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.
The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...
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
1) 1680 has more divisors than any smaller positive integer; thus for all m < n, d(1680^1) > d(m^1). 2) Since the exponents in 1680's prime factorization are (4,1,1,1), the k-th power of 1680 has (4k+1)(k+1)^3 = 4k^4 + 13k^3 + 15k^2 + 7k + 1 divisors. Comparison with the analogous formulas for all smaller members of A025487 shows the following: a) No number smaller than 1680 has a positive coefficient in its "power formula" for any exponent larger than k^4. b) The only power formula with a k^4 coefficient as high as 4 is that for 1260 (4k^4 + 12k^3 + 13k^2 + 6k + 1). c) The k^3 coefficient for 1680 is higher than for 1260. So for all sufficiently high values of k, d(1680^k) > d(m^k) for all m < 1680. 3) Careful comparison of 1680's "power formula" with the analogous formulas for smaller members of A025487 shows that no intermediate value of k can exist for which d(m^k) >= d(1680^k) if m < 1680.
Table[Range[Prime[n] - 1], {n, 9}] // Flatten (* or, per title definition: *)
#/Times @@ (FactorInteger[#][[All, 1]]) & /@ Flatten@ Table[Range[Prime[n + 1] - 1] Apply[Times, Prime@ Range@ n], {n, 0, 8}] (* Michael De Vlieger, Jul 26 2016 *)
A282472 := proc(n) option remember; local a,a1,rec ; if n = 1 then 1; else a1 := procname(n-1) ; rec := numtheory[tau](a1^2)/numtheory[tau](a1) ; for a from a1+1 do if numtheory[tau](a^2)/numtheory[tau](a) > rec then return a; end if; end do: end if; end proc: # R. J. Mathar, Mar 03 2017
s={}; rm=0; Do[r=DivisorSigma[0, n^2]/DivisorSigma[0, n]; If[r > rm, rm = r; AppendTo[s, n]], {n, 1, 10^4}]; s (* Amiram Eldar, Jul 17 2019 *)
lista(nn) = {rec = 0; for (n=1, nn, if ((newrec = numdiv(n^2)/numdiv(n)) > rec, rec = newrec; print1(n, ", ")););} \\ Michel Marcus, Feb 20 2017
use ntheory qw(:all);
for (my ($n, $m) = (1, 0) ; ; ++$n) {
my $d = divisors($n*$n) / divisors($n);
if ($m < $d) {
$m = $d;
print "$n\n";
}
}
a[0]=1; a[n_]:= a[n]=Prime[n]*LCM[a[n - 1],Prime[n] -1];
Table[a[i], {i, 1, 18}]
a(n)=if(n,my(v=primes(n),e=vector(n,i,1),f,j); forprime(p=2,v[n], f=factor(p-1); for(i=1,#f~,if(f[i,2]>1,j=primepi(f[i,1]);e[j]=max(e[j],f[i,2])))); prod(i=1,n,v[i]^e[i]),1) \\ Charles R Greathouse IV, Jun 28 2013
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
Comments