A002110 -id:A002110 - 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

A276150 Sum of digits when n is written in primorial base (A049345); minimal number of primorials (A002110) that add to n.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Aug 22 2016

The sum of digits of n in primorial base is odd if n is 1 or 2 (mod 4) and even if n is 0 or 3 (mod 4). Proof: primorials are 1 or 2 (mod 4) and a(n) can be constructed via the greedy algorithm. So if n = 4k + r where 0 <= r < 4, 4k needs an even number of primorials and r needs hammingweight(r) = A000120(r) primorials. Q.E.D. - David A. Corneth, Feb 27 2019

			For n=24, which is "400" in primorial base (as 24 = 4*(3*2*1) + 0*(2*1) + 0*1, see A049345), the sum of digits is 4, thus a(24) = 4.

Antti Karttunen, Table of n, a(n) for n = 0..30030
Index entries for sequences related to primorial base

Cf. A000120, A001222, A002110, A049345, A053589, A235168, A260188, A267263, A276084, A276086, A276151, A277022, A278226, A283477, A319713, A319715 (inverse Möbius transform), A321683, A324342, A324382, A324383, A324386, A324387, A371091, A373605, A373606, A373607.
Cf. A333426 [k such that a(k)|k], A339215 [numbers not of the form x+a(x) for any x], A358977 [k such that gcd(k, a(k)) = 1].
Cf. A338835, A366693.
Cf. A014601, A042963 (positions of even and odd terms), A343048 (positions of records).
Differs from analogous A034968 for the first time at n=24.

nn = 120; b = MixedRadix[Reverse@ Prime@ NestWhileList[# + 1 &, 1, Times @@ Prime@ Range[# + 1] <= nn &]]; Table[Total@ IntegerDigits[n, b], {n, 0, nn}] (* Version 10.2, or *)
nn = 120; f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Prime@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Prime@ Range[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[Total@ f@ n, {n, 0, 120}] (* Michael De Vlieger, Aug 26 2016 *)

A276150(n) = { my(s=0, p=2, d); while(n, d = (n%p); s += d; n = (n-d)/p; p = nextprime(1+p)); (s); }; \\ Antti Karttunen, Feb 27 2019

from sympy import prime, primefactors
def Omega(n): return 0 if n==1 else Omega(n//primefactors(n)[0]) + 1
def a276086(n):
    i=0
    m=pr=1
    while n>0:
        i+=1
        N=prime(i)*pr
        if n%N!=0:
            m*=(prime(i)**((n%N)/pr))
            n-=n%N
        pr=N
    return m
def a(n): return Omega(a276086(n))
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 23 2017

a(n) = 1 + a(A276151(n)) = 1 + a(n-A002110(A276084(n))), a(0) = 0.
or for n >= 1: a(n) = 1 + a(n-A260188(n)).
Other identities and observations. For all n >= 0:
a(n) = A001222(A276086(n)) = A001222(A278226(n)).
a(n) >= A371091(n) >= A267263(n).
From Antti Karttunen, Feb 27 2019: (Start)
a(n) = A000120(A277022(n)).
a(A283477(n)) = A324342(n).
(End)
a(n) = A373606(n) + A373607(n). - Antti Karttunen, Jun 19 2024

A143293 Partial sums of A002110, the primorial numbers.

Original entry on oeis.org

1, 3, 9, 39, 249, 2559, 32589, 543099, 10242789, 233335659, 6703028889, 207263519019, 7628001653829, 311878265181039, 13394639596851069, 628284422185342479, 33217442899375387209, 1955977793053588026279, 119244359152460559009549, 7977565910232727614888639

Offset: 0

Gary W. Adamson, Aug 05 2008

nonn

After 3, this is never prime because all values thereafter are multiples of 3. Starting from a(6) all are also multiples of 17. - Jonathan Vos Post, Feb 10 2010
Starting from a(162) all are also multiples of 967. - Alex Ratushnyak, May 14 2013
Repunits in primorial base, A049345. - Antti Karttunen, Aug 21 2016

			a(3) = 39 = (1 + 2 + 6 + 30), where A002110 = (1, 2, 6, 30, 210, 2310,...).

Soumyadeep Dhar, Table of n, a(n) for n = 0..350 (terms up to a(100) from T. D. Noe)
Index entries for sequences related to primorial base

Cf. A002110, A049345, A276085.

b:= proc(n) option remember; `if`(n=0, [1$2], (h->
      (p-> [p, p+h[2]])(ithprime(n)*h[1]))(b(n-1)))
    end:
a:= n-> b(n)[2]:
seq(a(n), n=0..19);  # Alois P. Heinz, Feb 23 2022

Table[s = 1; Do[s = 1 + s*Prime[i], {i, n, 1, -1}]; s, {n, 0, 20}] (* T. D. Noe, May 03 2013 *)
Accumulate[FoldList[Times,1,Prime[Range[20]]]] (* Harvey P. Dale, Feb 05 2015 *)

a(n)=if(n==0,return(1)); my(P=1,s=1); forprime(p=2,prime(n), s+=P*=p); s \\ Charles R Greathouse IV, Feb 05 2014

from itertools import chain, accumulate, count, islice
from operator import mul
from sympy import prime
def A143293_gen(): # generator of terms
    return accumulate(accumulate(chain((1,),(prime(n) for n in count(1))), mul))
A143293_list = list(islice(A143293_gen(),20)) # Chai Wah Wu, Feb 23 2022

a(n) = Sum_{k=0..n} prime(k)#, where prime(n)# = A002110(n).

a(n) = A276085(A002110(1+n)). - Antti Karttunen, Aug 21 2016

a(11)-a(19) from Jonathan Vos Post, Feb 10 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

A238745 a(1) = 1; for n > 1, if the first integer with the same prime signature as n is factorized into primorials as Product A002110(i)^e(i), then a(n) = Product prime(i)^e(i).

Original entry on oeis.org

1, 2, 2, 4, 2, 3, 2, 8, 4, 3, 2, 6, 2, 3, 3, 16, 2, 6, 2, 6, 3, 3, 2, 12, 4, 3, 8, 6, 2, 5, 2, 32, 3, 3, 3, 9, 2, 3, 3, 12, 2, 5, 2, 6, 6, 3, 2, 24, 4, 6, 3, 6, 2, 12, 3, 12, 3, 3, 2, 10, 2, 3, 6, 64, 3, 5, 2, 6, 3, 5, 2, 18, 2, 3, 6, 6, 3, 5, 2, 24, 16, 3, 2

Offset: 1

Matthew Vandermast, Apr 28 2014

nonn

Alternate definition: a(1) = 1; for n > 1, if row n of table A238744 is {k(1), k(2),...,k(A051903(n))}, then a(n) = Product {i = 1 to A051903(n)} prime(k(i)).
Since the first integer of each prime signature (A025487) is always a product of primorials (A002110), there is always a value for a(n). Every positive integer appears in the sequence.
a(m) = a(n) iff m and n have the same prime signature.  If the prime signatures of m and n are conjugate to each other when they are viewed as partitions, then a(n) = A181819(m) and a(m) = A181819(n).

			The first integer with the same prime signature as 40 is 24 = 2^3*3. Since the factorization of 24 into primorials is 24 = 2^2*6 = A002110(1)^2*A002110(2), a(24) = a(40) = prime(1)^2*prime(2) = 2^2*3 = 12.

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

Cf. A181815, A181819, A238744.

f[n_] := Block[{k = 1, d, a}, While[n - Times @@ Prime@ Range[k + 1] >= 0, k++]; If[n == Product[Prime@ i, {i, k}], Prime@ k, d = Select[Reverse@ FoldList[#1 #2 &, Prime@ Range@ k], Divisible[n, #] &]; If[AllTrue[#, IntegerQ], Times @@ Map[(FactorInteger[#1][[-1, 1]])^#2 & @@ # &, Reverse@ Tally@ #], False] &@ Rest@ NestWhileList[Function[P, {#1/P, P}]@ SelectFirst[d, Function[k, Divisible[#1, k]]] & @@ # &, {n, 1}, First@ # > 1 &][[All, -1]]]]; Table[f@ Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]]] - Boole[n == 1], {n, 83}] (* Michael De Vlieger, May 19 2017, Version 10.2 *)

a(n) = A181819(A124859(n)).

a(n) = A122111(A181819(n)).

A054640 a(n) is the sum of the divisors of the n-th primorial: a(n) = A000203(A002110(n)).

Original entry on oeis.org

1, 3, 12, 72, 576, 6912, 96768, 1741824, 34836480, 836075520, 25082265600, 802632499200, 30500034969600, 1281001468723200, 56364064623820800, 2705475101943398400, 146095655504943513600, 8765739330296610816000, 543475838478389870592000, 36956357016530511200256000

Offset: 0

Labos Elemer, May 15 2000

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Rafael Jakimczuk, Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts, International Mathematical Forum, Vol. 12, No. 7 (2017), pp. 331-338.

Cf. A000203, A002110, A005867, A008864, A028815, A051674, A054641, A070826, A073004, A074107.

[1/2*&*[(1+NthPrime(k)): k in [0..n-1]]: n in [1..19]]; // Vincenzo Librandi, May 08 2017

a:= n-> mul(1+ithprime(j), j=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, Aug 24 2008

Table[Product[1 + Prime[i], {i,n-1}], {n,100}] (* Geoffrey Critzer, Dec 01 2014 *)

a(n)=prod(i=1,n,prime(i)+1) \\ Charles R Greathouse IV, Feb 13 2013

def A054640(n): return product(nth_prime(j)+1 for j in range(1,n+1))
[A054640(n) for n in range(41)] # G. C. Greubel, Aug 05 2024

a(n+1) = a(n)*(prime(n) + 1) = a(n)*A028815(n) (quotient=n-th prime+1 starting with 2).
a(n) ~ (6/Pi^2) * exp(gamma) * A002110(n) * log(prime(n)) + O(A002110(n)) (Jakimczuk, 2017). - Amiram Eldar, Feb 17 2021
a(n) = a(n-1) * A008864(n). - Flávio V. Fernandes, Mar 20 2021
a(n) = A002110(n) + A074107(n), a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n). - Antti Karttunen, Nov 19 2024

a(0)=1 prepended by Alois P. Heinz, Apr 01 2021

A129912 Numbers that are products of distinct primorial numbers (see A002110).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, 4620, 6300, 12600, 13860, 27720, 30030, 37800, 60060, 69300, 75600, 138600, 180180, 360360, 415800, 485100, 510510, 831600, 900900, 970200, 1021020, 1801800, 2910600, 3063060, 5405400

Offset: 1

Bill McEachen, Jun 05 2007, Jun 06 2007, Jul 06 2007, Aug 07 2007

Conjecture: every odd prime p is either adjacent to a term of A129912 or a prime distance q from some term of A129912, where q < p. - Bill McEachen, Jun 03 2010, edited for clarity in Feb 26 2019

The first 2^20 terms k > 2 of A283477 all satisfy also the condition that the differences k-A151799(k) and A151800(k)-k are always either 1 or prime, like is also conjectured to hold for A002182 (cf. also the conjecture given in A117825). However, for A025487, which is a supersequence of both sequences, this is not always true: 512 is a member of A025487, but A151800(512) = 521, with 521 - 512 = 9, which is a composite number. - Antti Karttunen, Feb 26 2019

			For s = 4 there are 8 (generally 2^(s-1)) such numbers: 210 = 2*3*5*7, 420 = 2^2*3*5*7 = (2*3*5*7)*2, 1260 = 2^2*3^2*5*7 = (2*3*5*7)*(2*3), 6300 = 2^2*3^2*5^2*7 = (2*3*5*7)*(2*3*5), 2520 = 2^3*3^2*5*7 = (2*3*5*7)*(2*3)*2, 12600 = 2^3*3^2*5^2*7 = (2*3*5*7)*(2*3*5)*2, 37800 = 2^3*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3), 75600 = 2^4*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3)*2.

CRC Standard Mathematical Tables, 28th Ed., CRC Press

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Bill McEachen, Normalized A129912.
Robert Potter, Primorial Conjecture.
John Sokol, Sokol's Prime Conjecture, 2002.
Wikipedia, Primorial.
Index entries for sequences related to primorial numbers.

Subsequence of A025487. Sequence A283477 sorted into ascending order.

Cf. A002110, A117825, A151799, A151800.

Clear[f]; f[m_] := f[m] = Union[Times @@@ Subsets[FoldList[Times, 1, Prime[Range[m]]]]][[1 ;; 100]]; f[10]; f[m = 11]; While[f[m] != f[m-1], m++]; f[m] (* Jean-François Alcover, Mar 03 2014 *) (* or *)
pr[n_] := Product[Prime[n + 1 - i]^i, {i, n}]; upto[mx_] := Block[{ric, j = 1}, ric[n_, ip_, ex_] := If[n < mx, Block[{p = Prime[ip + 1]}, If[ex == 1, Sow@ n]; ric[n p^ex, ip + 1, ex]; If[ex > 1, ric[n p^(ex - 1), ip + 1, ex - 1]]]]; Sort@ Reap[ Sow[1]; While[pr[j] < mx, ric[2^j, 1, j]; j++]][[2, 1]]];
upto[10^30] (* faster, Giovanni Resta, Apr 02 2017 *)

is(n)=my(o=valuation(n,2),t); if(o<1||n<2, return(n==1)); n>>=o; forprime(p=3,, t=valuation(n,p); n/=p^t; if(t>o || tCharles R Greathouse IV, Oct 22 2015

Apart from 1 and 2, numbers of the form 2^k(1)*3^k(2)*5^k(3)*...*p(s)^k(s), where p(s) is s-th prime, k(i)>0 for i=1..s, k(i)-k(i-1) = 0 or 1 for i=2..s and |{k(1),k(2),..,k(s)}|=k(1). - Vladeta Jovovic, Jun 14 2007

Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/A002110(n)) = 1.8177952875... . - Amiram Eldar, Jun 03 2023

Edited by N. J. A. Sloane, Jun 09 2007, Aug 08 2007
I corrected the Potter link to reflect its relocation. - Bill McEachen, Sep 12 2009
I added link to Wikicommons image. - Bill McEachen, Sep 16 2009
I again corrected the Potter link for its relocation - Bill McEachen, May 30 2013

A329900 Primorial deflation of n: starting from x = n, repeatedly divide x by the largest primorial A002110(k) that divides it, until x is an odd number. Then a(n) = Product prime(k_i), for primorial indices k_1 >= k_2 >= ..., encountered in the process.

Original entry on oeis.org

1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 6, 1, 2, 1, 16, 1, 3, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4, 1, 5, 1, 32, 1, 2, 1, 9, 1, 2, 1, 8, 1, 3, 1, 4, 1, 2, 1, 24, 1, 2, 1, 4, 1, 3, 1, 8, 1, 2, 1, 10, 1, 2, 1, 64, 1, 3, 1, 4, 1, 2, 1, 18, 1, 2, 1, 4, 1, 3, 1, 16, 1, 2, 1, 6, 1, 2, 1, 8, 1, 5, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 3, 1, 8, 1

Offset: 1

Antti Karttunen, Dec 22 2019

nonn

When applied to arbitrary n, the "primorial deflation" (term coined by Matthew Vandermast in A181815) induces the splitting of n to two factors A328478(n)*A328479(n) = n, where we call A328478(n) the non-deflatable component of n (which is essentially discarded), while A328479(n) is the deflatable component. Only if n is in A025487, then the entire n is deflatable, i.e., A328478(n) = 1 and A328479(n) = n.

According to Daniel Suteu, also the ratio (A319626(n) / A319627(n)) can be viewed as a "primorial deflation". That definition coincides with this one when restricted to terms of A025487, as for all k in A025487, A319626(k) = a(k), and A319627(k) = 1. - Antti Karttunen, Dec 29 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537

A left inverse of A108951. Coincides with A319626 on A025487.

Cf. A002110, A002182, A111701, A181815, A181817, A181819, A181821, A276084, A304886, A319626, A319627, A328478, A328479, A329889, A329902, A330685, A330686, A330689.

Array[If[OddQ@ #, 1, Times @@ Prime@ # &@ Rest@ NestWhile[Append[#1, {#3, Drop[#, -LengthWhile[Reverse@ #, # == 0 &]] &[#2 - PadRight[ConstantArray[1, #3], Length@ #2]]}] & @@ {#1, #2, LengthWhile[#2, # > 0 &]} & @@ {#, #[[-1, -1]]} &, {{0, TakeWhile[If[# == 1, {0}, Function[f, ReplacePart[Table[0, {PrimePi[f[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #], # > 0 &]}}, And[FreeQ[#[[-1, -1]], 0], Length[#[[-1, -1]] ] != 0] &][[All, 1]] ] &, 105] (* Michael De Vlieger, Dec 28 2019 *)
Array[Times @@ Prime@(TakeWhile[Reap[FixedPointList[Block[{k = 1}, While[Mod[#, Prime@ k] == 0, k++]; Sow[k - 1]; #/Product[Prime@ i, {i, k - 1}]] &, #]][[-1, 1]], # > 0 &]) &, 105] (* Michael De Vlieger, Jan 11 2020 *)

A329900(n) = { my(m=1, pp=1); while(1, forprime(p=2, ,if(n%p, if(2==p, return(m), break), n /= p; pp = p)); m *= pp); (m); };

A111701(n) = forprime(p=2, , if(n%p, return(n), n /= p));
A276084(n) = { for(i=1,oo,if(n%prime(i),return(i-1))); }
A329900(n) = if(n%2,1,prime(A276084(n))*A329900(A111701(n)));

For odd n, a(n) = 1, for even n, a(n) = A000040(A276084(n)) * a(A111701(n)).
For even n, a(n) = A000040(A276084(n)) * a(n/A002110(A276084(n))).
A108951(a(n)) = A328479(n), for n >= 1.
a(A108951(n)) = n, for n >= 1.
a(A328479(n)) = a(n),  for n >= 1.
a(A328478(n)) = 1, for n >= 1.
a(A002110(n)) = A000040(n), for n >= 1.
a(A000142(n)) = A307035(n), for n >= 0.
a(A283477(n)) = A019565(n), for n >= 0.
a(A329886(n)) = A005940(1+n), for n >= 0.
a(A329887(n)) = A163511(n), for n >= 0.
a(A329602(n)) = A329888(n), for n >= 1.
a(A025487(n)) = A181815(n), for n >= 1.
a(A124859(n)) = A181819(n), for n >= 1.
a(A181817(n)) = A025487(n), for n >= 1.
a(A181821(n)) = A122111(n), for n >= 1.
a(A002182(n)) = A329902(n), for n >= 1.
a(A260633(n)) = A329889(n), for n >= 1.
a(A033833(n)) = A330685(n), for n >= 1.
a(A307866(1+n)) = A330686(n), for n >= 1.
a(A330687(n)) = A330689(n), for n >= 1.

A053589 Greatest primorial number (A002110) which divides n.

Original entry on oeis.org

1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 30, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 30, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 30, 1, 2, 1, 2, 1, 6, 1, 2, 1, 2, 1, 6

Offset: 1

Frederick Magata (frederick.magata(AT)uni-muenster.de), Jan 19 2000

			a(30) = 30 because 30=2*3*5, a(15) = 1 because 15=3*5.

Antti Karttunen, Table of n, a(n) for n = 1..2309
Index entries for sequences related to primorial numbers

Cf. A002110, A276084, A053669, A111701, A276151, A276152, A276157, A260188, A276086, A276150.

N:= 1000: # to get a(1)..a(N)
P:= 1: p:= 1:
A:= Vector(N,1):
do
  p:= nextprime(p);
  P:= P*p;
  if P > N then break fi;
  A[[seq(i,i=P..N,P)]]:= P;
od:
convert(A,list); # Robert Israel, Aug 30 2016

Table[k = 1; While[Divisible[n, Times @@ Prime@ Range@ k], k++]; Times @@ Prime@ Range[k - 1], {n, 120}] (* Michael De Vlieger, Aug 30 2016 *)

a(n)=my(f=factor(n), r = 1, k = 1, p); while(k<=matsize(f)[1], p=prime(k); if(f[k,1]!=p,return(r));r*=p; k++) ; r
a(n) = my(r = 1, p = 2); while(n/p==n\p, r*=p; p=nextprime(p+1));r
\\ list of all terms up to n#.
lista(n) = my(l = List([1]),k,s=1); forprime(i=2,n, for(j=1,i-1, for(k=1,s, listput(l,l[k]))); l[#l]*=i; s=#l); l \\ David A. Corneth, Aug 30 2016

a(n)=my(s=1); forprime(p=2,, if(n%p, return(s), s *= p)) \\ Charles R Greathouse IV, Sep 07 2016

(define (A053589 n) (A002110 (A276084 n))) ;; Antti Karttunen, Aug 30 2016

From Antti Karttunen, Aug 30 2016: (Start)
a(n) = A002110(A276084(n)).
a(n) = n/A111701(n).
A276157(n) = A260188(n)/a(n).
(End)

More terms from Larry Reeves (larryr(AT)acm.org), Oct 02 2000

A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 1, 2, 4, 6, 7, 30, 12, 12, 17, 210, 20, 2310, 67, 28, 32, 30030, 33, 510510, 44, 104, 431, 9699690, 52, 60, 4633, 54, 148, 223092870, 71, 6469693230, 80, 652, 60077, 192, 84, 200560490130, 1021039, 6956, 108, 7420738134810, 229, 304250263527210, 884, 114, 19399403, 13082761331670030, 128, 420, 145, 90124, 9292, 614889782588491410, 135, 1116, 324

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001
Index entries for sequences related to primorial numbers

Cf. A002110, A108951, A276085, A276150, A328771, A373982, A374211.
Cf. A042965 (indices of even terms), A016825 (of odd terms), A152822 (antiparity of terms), A373992 (indices of multiples of 3), A374212 (2-adic valuation), A374213 (3-adic valuation), A374123 [a(n) mod 360].
Cf. A374031 [gcd(a(n), A276085(n))], A374116 [gcd(a(n), A328845(n))].
For variants of the same formula, see A003415, A258851, A328769, A328845, A328846, A371192.

A002110(n) = prod(i=1,n,prime(i));
A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i,1])-1)/f[i, 1]));

A328768(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*A002110(primepi(f[i, 1]))/(f[i, 1]^2)));

a(n) = n * Sum e_j * A276085(p_j)/p_j for n = Product p_j^e_j, where for primes p, A276085(p) = A002110(A000720(p)-1).
a(n) = n * Sum e_j * (p_j)#/(p_j^2) for n = Product p_j^e_j with (p_j)# = A034386(p_j).
For all n >= 0, A276150(a(n)) = A328771(n).

cp's OEIS Frontend

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

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A276150 Sum of digits when n is written in primorial base (A049345); minimal number of primorials (A002110) that add to n.

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A143293 Partial sums of A002110, the primorial numbers.

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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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A238745 a(1) = 1; for n > 1, if the first integer with the same prime signature as n is factorized into primorials as Product A002110(i)^e(i), then a(n) = Product prime(i)^e(i).

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A054640 a(n) is the sum of the divisors of the n-th primorial: a(n) = A000203(A002110(n)).

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A328768 The first primorial based variant of arithmetic derivative: a(prime(i)) = A002110(i-1), where prime(i) = A000040(i), a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.