7, 210, 13, 210, 7, 210, 13, 30030, 13, 210, 13, 210, 7, 6, 7, 210, 7, 210, 13, 210, 7, 6, 7, 210, 7, 6, 7, 6, 3, 6, 7, 210, 7, 210, 13, 210, 7, 210, 13, 30030, 13, 210, 13, 210, 7, 210, 13, 30030, 13, 30030, 19, 30030, 13

Offset: 0

Antti Karttunen, Jan 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A322827, A323504.

Bisection of A323507.

A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));
A323503(n) = A322827(2*n)/A322827(n);

a(n) = A322827(2*n) / A322827(n).

A323504 a(n) = A322827(2n+1) / A322827(n).

Original entry on oeis.org

2, 2, 5, 2, 5, 30, 5, 2, 5, 30, 11, 30, 5, 30, 5, 2, 5, 30, 11, 30, 11, 2310, 11, 30, 5, 30, 11, 30, 5, 30, 5, 2, 5, 30, 11, 30, 11, 2310, 11, 30, 11, 2310, 17, 2310, 11, 2310, 11, 30, 5, 30, 11, 30, 11, 2310, 11, 30, 5, 30, 11, 30, 5, 30, 5, 2, 5, 30, 11, 30, 11, 2310, 11, 30, 11, 2310, 17, 2310, 11, 2310, 11, 30, 11, 2310, 17, 2310, 17

Offset: 0

Antti Karttunen, Jan 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A322827, A323503.

Bisection of A323507.

A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));
A323504(n) = A322827(n+n+1)/A322827(n);

a(n) = A322827(2n+1) / A322827(n).

A323507 a(n) = A322827(n) / A322827(floor(n/2)).

Original entry on oeis.org

1, 2, 3, 2, 6, 5, 3, 2, 6, 5, 7, 30, 6, 5, 3, 2, 6, 5, 7, 30, 210, 11, 7, 30, 6, 5, 7, 30, 6, 5, 3, 2, 6, 5, 7, 30, 210, 11, 7, 30, 210, 11, 13, 2310, 210, 11, 7, 30, 6, 5, 7, 30, 210, 11, 7, 30, 6, 5, 7, 30, 6, 5, 3, 2, 6, 5, 7, 30, 210, 11, 7, 30, 210, 11, 13, 2310, 210, 11, 7, 30, 210, 11, 13, 2310, 30030, 17, 13, 2310, 210, 11, 13, 2310

Offset: 0

Antti Karttunen, Jan 17 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A322827.

Cf. A323503, A323504 (bisections).

A322827(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vecrev(Vec(o)), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));
A323507(n) = A322827(n)/A322827(n>>1);

a(n) = A322827(n) / A322827(floor(n/2)).

A322825 A variant of A322827.

Original entry on oeis.org

1, 2, 6, 4, 36, 30, 18, 8, 216, 450, 210, 900, 108, 150, 54, 16, 1296, 6750, 7350, 13500, 44100, 2310, 22050, 27000, 648, 2250, 1470, 4500, 324, 750, 162, 32, 7776, 101250, 257250, 202500, 1543500, 177870, 771750, 405000, 9261000, 2668050, 30030, 5336100, 4630500, 889350, 2315250, 810000, 3888, 33750, 51450, 67500

Offset: 0

Antti Karttunen, Jan 16 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for sequences related to binary expansion of n

Cf. A005811, A227183, A322827 (the main entry).

A322825(n) = if(!n,1, my(rl=1, m=1, p=2, eb = (n%2)); n>>=1; while(n,if((n%2)==eb, rl++, eb = 1-eb; m *= p^rl; p = nextprime(1+p)); n >>= 1); m *= p^rl; (m));

A322825(n) = if(!n,1,my(bits = Vecrev(binary(n)), rl=1, o = List([])); for(i=2,#bits,if(bits[i]==bits[i-1], rl++, listput(o,rl))); listput(o,rl); my(es=Vec(o), m=1); for(i=1,#es,m *= prime(i)^es[i]); (m));

A001221(a(n)) = A005811(n).
A001222(a(n)) = A227183(n).
A046523(a(n)) = A322827(n).

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

Original entry on oeis.org

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

A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 360, 210, 420, 1260, 2520, 6300, 12600, 37800, 75600, 2310, 4620, 13860, 27720, 69300, 138600, 415800, 831600, 485100, 970200, 2910600, 5821200, 14553000, 29106000, 87318000, 174636000, 30030, 60060, 180180, 360360, 900900, 1801800, 5405400, 10810800, 6306300, 12612600, 37837800, 75675600

Offset: 0

Antti Karttunen, Mar 16 2017

nonn

a(n) = Product of distinct primorials larger than one, obtained as Product_{i} A002110(1+i), where i ranges over the zero-based positions of the 1-bits present in the binary representation of n.
This sequence can be represented as a binary tree. Each child to the left is obtained as A283980(k), and each child to the right is obtained as 2*A283980(k), when their parent contains k:
                                      1
                                      |
                   ...................2....................
                  6                                       12
       30......../ \........60                 180......../ \......360
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
  210       420      1260       2520     6300       12600   37800       75600
etc.

Cf. A129912 (same terms, but sorted into ascending order).
Cf. A000120, A001221, A001222, A002110, A005187, A005361, A006068, A007814, A007913, A019565, A029931, A030308, A046660, A048675, A051903, A056169, A065120, A070939, A072411, A090880, A097248, A108951, A124757, A248663, A276075, A276085, A283475, A283483, A283980, A283984, A283985, A284001, A284002, A284003, A284005, A324287, A324289, A324341, A324342, A324343.
Cf. A005940, A052330, A322827, A323505 for other similar trees.
Cf. also A260443.

Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e == 1 :> {Times @@ Prime@ Range@ PrimePi@ p, e}] &[Times @@ Prime@ Flatten@ Position[#, 1] &@ Reverse@ IntegerDigits[n, 2]], {n, 0, 43}] (* Michael De Vlieger, Mar 18 2017 *)

A283477(n) = prod(i=0,exponent(n),if(bittest(n,i),vecprod(primes(1+i)),1)) \\ Edited by M. F. Hasler, Nov 11 2019

from sympy import prime, primerange, factorint
from operator import mul
from functools import reduce
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a108951(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # after Chai Wah Wu
def a(n): return a108951(a019565(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 22 2017

from sympy import primorial
from math import prod
def A283477(n): return prod(primorial(i) for i, b in enumerate(bin(n)[:1:-1],1) if b =='1') # Chai Wah Wu, Dec 08 2022

(define (A283477 n) (A108951 (A019565 n)))
;; Recursive "binary tree" implementation, using memoization-macro definec:
(definec (A283477 n) (cond ((zero? n) 1) ((even? n) (A283980 (A283477 (/ n 2)))) (else (* 2 (A283980 (A283477 (/ (- n 1) 2)))))))

a(0) = 1; a(2n) = A283980(a(n)), a(2n+1) = 2*A283980(a(n)).
Other identities. For all n >= 0 (or for n >= 1):
a(2n+1) = 2*a(2n).
a(n) = A108951(A019565(n)).
A097248(a(n)) = A283475(n).
A007814(a(n)) = A051903(a(n)) = A000120(n).
A001221(a(n)) = A070939(n).
A001222(a(n)) = A029931(n).
A048675(a(n)) = A005187(n).
A248663(a(n)) = A006068(n).
A090880(a(n)) = A283483(n).
A276075(a(n)) = A283984(n).
A276085(a(n)) = A283985(n).
A046660(a(n)) = A124757(n).
A056169(a(n)) = A065120(n). [seems to be]
A005361(a(n)) = A284001(n).
A072411(a(n)) = A284002(n).
A007913(a(n)) = A284003(n).
A000005(a(n)) = A284005(n).
A324286(a(n)) = A324287(n).
A276086(a(n)) = A324289(n).
A267263(a(n)) = A324341(n).
A276150(a(n)) = A324342(n). [subsequences in the latter are converging towards this sequence]
G.f.: Product_{k>=0} (1 + prime(k + 1)# * x^(2^k)), where prime()# = A002110. - Ilya Gutkovskiy, Aug 19 2019

More formulas and the binary tree illustration added by Antti Karttunen, Mar 19 2017

Four more linking formulas added by Antti Karttunen, Feb 25 2019

A329886 Primorial inflation of Doudna-tree: a(0) = 1, a(1) = 2; for n > 1, if n is even, a(n) = A283980(a(n/2)), and if n is odd, then a(n) = 2*a((n-1)/2).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 36, 8, 210, 60, 180, 24, 900, 72, 216, 16, 2310, 420, 1260, 120, 6300, 360, 1080, 48, 44100, 1800, 5400, 144, 27000, 432, 1296, 32, 30030, 4620, 13860, 840, 69300, 2520, 7560, 240, 485100, 12600, 37800, 720, 189000, 2160, 6480, 96, 5336100, 88200, 264600, 3600, 1323000, 10800, 32400, 288, 9261000

Offset: 0

Antti Karttunen, Dec 24 2019

			This irregular table can be represented as a binary tree. Each child to the left is obtained by applying A283980 to the parent, and each child to the right is obtained by doubling the parent:
                                     1
                                     |
                  ...................2...................
                 6                                       4
      30......../ \........12                 36......../ \........8
      / \                 / \                 / \                 / \
     /   \               /   \               /   \               /   \
    /     \             /     \             /     \             /     \
  210      60         180     24          900      72         216      16
etc.
A329887 is the mirror image of the same tree. See also A342000.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Michael De Vlieger, Tree showing levels 0 <= j <= 5
Michael De Vlieger, Tree showing levels 0 <= j <= 7

Permutation of A025487.
Cf. A005940, A054429, A108951, A283980, A329900, A342000.
Cf. also A283477, A322827, A329887, A337376/A337377.

Block[{a}, a[0] = 1; a[1] = 2; a[n_] := a[n] = If[EvenQ@ n, (Times @@ Map[Prime[PrimePi@#1 + 1]^#2 & @@ # &, FactorInteger[#]] - Boole[# == 1])*2^IntegerExponent[#, 2] &[a[n/2]], 2 a[(n - 1)/2]]; Array[a, 57, 0]]
(* or, via Doudna *)
Map[Times @@ Flatten@ MapIndexed[ConstantArray[Prime[First[#2]], #1] &, Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &@ Sort[Flatten[ConstantArray[PrimePi@#1, #2] & @@@ FactorInteger[#]], Greater]] &, Nest[Append[#1, Prime[1 + BitLength[#2] - DigitCount[#2, 2, 1]]*#1[[#2 - 2^Floor@ Log2@ #2 + 1]]] & @@ {#, Length@ #} &, {1}, 57] ] (* Michael De Vlieger, Mar 05 2021 *)

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
A329886(n) = if(n<2,1+n,if(!(n%2),A283980(A329886(n/2)),2*A329886(n\2)));

a(0) = 1, a(1) = 2; for n > 1, if n is even, a(n) = A283980(a(n/2)), and if n is odd, then a(n) = 2*a((n-1)/2).
a(n) = A108951(A005940(1+n)).
For n >= 1, a(n) = A329887(A054429(n)).

Name amended by Antti Karttunen, Mar 05 2021

A106831 Define a triangle in which the entries are of the form +-1/(b!c!d!e!...), where the order of the factorials is important; read the triangle by rows and record and expand the denominators.

Original entry on oeis.org

2, 6, 4, 24, 12, 12, 8, 120, 48, 36, 24, 48, 24, 24, 16, 720, 240, 144, 96, 144, 72, 72, 48, 240, 96, 72, 48, 96, 48, 48, 32, 5040, 1440, 720, 480, 576, 288, 288, 192, 720, 288, 216, 144, 288, 144, 144, 96, 1440, 480, 288, 192, 288, 144, 144, 96, 480, 192, 144, 96, 192

Offset: 0

N. J. A. Sloane, May 22 2005

Row n has 2^n terms. Row 0 is +1/2!. An entry +-1/b!c!d!... has two children, a left child -+1/(a+1)!b!c!... and a right child +-1/2!b!c!d!...

Let S_n = sum of entries in row n of the triangle. Then for n > 0, n!S_{n-1} is the Bernoulli number B_n.

			Woon's "Bernoulli Tree" begins like this (see also the given Wikipedia-link). This sequence gives the values of the denominators:
                                     +1
                                    ────
                                     2!
                 -1                 /  \                  +1
                ──── ............../    \.............. ─────
                 3!                                      2!2!
        +1        .         -1                 -1         .         +1
       ────      / \       ────               ────       / \      ──────
        4! ...../   \..... 2!3!               3!2! ...../   \.... 2!2!2!
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
    -1      +1         +1       -1         +1      -1          -1       +1
   ────    ────       ────     ──────     ────   ──────      ──────  ────────
    5!     2!4!       3!3!     2!2!3!     4!2!   2!3!2!      3!2!2!  2!2!2!2!
etc.

Reinhard Zumkeller, Rows n = 0..13 of table, flattened
Wikipedia, Bernoulli number: A binary tree representation
S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56.
Index entries for sequences related to Bernoulli numbers.

Cf. A242179 (numerators), A050925, A050932, A000142.
Cf. A001013, A060054, A075180, A164555, A027642, A246660.
Cf. A323505 (mirror image), and also A005940, A283477, A322827 for other similar trees.

a106831 n k = a106831_tabf !! n !! n
a106831_row n = a106831_tabf !! n
a106831_tabf = map (map (\(, , left, right) -> left * right)) $
   iterate (concatMap (\(x, f, left, right) -> let f' = f * x in
   [(x + 1, f', f', right), (3, 2, 2, left * right)])) [(3, 2, 2, 1)]
-- Reinhard Zumkeller, May 05 2014

Contribution from Peter Luschny, Jun 12 2009: (Start)
The routine computes the triangle row by row and gives the numbers with their sign.
Thus A(1)=[2]; A(2)=[ -6,4]; A(3)=[24,-12,-12,8]; etc.
A := proc(n) local k, i, j, m, W, T; k := 2;
W := array(0..2^n); W[1] := [1,`if`(n=0,1,2)];
for i from 1 to n-1 do for m from k by 2 to 2*k-1 do
T := W[iquo(m,2)]; W[m] := [ -T[1],T[2]+1,seq(T[j],j=3..nops(T))];
W[m+1] := [T[1],2,seq(T[j],j=2..nops(T))]; od; k := 2*k; od;
seq(W[i][1]*mul(W[i][j]!,j=2..nops(W[i])),i=iquo(k,2)..k-1) end:
seq(print(A(i)),i=1..5); (End)

a [n_] := Module[{k, i, j, m, w, t}, k = 2; w = Array[0&, 2^n]; w[[1]] := {1, If[n == 0, 1, 2]}; For[i = 1, i <= n-1, i++, For[m = k, m <= 2*k-1 , m = m+2, t = w[[Quotient[m, 2]]]; w[[m]] = {-t[[1]], t[[2]]+1, Sequence @@ Table[t[[j]], {j, 3, Length[t]}]}; w[[m+1]] = {t[[1]], 2, Sequence @@ Table[t[[j]], {j, 2, Length[t]}]}]; k = 2*k]; Table[w[[i, 1]]*Product[w[[i, j]]!, {j, 2, Length[w[[i]]]}], {i, Quotient[k, 2], k-1}]]; Table[a[i] , {i, 1, 6}] // Flatten // Abs (* Jean-François Alcover, Dec 20 2013, translated from Maple *)

A106831off1(n) = if(!n,1, my(rl=1, m=1); while(n,if(!(n%2), rl++, m *= ((1+rl)!); rl=1); n >>= 1); (m));
A106831(n) = A106831off1(1+n); \\ Antti Karttunen, Jan 16 2019

A001511(n) = (1+valuation(n,2));
A106831r1(n) = if(!n,1,if(n%2, 2*A106831r1((n-1)/2), (1+A001511(n))*A106831r1(n/2))); \\ Implements the given recurrence.
A106831(n) = A106831r1(1+n); \\ Antti Karttunen, Jan 16 2019

From Antti Karttunen, Jan 16 2019: (Start)
If sequence is shifted one term to the right, then the following recurrence works:
a(0) = 1; and for n > 0, a(2n) = (1+A001511(2n))*a(n), a(2n+1) = 2*a(n).
(End)

More terms from Franklin T. Adams-Watters, Apr 28 2006

Example section reillustrated by Antti Karttunen, Jan 16 2019

A323505 Mirror image of (denominators of) Bernoulli tree, A106831.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 12, 24, 16, 24, 24, 48, 24, 36, 48, 120, 32, 48, 48, 96, 48, 72, 96, 240, 48, 72, 72, 144, 96, 144, 240, 720, 64, 96, 96, 192, 96, 144, 192, 480, 96, 144, 144, 288, 192, 288, 480, 1440, 96, 144, 144, 288, 144, 216, 288, 720, 192, 288, 288, 576, 480, 720, 1440, 5040, 128, 192, 192, 384, 192, 288, 384, 960, 192, 288, 288

Offset: 0

Antti Karttunen, Jan 16 2019

In contrast to A106831 which follows Woon's original indexing (and orientation), this variant starts with value a(0) = 1, with the rest of terms having an index incremented by one, thus allowing for a simple recurrence.

Sequence contains only terms of A001013 and each a(n) is a multiple of A246660(n).

			This sequence can be represented as a binary tree:
                                       1
                                       |
                    ...................2....................
                   4                                        6
         8......../ \........12                 12........./ \.......24
        / \                 / \                 / \                 / \
       /   \               /   \               /   \               /   \
      /     \             /     \             /     \             /     \
    16       24         24       48         24       36         48      120
  32  48   48  96     48  72   96  240    48  72   72  144    96  144 240  720
etc.

Antti Karttunen, Table of n, a(n) for n = 0..16383
S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56.
Index entries for sequences related to Bernoulli numbers.

Cf. A000079 (left edge), A000142 (right edge), A001013, A001511, A036987, A054429, A246660, A323506, A323508.

Cf. A106831 and also A005940, A283477, A322827 for other similar trees.

A001511(n) = (1+valuation(n,2));
A036987(n) = !bitand(n,1+n);
A323505(n) = if(!n,1,if(!(n%2), 2*A323505(n/2), (A001511(n+1)+1-A036987(n))*A323505((n-1)/2)));

A054429(n) = if(!n,n,((3<<#binary(n\2))-n-1)); \\ From A054429
A106831r1(n) = if(!n,1,if(n%2, 2*A106831r1((n-1)/2), (1+A001511(n))*A106831r1(n/2))); \\ Recurrence for A106831, when prepended with 1, thus shifted one term right
A323505(n) = A106831r1(A054429(n));

a(0) = 1; and for n > 0, if n is even, a(n) = 2*a(n/2), and if n is odd, a(n) = (A001511(n+1)+1-A036987(n)) * a((n-1)/2).
For n > 0, a(n) = b(A054429(n)), where b(n) = A106831(n-1).
a(n) = A246660(n) * A323506(n).
a(n) = A323508(A005940(1+n)).

cp's OEIS Frontend

A324383 a(n) is the minimal number of primorials that add to A322827(n).

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A323503 a(n) = A322827(2n) / A322827(n).

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A323504 a(n) = A322827(2n+1) / A322827(n).

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A323507 a(n) = A322827(n) / A322827(floor(n/2)).

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A322825 A variant of A322827.

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

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A283477 If 2n = 2^e1 + 2^e2 + ... + 2^ek [e1 .. ek distinct], then a(n) = A002110(e1) * A002110(e2) * ... * A002110(ek).

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