cp's OEIS Frontend

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.

Showing 1-7 of 7 results.

A329905 a(1) = 0; a(2) = 1; and for n > 2, a(n) = A330682(n) + 2*a(A329904(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 12, 7, 16, 5, 24, 14, 32, 10, 48, 28, 64, 20, 13, 96, 15, 9, 56, 128, 40, 26, 192, 30, 18, 112, 256, 80, 52, 384, 60, 36, 11, 224, 512, 25, 160, 29, 17, 104, 768, 120, 72, 22, 448, 1024, 50, 320, 31, 58, 34, 208, 1536, 240, 144, 44, 896, 2048, 100, 640, 62, 116, 68, 21, 416, 3072, 27, 49, 480, 288, 88, 57
Offset: 1

Views

Author

Antti Karttunen, Dec 24 2019

Keywords

Comments

Note the indexing: domain begins from one, but the range contains also zero.

Crossrefs

Programs

Formula

a(1) = 0; a(2) = 1; and for n > 2, if A181815(n) is odd, a(n) = 1 + 2*a(A329904(n)), otherwise a(n) = 2*a(A329904(n)).
a(n) = A243071(A181815(n)).
For all n >= 1, A000120(a(n)) = A061394(n).
For all n >= 2, A070939(a(n)) = A329907(n).

A329907 Number of iterations of A329904 needed to reach 1.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 4, 3, 5, 3, 5, 4, 6, 4, 6, 5, 7, 5, 4, 7, 4, 4, 6, 8, 6, 5, 8, 5, 5, 7, 9, 7, 6, 9, 6, 6, 4, 8, 10, 5, 8, 5, 5, 7, 10, 7, 7, 5, 9, 11, 6, 9, 5, 6, 6, 8, 11, 8, 8, 6, 10, 12, 7, 10, 6, 7, 7, 5, 9, 12, 5, 6, 9, 9, 7, 6, 11, 6, 13, 8, 11, 7, 8, 8, 6, 10, 13, 6, 7, 10, 10, 6, 8, 7, 12, 7, 14, 9, 12, 8, 9, 9, 7, 11
Offset: 1

Views

Author

Antti Karttunen, Dec 24 2019

Keywords

Comments

Equally, starting from A025487(n), number of iterations of A329899 needed to reach 1.
Any k > 0 occurs 2^(k-1) times in total in this sequence.

Crossrefs

Programs

Formula

a(1) = 0; for n > 1, a(n) = 1 + a(A329904(n)).
a(1) = 0; for n > 1, a(n) = A070939(A329905(n)).
a(n) = A252464(A181815(n)).
For all n >= 1, a(n) >= A061394(n).

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

Original entry on oeis.org

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

Views

Author

Keywords

Comments

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

Examples

			The first few terms are 1, 2, 2^2, 2*3, 2^3, 2^2*3, 2^4, 2^3*3, 2*3*5, ...
		

Crossrefs

Subsequence of A055932, A191743, and A324583.
Cf. A085089, A101296 (left inverses).
Equals range of values taken by A046523.
Cf. A178799 (first differences), A247451 (squarefree kernel), A146288 (number of divisors).
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.

Programs

  • Haskell
    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
    
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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
    
  • PARI
    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 */
    
  • PARI
    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
    
  • PARI
    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
    
  • PARI
    \\ 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
    
  • Sage
    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

Formula

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

Extensions

Offset corrected by Matthew Vandermast, Oct 19 2008
Minor correction by Charles R Greathouse IV, Sep 03 2010

A181815 a(n) = largest integer such that, when any of its divisors divides A025487(n), the quotient is a member of A025487.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 12, 5, 32, 9, 24, 10, 64, 18, 48, 20, 128, 36, 15, 96, 7, 27, 40, 256, 72, 30, 192, 14, 54, 80, 512, 144, 60, 384, 28, 108, 25, 160, 1024, 45, 288, 21, 81, 120, 768, 56, 216, 50, 320, 2048, 90, 576, 11, 42, 162, 240, 1536, 112, 432, 100, 640, 4096, 180, 1152
Offset: 1

Views

Author

Matthew Vandermast, Nov 30 2010

Keywords

Comments

A permutation of the natural numbers.
The number of divisors of a(n) equals the number of ordered factorizations of A025487(n) as A025487(j)*A025487(k). Cf. A182762.
From Antti Karttunen, Dec 28 2019: (Start)
Rearranges terms of A108951 into ascending order, as A108951(a(n)) = A025487(n).
The scatter plot looks like a curtain of fractal spray, which is typical for many of the so-called "entanglement permutations". Indeed, according to the terminology I use in my 2016-2017 paper, this sequence is obtained by entangling the complementary pair (A329898, A330683) with the complementary pair (A005843, A003961), which gives the following implicit recurrence: a(A329898(n)) = 2*a(n) and a(A330683(n)) = A003961(a(n)). An explicit form is given in the formula section.
(End)

Examples

			For any divisor d of 9 (d = 1, 3, 9), 36/d (36, 12, 4) is a member of A025487. 9 is the largest number with this relationship to 36; therefore, since 36 = A025487(11), a(11) = 9.
		

Crossrefs

If this sequence is considered the "primorial deflation" of A025487(n) (see first formula), the primorial inflation of n is A108951(n), and the primorial inflation of A025487(n) is A181817(n).
A181820(n) is another mapping from the members of A025487 to the positive integers.

Programs

  • Mathematica
    (* First, load the program at A025487, then: *)
    Map[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[g, ReplacePart[Table[0, {PrimePi[g[[-1, 1]]]}], #] &@ Map[PrimePi@ First@ # -> Last@ # &, g]]@ FactorInteger@ #], # > 0 &]}}, And[FreeQ[#[[-1, -1]], 0], Length[#[[-1, -1]] ] != 0] &][[All, 1]] ] &, Union@ Flatten@ f@ 6] (* Michael De Vlieger, Dec 28 2019 *)
  • PARI
    A181815(n) = A329900(A025487(n)); \\ Antti Karttunen, Dec 24 2019

Formula

If A025487(n) is considered in its form as Product A002110(i)^e(i), then a(n) = Product p(i)^e(i). If A025487(n) is instead considered as Product p(i)^e(i), then a(n) = Product (p(i)/A008578(i))^e(i).
a(n) = A122111(A181820(n)). - Matthew Vandermast, May 21 2012
From Antti Karttunen, Dec 24-29 2019: (Start)
a(n) = Product_{i=1..A051282(n)} A000040(A304886(i)).
a(n) = A329900(A025487(n)) = A319626(A025487(n)).
a(n) = A163511(A329905(n)).
For n > 1, if A330682(n) = 1, then a(n) = A003961(a(A329904(n))), otherwise a(n) = 2*a(A329904(n)).
A252464(a(n)) = A329907(n).
A330690(a(n)) = A050378(n).
a(A306802(n)) = A329902(n).
(End)

A329898 a(n) is the position of 2*A025487(n) in A025487.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 42, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 71, 74, 75, 76, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 94, 95, 96, 97, 98, 99, 100
Offset: 1

Views

Author

Antti Karttunen, Dec 24 2019

Keywords

Comments

Numbers k for which A007814(A025487(k)) > A007949(A025487(k)), i.e., numbers k for which the 2-adic valuation of A025487(k) is larger than its 3-adic valuation.
Numbers k for which A181815(k) is even.

Crossrefs

Cf. A329897 (complement), A330683 (and its permutation).
Cf. A007814, A007949, A025487, A329904 (a left inverse), A329906.
Positions of even terms in A181815, zeros in A330682.

Programs

  • Mathematica
    (* First, load the function f at A025487, then: *)
    With[{s = Union@ Flatten@ f@ 6}, Map[If[2 # > Max@ s, Nothing, FirstPosition[s, 2 #][[1]] ] &, s]] (* Michael De Vlieger, Jan 11 2020 *)
  • PARI
    upto_e = 64; \\ 64 -> 43608 terms.
    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
    A329898list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, v025487); 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++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,oo,if(!(t=vecsearch(v025487,2*(v025487[i]))),return(Vec(lista)), listput(lista,t))); };
    v329898 = A329898list(upto_e);
    A329898(n) = v329898[n];

Formula

For all n >= 1, A329904(a(n)) = n.

A330683 a(n) is the position of A283980(A025487(n)) in A025487.

Original entry on oeis.org

1, 4, 11, 9, 23, 20, 44, 41, 22, 79, 38, 73, 43, 131, 69, 124, 77, 212, 118, 72, 201, 54, 110, 129, 327, 191, 123, 312, 93, 181, 209, 493, 300, 199, 474, 154, 286, 128, 324, 725, 190, 454, 147, 272, 310, 697, 245, 434, 208, 490, 1044, 299, 671, 114, 232, 416, 469, 1008, 374, 646, 321, 721, 1481, 451, 974, 186, 359
Offset: 1

Views

Author

Antti Karttunen, Dec 26 2019

Keywords

Crossrefs

Permutation of A329897.
Cf. A025487, A085089, A101296, A181815, A283980, A329898 (positive integers not in this sequence), A329904 (a left inverse), A329906, A330681.

Programs

  • Mathematica
    (* First, load the function f at A025487, then: *)
    With[{s = Union@ Flatten@ f@ 10}, TakeWhile[#, # != 0 &] &@ Map[If[# > Max@ s, 0, FirstPosition[s, #][[1]] ] &[(Times @@ Map[#1^#2 & @@ # &, FactorInteger[#] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}] - Boole[# == 1])*2^IntegerExponent[#, 2]] &, s]] (* Michael De Vlieger, Jan 11 2020 *)
  • PARI
    upto_e = 101;
    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
    A330683list(e) = { my(lista = List([1, 2]), i=2, u = 2^e, t, v025487); 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++); v025487 = vecsort(Vec(lista)); lista = List([]); for(i=1,oo,if(!(t=vecsearch(v025487,A283980(v025487[i]))),return(Vec(lista)), listput(lista,t))); };
    v330683 = A330683list(upto_e);
    A330683(n) = v330683[n];

Formula

For all n >= 1, A329904(a(n)) = n.

A329899 If A181815(n) is odd, a(n) = A064989(A025487(n)), otherwise a(n) = A025487(n)/2.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 8, 12, 6, 16, 4, 24, 30, 32, 36, 48, 60, 64, 72, 12, 96, 30, 8, 120, 128, 144, 180, 192, 210, 216, 240, 256, 288, 360, 384, 420, 432, 36, 480, 512, 24, 576, 60, 16, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 210, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2310, 2520, 2592, 72
Offset: 1

Views

Author

Antti Karttunen, Dec 24 2019

Keywords

Comments

If 2-adic and 3-adic valuations of A025487(n) are equal, then a(n) = A064989(A025487(n)), otherwise a(n) = A025487(n)/2.
Only terms of A025487 occur, and each one of them occurs exactly twice.

Crossrefs

Programs

Formula

If A181815(n) is odd, a(n) = A064989(A025487(n)), otherwise a(n) = A025487(n)/2.
a(n) = A025487(A329904(n)).
Showing 1-7 of 7 results.