This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(6)=4 because A095848(6)=24 and 24 has 2 and 3 as prime divisors; the smallest number in A095848 divisible by 3 is 6, and 24/6=4.
lcm[1,...,n] is 2520 for n=9 and 10. The smallest such n's are always prime powers, where A003418 jumps.
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
72 is a multiple of seven of the first nine positive integers (namely, 1, 2, 3, 4, 6, 8 and 9). It is the smallest positive integer for which this is true.
\\ Computes the first 50 terms of A094348 A094348() = {local(n,i,R,A,len,count,change,high,lim); lim = 7208000; R = vector(500); A = vector(50); A[1]=1; A[2]=2; A[3]=4; A[4]=6; A[5]=12; count=5; high=0; n=12; while(n < lim, d=divisors(n); len=length(d); change=0; for(i=1,min(len,high),if(R[i]>d[i],R[i]=d[i];change=1)); if(len>high,for(i=high+1,len,R[i]=d[i]); high=len); if(change, count++; A[count] = n); n += 12; ); write("A094348.txt", vector(count, i, A[i])); } \\ Peter Luschny, Dec 29 2010
After 6, none of 7,8,9,10 or 11 are in the sequence since they are not divisible by 1,2 and 3 as 6 is. 12 is a term, but is now divisible by 1,2,3 and 4, adding a new constraint on subsequent terms. After 24, 30 is not in the sequence because 24 is divisible by all numbers from 1 to 4 and 30 is not divisible by 4. But 36, which is divisible by all of 1 through 4, is a term. As an irregular table, the n-th row consists of all numbers divisible by A051451(n) but not by A051451(n+1). - _Tom Edgar_, May 22 2014
consecd(a) = {d = divisors(a); for (i=2, #d, if (d[i] - d[i-1] > 1, return(i-1));); return(a);}
findnext(a) = {nconsd = consecd(a); na = a + 1; while (consecd(na) < nconsd, na++); na;}
lista(nn) = {a = 1; print1(a, ", "); for (n=1, nn, a = findnext(a); print1(a, ", "););} \\ Michel Marcus, May 11 2014
first(n) = {
my(res = vector(n), step = 1, oldm = 1, newm = 1);
res[1] = 1;
for(i = 2, n,
while(res[i-1] % (newm+1) == 0,
newm++;
);
if(newm > oldm,
step = lcm([step, lcm([oldm..newm])]);
oldm = newm
);
res[i] = res[i-1]+step
);
res
} \\ David A. Corneth, Jan 28 2024
8 is a term of A_(2/3) and therefore of this sequence as well as A_x for x > 2/3, even though 8 is not a term of A384669, because 3^(2/3) (corresponding to 8) > 2 = 1^(2/3) + 1^(2/3) (corresponding to 6). Thus 8 qualifies to be a term in B_2, this sequence.
s(n, q) = my(f=factor(n)); sum(k=1, #f~, f[k, 2]^q); listaq(nn, q) = my(r=-oo, list=List()); for (n=1, nn, my(ss=s(n, q)); if (ss > r, r = ss; listput(list, n)); ); Vec(list); putlist(list, elems) = for (i=1, #elems, listput(list, elems[i])); list; lista(nn) = my(list=List(), vq=[1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/6]); for (i=1, #vq, list = putlist(list, listaq(nn, vq[i]))); Set(Vec(list)); \\ Michel Marcus, Jul 08 2025
f_(1/2)(24) = sqrt(3) + sqrt(1), because 24 = (2^3)(3^1). This is a record value for f_(1/2), so 24 is in the sequence. f_(1/2)(30) = sqrt(1) + sqrt(1) + sqrt(1) (because 30 = (2^1)(3^1)(5^1)), which is larger still, putting 30 in the sequence. However, f_(1/2)(32) = sqrt(5) (because 32 = 2^5), smaller than the previous value, so 32 is not in the sequence. g_216(x) = 3^x + 3^x, because 216 = (2^3)(3^3).
s[n_] := Total[Sqrt[FactorInteger[n][[;; , 2]]]]; s[1] = 0; With[{lps = Cases[Import[ "https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]}, sm = -1; seq = {}; Do[s1 = s[lps[[i]]]; If[s1 > sm, sm = s1; AppendTo[seq, lps[[i]]]], {i, 1, Length[lps]}]; seq] (* Amiram Eldar, Jun 08 2025 *)
s(n) = my(f=factor(n)); sum(k=1, #f~, sqrt(f[k,2])); lista(nn) = my(r=-oo, list=List()); for (n=1, nn, my(ss=s(n)); if (ss > r, r = ss; listput(list, n));); Vec(list); \\ Michel Marcus, Jun 15 2025
Once we get to 60, all succeeding terms in the sequence must be divisible by 1, 2, 3, 4, 5, and 6. However, not all multiples of 60 that are less than 420 (LCM(1,...,7)) have least prime signature, so 300 is excluded. All other multiples of 60 between 60 and 420 have least prime signature, so they are terms of this sequence.
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