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.

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A049417 a(n) = isigma(n): sum of infinitary divisors of n.

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 15, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 60, 26, 42, 40, 40, 30, 72, 32, 51, 48, 54, 48, 50, 38, 60, 56, 90, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 120, 72, 120, 80, 90, 60, 120, 62, 96, 80, 85, 84, 144, 68, 90
Offset: 1

Views

Author

Yasutoshi Kohmoto, Dec 11 1999

Keywords

Comments

A divisor of n is called infinitary if it is a product of divisors of the form p^{y_a 2^a}, where p^y is a prime power dividing n and sum_a y_a 2^a is the binary representation of y.
This sequence is an infinitary analog of the Dedekind psi function A001615. Indeed, a(n) = Product_{q in Q_n}(q+1) = n*Product_{q in Q_n} (1+1/q), where {q} are terms of A050376 and Q_n is the set of distinct q's whose product is n. - Vladimir Shevelev, Apr 01 2014
1/a(n) is the asymptotic density of numbers that are infinitarily divided by n (i.e., numbers whose set of infinitary divisors includes n). - Amiram Eldar, Jul 23 2025

Examples

			If n = 8: 8 = 2^3 = 2^"11" (writing 3 in binary) so the infinitary divisors are 2^"00" = 1, 2^"01" = 2, 2^"10" = 4 and 2^"11" = 8; so a(8) = 1+2+4+8 = 15.
n = 90 = 2*5*9, where 2, 5, 9 are in A050376; so a(n) = 3*6*10 = 180. - _Vladimir Shevelev_, Feb 19 2011

Links

Crossrefs

Cf. A004607, A037445, A050376, A077609, A091732, A127661, A293355.
Cf. A049418 (3-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).
Cf. A000079, A001615, A027748, A030308, A124010, A126168.

Programs

  • Haskell
    a049417 1 = 1
a049417 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = product $ zipWith div
           (map (subtract 1 . (p ^)) $
                zipWith (*) a000079_list $ map (+ 1) $ a030308_row e)
           (map (subtract 1 . (p ^)) a000079_list)
-- Reinhard Zumkeller, Sep 18 2015
  • Maple
    isidiv := proc(d, n)
    local n2, d2, p, j;
    if n mod d <> 0 then
        return false;
    end if;
    for p in numtheory[factorset](n) do
        padic[ordp](n,p) ;
        n2 := convert(%, base, 2) ;
        padic[ordp](d,p) ;
        d2 := convert(%, base, 2) ;
        for j from 1 to nops(d2) do
            if op(j, n2) = 0 and op(j, d2) <> 0 then
                return false;
            end if;
        end do:
    end do;
    return true;
end proc:
idivisors := proc(n)
    local a, d;
    a := {} ;
    for d in numtheory[divisors](n) do
        if isidiv(d, n) then
            a := a union {d} ;
        end if;
    end do:
    a ;
end proc:
A049417 := proc(n)
    local d;
    add(d, d=idivisors(n)) ;
end proc:
seq(A049417(n),n=1..100) ; # R. J. Mathar, Feb 19 2011
  • Mathematica
    bitty[k_] := Union[Flatten[Outer[Plus, Sequence @@ ({0, #1} & ) /@ Union[2^Range[0, Floor[Log[2, k]]]*Reverse[IntegerDigits[k, 2]]]]]]; Table[Plus@@((Times @@ (First[it]^(#1 /. z -> List)) & ) /@ Flatten[Outer[z, Sequence @@ bitty /@ Last[it = Transpose[FactorInteger[k]]], 1]]), {k, 2, 120}]
(* Second program: *)
a[n_] := If[n == 1, 1, Sort @ Flatten @ Outer[ Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] // Total;
Array[a, 100] (* Jean-François Alcover, Mar 23 2020, after Paul Abbott in A077609 *)
  • PARI
    A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[,2], b = binary(f[k,2]); prod(j=1, #b, if(b[j], 1+f[k,1]^(2^(#b-j)), 1)))} \\ Andrew Lelechenko, Apr 22 2014
  • PARI
    isigma(n)=vecprod([vecprod([f[1]^2^k+1|k<-[0..exponent(f[2])], bittest(f[2],k)])|f<-factor(n)~]) \\ M. F. Hasler, Oct 20 2022
  • Python
    from math import prod
from sympy import factorint
def A049417(n): return prod(p**(1<Chai Wah Wu, Jul 11 2024

Formula

Multiplicative: If e = Sum_{k >= 0} d_k 2^k (binary representation of e), then a(p^e) = Product_{k >= 0} (p^(2^k*{d_k+1}) - 1)/(p^(2^k) - 1). - Christian G. Bower and Mitch Harris, May 20 2005 [This means there is a factor p^2^k + 1 if d_k = 1, otherwise the factor is 1. - M. F. Hasler, Oct 20 2022]
Let n = Product(q_i) where {q_i} is a set of distinct terms of A050376. Then a(n) = Product(q_i + 1). - Vladimir Shevelev, Feb 19 2011
If n is squarefree, then a(n) = A001615(n). - Vladimir Shevelev, Apr 01 2014
a(n) = Sum_{k>=1} A077609(n,k). - R. J. Mathar, Oct 04 2017
a(n) = A126168(n)+n. - R. J. Mathar, Oct 05 2017
Multiplicative with a(p^e) = Product{k >= 0, e_k = 1} p^2^k + 1, where e = Sum e_k 2^k, i.e., e_k is bit k of e. - M. F. Hasler, Oct 20 2022
a(n) = iphi(n^2)/iphi(n), where iphi(n) = A091732(n). - Amiram Eldar, Sep 21 2024

Extensions

More terms from Wouter Meeussen, Sep 02 2001

A127661 Lengths of the infinitary aliquot sequences.

Original entry on oeis.org

2, 3, 3, 3, 3, 1, 3, 4, 3, 5, 3, 5, 3, 6, 4, 3, 3, 6, 3, 6, 4, 7, 3, 8, 3, 4, 4, 6, 3, 6, 3, 4, 5, 7, 4, 7, 3, 8, 4, 8, 3, 5, 3, 4, 5, 5, 3, 7, 3, 7, 5, 7, 3, 4, 4, 6, 4, 5, 3, 1, 3, 8, 4, 5, 4, 3, 3, 8, 5, 10, 3, 3, 3, 9, 4, 9, 4, 2, 3, 8, 3, 5, 3, 10, 4, 6, 6, 8, 3, 1, 5, 7, 5, 8, 4, 9, 3, 8, 5, 7
Offset: 1

Views

Author

Ant King, Jan 26 2007

Keywords

Comments

An infinitary aliquot sequence is defined by the map x->A049417(x)-x. The map usually terminates with a zero, but may enter cycles (if n in A127662 for example).
The length of an infinitary aliquot sequence is defined to be the length of its transient part + the length of its terminal cycle.
The value of a(840) starting the infinitary aliquot sequence 840 -> 2040 -> 4440 -> 9240 -> 25320,... is >1500. - R. J. Mathar, Oct 05 2017

Examples

			a(4)=3 because the infinitary aliquot sequence generated by 4 is 4 -> 1 -> 0 and it has length 3.
a(6) = 1 because 6 -> 6 -> 6 ->... enters a cycle after 1 term.
a(8) = 4 because 8 -> 7 -> 1 -> 0 terminates after 4 terms.
a(30) = 6 because 30 ->42 -> 54 -> 66 -> 78 -> 90 -> 90 -> 90 -> ...enters a cycle after 6 terms.
a(126)=2 because 126 -> 114 -> 126 enters a cycle after 2 terms.

Links

Crossrefs

Cf. A126168, A127662 - A127667, A293355, A098007.

Programs

  • Maple
    # Uses code snippets of A049417
A127661 := proc(n)
    local trac,x;
    x := n ;
    trac := [x] ;
    while true do
        x := A049417(x)-trac[-1] ;
        if x = 0 then
            return 1+nops(trac) ;
        elif x in trac then
            return nops(trac) ;
        end if;
        trac := [op(trac),x] ;
    end do:
end proc:
seq(A127661(n),n=1..100) ; # R. J. Mathar, Oct 05 2017
  • Mathematica
    ExponentList[n_Integer,factors_List]:={#,IntegerExponent[n,# ]}&/@factors;InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f,g}, BitOr[f,g]==g][ #,Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #,factors]&/@d]],?(And@@#&),{1}]] ]] ]; properinfinitarydivisorsum[k]:=Plus@@InfinitaryDivisors[k]-k;g[n_] := If[n > 0,properinfinitarydivisorsum[n], 0];iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];Length[iTrajectory[ # ]] &/@ Range[100]
(* Second program: *)
A049417[n_] := If[n == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] // Total;
A127661[n_] := Module[{trac, x}, x = n; trac = {x}; While[True, x = A049417[x] - trac[[-1]]; If[x == 0, Return[1 + Length[trac]], If[MemberQ[trac, x], Return[Length[trac]]]]; trac = Append[trac, x]]];
Table[A127661[n], {n, 1, 100}] (* Jean-François Alcover, Aug 28 2023, after R. J. Mathar *)

A361421 Infinitary aliquot sequence starting at 840: a(1) = 840, a(n) = A126168(a(n-1)), for n >= 2.

Original entry on oeis.org

840, 2040, 4440, 9240, 25320, 51000, 117480, 271320, 765480, 1531320, 3721800, 5956440, 12295560, 25086840, 54141960, 108284280, 250301640, 502213560, 1007626440, 2017856760, 4039750920, 8079502200, 19596145800, 44369345400, 71495068200, 115576350360, 231152701080
Offset: 1

Views

Author

Amiram Eldar, Mar 11 2023

Keywords

Comments

First differs from A045477 at n = 12.
840 is the least number whose infinitary aliquot sequence is not known to be finite or eventually periodic.
R. J. Mathar found that this sequence does not reach 0 or enter a cycle before the 1500th term (see A127661). This limit was extended to beyond the 3000th term (see the b-file).

Examples

			a(1) = 840 by definition.
a(2) = A126168(a(1)) = A126168(840) = 2040.
a(3) = A126168(a(2)) = A126168(2040) = 4440.

Links

Crossrefs

Cf. A008892, A045477, A126168, A127661, A293355.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]];
infs[n_] := If[n==1, 1, Times @@ f @@@ FactorInteger[n]] - n; infs[0] = 0;
seq[len_, init_] := NestWhileList[infs, init, UnsameQ, All, len];
seq[27, 840]
  • PARI
    s(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], f[i, 1]^(2^(#b-k)) + 1, 1))) - n; }
lista(nmax) = {my(k = 840); for(n = 1, nmax, print1(k, ", "); if(k == 0, break); k = s(k)); }
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