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.
A025487(5) = 8 and A181822(5) = 30 have the prime signatures (3) and (1,1,1) respectively. 8 is the smaller member of the pair and is therefore included in this sequence.
A025487(9) = 30 and A181822(9) = 8 have the prime signatures (1,1,1) and (3) respectively. 30 is the larger member of the pair and is therefore included in this sequence.
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
If p,q,... are different primes, a(p)=2, a(p^2)=4, a(pq)=6, a(p^2*q)=12, etc. n = 108 = 2*2*3*3*3 is replaced by a(n) = 2*2*2*3*3 = 72; n = 105875 = 5*5*5*7*11*11 is represented by a(n) = 2*2*2*3*3*5 = 360. Prime-powers are replaced by corresponding powers of 2, primes by 2. Factorials, primorials and lcm[1..n] are in the sequence. A000005(a(n)) = A000005(n) remains invariant; least and largest prime factors of a(n) are 2 or p[A001221(n)] resp.
import Data.List (sort)
a046523 = product .
zipWith (^) a000040_list . reverse . sort . a124010_row
-- Reinhard Zumkeller, Apr 27 2013
a:= n-> (l-> mul(ithprime(i)^l[i][2], i=1..nops(l)))
(sort(ifactors(n)[2], (x, y)->x[2]>y[2])):
seq(a(n), n=1..100); # Alois P. Heinz, Aug 18 2014
Table[Apply[Times, p[w]^Reverse[Sort[ex[w]]]], {w, 1, 1000}] p[x_] := Table[Prime[w], {w, 1, lf[x]}] ex[x_] := Table[Part[ffi[x], 2*w], {w, 1, lf[x]}] ffi[x_] := Flatten[FactorInteger[x]] lf[x_] := Length[FactorInteger[x]]
ps[n_] := Sort[Last /@ FactorInteger[n]]; Join[{1}, Table[i = 2; While[ps[n] != ps[i], i++]; i, {n, 2, 89}]] (* Jayanta Basu, Jun 27 2013 *)
a(n)=my(f=vecsort(factor(n)[,2],,4),p);prod(i=1,#f,(p=nextprime(p+1))^f[i]) \\ Charles R Greathouse IV, Aug 17 2011
A046523(n)=factorback(primes(#n=vecsort(factor(n)[,2],,4)),n) \\ M. F. Hasler, Oct 12 2018, improved Jul 18 2019
from sympy import factorint
def P(n):
f = factorint(n)
return sorted([f[i] for i in f])
def a(n):
x=1
while True:
if P(n) == P(x): return x
else: x+=1 # Indranil Ghosh, May 05 2017
from math import prod from sympy import factorint, prime def A046523(n): return prod(prime(i+1)**e for i,e in enumerate(sorted(factorint(n).values(),reverse=True))) # Chai Wah Wu, Feb 04 2022
a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24 a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).
a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)
primorial(n)=prod(i=1,primepi(n),prime(i)) a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015
from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
f = factorint(n)
return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017
def sharp_primorial(n): return sloane.A002110(prime_pi(n)) def p(f): return sharp_primorial(f[0])^f[1] [prod(p(f) for f in factor(n)) for n in range (1,51)] # Giuseppe Coppoletta, Feb 07 2015
The canonical factorization of 24 is 2^3*3^1. Therefore, p(e(i)) = prime(3)*prime(1)(i.e., A000040(3)*A000040(1)), which equals 5*2 = 10. Since 24 is the smallest integer for which p(e(i)) = 10, a(10) = 24.
a:= n-> (l-> mul(ithprime(i)^l[i], i=1..nops(l)))(sort(map(i->
numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..70); # Alois P. Heinz, Sep 05 2018
With[{s = Array[If[# == 1, 1, Times @@ Map[Prime@ Last@ # &, FactorInteger@ #]] &, 2^16]}, Array[First@ FirstPosition[s, #] &, LengthWhile[Differences@ Union@ s, # == 1 &]]] (* Michael De Vlieger, Dec 17 2018 *)
Table[Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,30}] (* Gus Wiseman, Jan 02 2019 *)
A181821(n) = { my(f=factor(n),p=0,m=1); forstep(i=#f~,1,-1,while(f[i,2], f[i,2]--; m *= (p=nextprime(p+1))^primepi(f[i,1]))); (m); }; \\ Antti Karttunen, Dec 10 2018
from math import prod from sympy import prime, primepi, factorint def A181821(n): return prod(prime(i)**e for i, e in enumerate(sorted(map(primepi,factorint(n,multiple=True)),reverse=True),1)) # Chai Wah Wu, Sep 15 2023
import Data.List (elemIndex); import Data.Maybe (fromJust) a061394 = fromJust . (`elemIndex` a002110_list) . a247451 -- Reinhard Zumkeller, Sep 17 2014
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))) [omega(n) | n <- [1..1000], isA025487(n)] \\ Or, for older versions: apply(omega, select(isA025487, [1..1000])) \\ Charles R Greathouse IV, Nov 07 2014
The sequence can be represented as a binary tree:
1
|
...................2...................
6 4
36......../ \........30 12......../ \........8
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
216 180 210 900 72 60 24 16
etc.
Both children are multiples of their common parent, see A323503, A323504 and A323507.
The value of a(n) is computed from the binary expansion of n as follows: Starting from the least significant end of the binary expansion of n (A007088), we record the successive run lengths, subtract one from all lengths except the first one, and use the reversed partial sums of these adjusted values as the exponents of successive primes.
For 11, which is "1011" in base 2, we have run lengths [2, 1, 1] when scanned from the right, and when one is subtracted from all except the first, we have [2, 0, 0], partial sums of which is [2, 2, 2], which stays same when reversed, thus a(11) = 2^2 * 3^2 * 5^2 = 900.
For 13, which is "1101" in base 2, we have run lengths [1, 1, 2] when scanned from the right, and when one is subtracted from all except the first, we have [1, 0, 1], partial sums of which is [1, 1, 2], reversed [2, 1, 1], thus a(13) = 2^2 * 3^1 * 5^1 = 60.
Sequence A227183 is based on the same algorithm.
{1}~Join~Array[Times @@ MapIndexed[Prime[First@ #2]^#1 &, Reverse@ Accumulate@ MapIndexed[Length[#1] - Boole[First@ #2 > 1] &, Split@ Reverse@ IntegerDigits[#, 2]]] &, 54] (* Michael De Vlieger, Feb 05 2020 *)
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));
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