A049536 -id:A049536 - cp's OEIS frontend

A051451 a(n) = lcm{ 1,2,...,x } where x is the n-th prime power (A000961).

1, 2, 6, 12, 60, 420, 840, 2520, 27720, 360360, 720720, 12252240, 232792560, 5354228880, 26771144400, 80313433200, 2329089562800, 72201776446800, 144403552893600, 5342931457063200, 219060189739591200, 9419588158802421600, 442720643463713815200

Offset: 1

Labos Elemer, Dec 11 1999

This sequence is the list of distinct terms in A003418.
This may be the "smallest" product-based numbering system that has a unique finite representation for every rational number. In this base 1/2 = .1 (1*1/2), 1/3 = .02 (0*1/2 + 2*1/6), 1/5 = .0102 (0*1/2 + 1*1/6 + 0*1/12 + 2*1/60). - Russell Easterly, Oct 03 2001
Partial products of A025473, prime roots of the prime powers.
Conjecture: For every n > 2, there exists a twin prime pair [p, p+2] with p < a(n), such that [a(n)+p, a(n)+p+2] is also a twin prime pair. Example: For n=6 we can take p=11, because for a(6) = 420 is [420+11, 420+13] = [431, 433] also a twin prime pair. This has been verified for 2 < n <= 200. - Mike Winkler, Sep 12 2013, May 09 2014
The prime powers give all values, and do so uniquely. (Other positive integers give repeated values.) - Daniel Forgues, Apr 28 2014
"LCM numeral system": a(n+1) is place value for index n, n >= 0; a(-n+1) is (place value)^(-1) for index n, n < 0. - Daniel Forgues, May 03 2014
Repetitions removed from slowest growing integer series A003418 with integers > 0 converging to 0 in the ring Z^ of profinite integers. Both A003418 and the present sequence may be used as a replacement for the usual "factorial system" for coding profinite integers. - Herbert Eberle, May 01 2016
Every term of this sequence is deeply composite (A095848). Moreover, the terms of this sequence are the "special deeply composite numbers", in analogy to the special highly composite numbers (A106037). A special highly composite number is a highly composite number (A002182) that divides every larger highly composite number. In the same fashion, the deeply composite numbers that divide every larger deeply composite number are just the terms of this sequence. This follows from the formula for deeply composite numbers. - Hal M. Switkay, Jun 08 2021
From Bill McEachen, Apr 28 2023: (Start)
Every term belongs to A025487.
Conjecture: Every term = A001013(j)*A129912(k) for some j,k. (End)

			lcm[1,...,n] is 2520 for n=9 and 10. The smallest such n's are always prime powers, where A003418 jumps.

Amiram Eldar, Table of n, a(n) for n = 1..377 (terms 1..100 from T. D. Noe)
Thomas Baruchel and Carsten Elsner, On error sums formed by rational approximations with split denominators, arXiv preprint arXiv:1602.06445 [math.NT], 2016.
Russell Easterly, Product Bases A Million Ways to Count [Archive link]
OEIS Wiki, LCM numeral system
Mike Winkler, Table of n, a(n), p for n = 3..200, 2013.
Index entries for sequences related to lcm's

Cf. A000961, A003418, A025473, A049536, A049537, A064890, A025487.

a051451 n = a051451_list !! (n-1)
a051451_list = scanl1 lcm a000961_list
-- Reinhard Zumkeller, Mar 01 2012

f[n_] := LCM @@ Range@ n; Union@ Array[f, 41] (* Robert G. Wilson v, Jul 11 2011 *)
Join[{1},LCM@@Range[#]&/@Select[Range[50],PrimePowerQ]] (* Harvey P. Dale, Feb 06 2020 *)

do(lim)=my(v=primes(primepi(lim)), u=List([1])); forprime(p=2, sqrtint(lim\1), for(e=2, log(lim+.5)\log(p), listput(u, p^e))); v=vecsort(concat(v, Vec(u))); for(i=2,#v,v[i]=lcm(v[i],v[i-1])); v \\ Charles R Greathouse IV, Nov 20 2012

{lim=100; n=1; i=1; j=1; until(n==lim, until(a!=j, a=lcm(j,i+1); i++;); j=a; n++; print(n" "a););} \\ Mike Winkler, Sep 07 2013

x=1;for(i=1,100,if(omega(i)==1,x*=factor(i)[1,1])) \\ Florian Baur, Apr 11 2022

from math import prod
from sympy import primepi, integer_nthroot, integer_log, primerange
def A051451(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return prod(p**integer_log(m, p)[0] for p in primerange(m+1)) # Chai Wah Wu, Aug 15 2024

def A051451_list(n):
    a = [ ]
    L = [1]
    for i in (1..n):
       a.append(i)
       if (is_prime_power(i) == 1):
           L.append(lcm(a))
    return(L)
A051451_list(42) # Jani Melik, Jul 07 2022

a(n) = A003418(A000961(n)).
a(n) = A208768(n) + 1. - Reinhard Zumkeller, Mar 01 2012
Sum_{n>=1} 1/a(n) = A064890. - Amiram Eldar, Nov 16 2020

Minor edits by Ray Chandler, Jan 16 2009

A049537 Values of k for which A075059(k) = A003418(k) + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 19, 20, 21, 22, 25, 26, 31, 47, 48, 89, 90, 91, 92, 93, 94, 95, 96, 127, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 1369, 1370, 1371, 1372, 2251, 2252, 2253, 2254, 2255, 2256, 2257, 2258, 2259, 2260, 2261, 2262, 2263, 2264, 2265, 2266, 3271, 3272, 3273, 3274, 3275, 3276, 3277, 3278, 3279, 3280, 3281, 3282, 3283, 3284, 3285, 3286, 3287

Offset: 1

Labos Elemer

nonn

			8 is not in the sequence because A075059(8) = 1 + A003418(8) = 1 + lcm(1, 2, ..., 8) = 841 = 29^2 is not prime.
127 is in the sequence because A075059(127) = 1 + A003418(127) = 1 + lcm(1, 2, ..., 127) = 6676878045498705789701874602220118271269436344024536001 is prime.

Jinyuan Wang, Table of n, a(n) for n = 1..150

Cf. A000040, A003418, A049536, A057825, A075059.

Join[{0}, Select[Range[250], PrimeQ[LCM@@Range[#]+1]&]] (* Harvey P. Dale, Nov 15 2011 *)

isok(n) = isprime(lcm(vector(n, i, i))+1); \\ Michel Marcus, Feb 25 2014

a(43)-a(57) from Ray Chandler, Jan 16 2009

a(1)=0 prepended and a(58)-a(86) added by Max Alekseyev, Sep 04 2015

A045948 a(n) = A003418(n)/A034386(n).

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 12, 12, 12, 12, 12, 12, 12, 24, 24, 24, 24, 24, 24, 24, 24, 24, 120, 120, 360, 360, 360, 360, 360, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 720, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040, 5040

Offset: 1

Labos Elemer

nonn

			n=11: lcm(1..11) = 27720 = 8*9*5*7*11 = 2310*12. A034386(11)=2310, so the quotient is 12. Thus a(11) = 12.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to lcm's

Cf. A003418, A034386, A002110, A000142, A049536, A048148.

Table[Exp[Sum[MangoldtLambda[n], {n, 1, m}]]/ Product[x, {x, Prime[Range[PrimePi[m]]]}], {m, 1, 57}] (* Fred Daniel Kline, Apr 02 2015 *)

a(n)=lcm([1..n])/prod(i=1,primepi(n),prime(i)) \\ Charles R Greathouse IV, Apr 02 2015; corrected by Michel Marcus, Dec 26 2020

A057824 Primes p such that p+1 is LCM(1,...,m) for some (usually more than one) m.

Original entry on oeis.org

5, 11, 59, 419, 839, 232792559, 5354228879, 2329089562799, 144403552893599, 442720643463713815199, 591133442051411133755680799, 69720375229712477164533808935312303556799

Offset: 1

Labos Elemer, Nov 08 2000

nonn

Prime numbers of the form A051451(k)-1.
Primes p = LCM(1,...,m) - 1, where m is a prime power. (The prime powers give all values, and do so uniquely.) - Daniel Forgues, Apr 28 2014
This unique prime power m associated with a(n) is m = A385564(n). - Jeppe Stig Nielsen, Jul 09 2025

			232792559 + 1 = LCM(1,...,m) for m = 19, 20, 21, 22.

Amiram Eldar, Table of n, a(n) for n = 1..23

Cf. A003418, A051451, A049536, A049537, A385564.

Select[FoldList[LCM, Select[Range[100], PrimePowerQ]] - 1, PrimeQ] (* Amiram Eldar, Aug 18 2024 *)

L=1; for(n=2,1e3,if(isprimepower(n,&p) && ispseudoprime((L*=p)-1), print1(L-1", "))) \\ Charles R Greathouse IV, Apr 28 2014

A057825 Values of k for which A003418(k) - 1 is prime.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 19, 20, 21, 22, 23, 24, 29, 30, 32, 33, 34, 35, 36, 47, 48, 61, 62, 63, 97, 98, 99, 100, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 233, 234, 235, 236, 237, 238, 307, 308, 309, 310, 401, 402, 403, 404, 405, 406, 407, 408, 887, 888, 889, 890

Offset: 1

Labos Elemer, Nov 08 2000

nonn

Fewer distinct primes than distinct values of a(n) are generated. So, e.g., k = 97, 98, 99, 100 all correspond to lcm([1..97]) - 1 = 69720375229712477164533808935312303556799, a prime.

Jinyuan Wang, Table of n, a(n) for n = 1..125

Cf. A003418, A049536, A049537, A051451.

isok(k) = ispseudoprime(lcm(vector(k, i, i))-1); \\ Jinyuan Wang, May 02 2020

A048148 Distinct elements of A045948.

Original entry on oeis.org

1, 2, 4, 12, 24, 120, 360, 720, 5040, 10080, 30240, 332640, 1663200, 3326400, 43243200, 129729600, 259459200, 4410806400, 30875644800, 586637251200, 1173274502400, 26985313555200, 134926567776000, 404779703328000, 11738611396512000, 363896953291872000

Offset: 1

Labos Elemer

nonn

A001222(a(n)) = n - 1. - Eric Desbiaux, Apr 05 2018

Chai Wah Wu, Table of n, a(n) for n = 1..412 (terms 1..237 from Michael De Vlieger)

Cf. A003418, A034386, A002110, A000142, A049536.

Map[Function[m, Exp[Sum[MangoldtLambda@ n, {n, m}]]/Product[x, {x, Prime@ Range@ PrimePi@ m}]], Select[Range[10^3], Or[# == 1, And[PrimePowerQ@ #, ! PrimeQ@ #]] &] ] (* Michael De Vlieger, Aug 01 2017, after Fred Daniel Kline at A045948 *)

# uses program from A025476
from math import prod
def A048148(n): return prod(A025476(i) for i in range(1,n)) # Chai Wah Wu, Aug 15 2024

A025476(n) = a(n+1) / a(n). - Eric Desbiaux, Jun 22 2013

More terms from James Sellers, May 03 2000

More terms from Michael De Vlieger, Aug 01 2017

A051452 a(n) = 1 + lcm(1..k) where k is the n-th prime power A000961(n).

Original entry on oeis.org

2, 3, 7, 13, 61, 421, 841, 2521, 27721, 360361, 720721, 12252241, 232792561, 5354228881, 26771144401, 80313433201, 2329089562801, 72201776446801, 144403552893601, 5342931457063201, 219060189739591201

Offset: 1

Labos Elemer

nonn

From Daniel Forgues, Apr 27 2014: (Start)
Factorizations:
  2, 3, 7, 13, 61, 421 are primes;
  841 = 29^2;
  2521 is prime;
  27721 = 19*1459, 360361 = 89*4049, 720721 = 71*10151,
  12252241 = 1693*7237;
  232792561 is prime;
  5354228881 = 6659*804059;
  26771144401 is prime;
  80313433201 = 331*11239*21589, 2329089562801 = 101*271*2311*36821;
  72201776446801 is prime.
Very likely contains an infinite number of primes (see A049536). (End)

1 + A003418(A000961(n)), corresponds to distinct values of 1 + A003418.

Cf. A049536, A049537, A051454, A208768.

print1(2);t=1;for(n=2,100,if(t%n, t=lcm(t,n); print1(", "t+1))) \\ Charles R Greathouse IV, Jan 04 2013

from math import prod
from sympy import primepi, integer_nthroot, integer_log, primerange
def A051452(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return 1+prod(p**integer_log(m, p)[0] for p in primerange(m+1)) # Chai Wah Wu, Aug 15 2024

A051454 a(n) is the smallest prime factor of 1 + lcm(1..k) where k is the n-th prime power A000961(n).

Original entry on oeis.org

2, 3, 7, 13, 61, 421, 29, 2521, 19, 89, 71, 1693, 232792561, 6659, 26771144401, 331, 101, 72201776446801, 1801, 173, 54941, 89, 442720643463713815201, 593, 5171, 239, 1222615931, 103, 7265496855919, 6562349363, 4447, 147099357127, 1931

Offset: 1

Labos Elemer

nonn

			1 + lcm(1..8) = 29^2, so its smallest prime divisor is 29; it occurs as the 7th term in the sequence because 8 is the 7th prime power: A000961(7) = 8.

Sean A. Irvine, Table of n, a(n) for n = 1..82

Cf. A000961, A003418, A049536, A049537, A051301, A051342, A051452.

a:=[]; lcm:=1; for k in [1..83] do if (k eq 1) or IsPrimePower(k) then lcm:=Lcm(lcm,k); a:=a cat [Factorization(1+lcm)[1][1]]; end if; end for; a; // Jon E. Schoenfield, May 28 2018

Join[{2},With[{ppwr=Select[Range[200],PrimePowerQ]},Table[FactorInteger[LCM@@Take[ ppwr,n]+ 1][[1,1]],{n,40}]]] (* Harvey P. Dale, May 28 2024 *)

a(n) = {my(nb = 1, lc = 1, k = 2); while (nb != n, if (isprimepower(k), nb++; lc = lcm(lc, k)); k++;); vecmin(factor(lc +1)[,1]);} \\ Michel Marcus, May 29 2018

from math import prod
from sympy import primepi, integer_nthroot, integer_log, primerange, primefactors
def A051454(n):
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return min(primefactors(1+prod(p**integer_log(m, p)[0] for p in primerange(m+1)))) # Chai Wah Wu, Aug 15 2024

A051453 Prime powers such that 1 + lcm(1,2,...,p^w) is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 19, 25, 31, 47, 89, 127, 139, 1369, 2251, 3271, 4253, 4373, 4549, 5449, 13331, 37123, 55291, 79633, 105653

Offset: 1

Labos Elemer

			1 + lcm(1,2,..,89) = 718766754945489455304472257065075294401 is prime.

Cf. A000961, A003418, A049536, A049537.

ispp(n) = isprimepower(n) || n==1;
lista(nn) = {for (n=1, nn, if (ispp(n) && ispseudoprime(1+lcm([1..n])), print1(n, ", ")););} \\ Michel Marcus, Aug 26 2019

Corrected and extended by Wouter Meeussen, Apr 18 2003
a(18)-a(22) from Jinyuan Wang, May 02 2020
a(23)-a(24) from Michael S. Branicky, Apr 22 2023
a(25)-a(26) from Michael S. Branicky, Jul 04 2025

A053608 Numbers x = LCM(1,2,...,k) such that x^2 + 1 is prime.

Original entry on oeis.org

1, 2, 6, 420, 360360, 718766754945489455304472257065075294400

Offset: 1

Labos Elemer, Feb 09 2000

The next term has k > 10^4, if it exists. - Amiram Eldar, Aug 23 2024

Cf. A003418, A049536, A046029, A051418, A051451, A051739, A053609.

Select[FoldList[LCM, 1, Select[Range[100], PrimePowerQ]], PrimeQ[#^2 + 1] &] (* Amiram Eldar, Aug 23 2024 *)

cp's OEIS Frontend

A051451 a(n) = lcm{ 1,2,...,x } where x is the n-th prime power (A000961).

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A049537 Values of k for which A075059(k) = A003418(k) + 1 is prime.

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A045948 a(n) = A003418(n)/A034386(n).

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A057824 Primes p such that p+1 is LCM(1,...,m) for some (usually more than one) m.

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A057825 Values of k for which A003418(k) - 1 is prime.

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A048148 Distinct elements of A045948.

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A051452 a(n) = 1 + lcm(1..k) where k is the n-th prime power A000961(n).

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