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%I #107 Sep 21 2024 08:42:00
%S 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,
%T 42,40,40,30,72,32,51,48,54,48,50,38,60,56,90,42,96,44,60,60,72,48,68,
%U 50,78,72,70,54,120,72,120,80,90,60,120,62,96,80,85,84,144,68,90
%N a(n) = isigma(n): sum of infinitary divisors of n.
%C 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.
%C 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, else the factor is 1. - _M. F. Hasler_, Oct 20 2022]
%C 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
%H Amiram Eldar, <a href="/A049417/b049417.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..7417 from R. J. Mathar)
%H Graeme L. Cohen, <a href="http://dx.doi.org/10.1090/S0025-5718-1990-0993927-5">On an integer's infinitary divisors</a>, Math. Comp. 54 (189) (1990) 395-411.
%H Steven R. Finch, <a href="/A007947/a007947.pdf">Unitarism and Infinitarism</a>, February 25, 2004. [Cached copy, with permission of the author]
%H J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Broken link]
%H J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Via Internet Archive Wayback-Machine]
%H J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a> [Cached copy, pdf file only]
%H Tomohiro Yamada, <a href="https://arxiv.org/abs/1705.10933">Infinitary superperfect numbers</a>, arXiv:1705.10933 [math.NT], 2017.
%F 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
%F If n is squarefree, then a(n) = A001615(n). - _Vladimir Shevelev_, Apr 01 2014
%F a(n) = Sum_{k>=1} A077609(n,k). - _R. J. Mathar_, Oct 04 2017
%F a(n) = A126168(n)+n. - _R. J. Mathar_, Oct 05 2017
%F 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
%F a(n) = iphi(n^2)/iphi(n), where iphi(n) = A091732(n). - _Amiram Eldar_, Sep 21 2024
%e 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.
%e n = 90 = 2*5*9, where 2, 5, 9 are in A050376; so a(n) = 3*6*10 = 180. - _Vladimir Shevelev_, Feb 19 2011
%p isidiv := proc(d, n)
%p local n2, d2, p, j;
%p if n mod d <> 0 then
%p return false;
%p end if;
%p for p in numtheory[factorset](n) do
%p padic[ordp](n,p) ;
%p n2 := convert(%, base, 2) ;
%p padic[ordp](d,p) ;
%p d2 := convert(%, base, 2) ;
%p for j from 1 to nops(d2) do
%p if op(j, n2) = 0 and op(j, d2) <> 0 then
%p return false;
%p end if;
%p end do:
%p end do;
%p return true;
%p end proc:
%p idivisors := proc(n)
%p local a, d;
%p a := {} ;
%p for d in numtheory[divisors](n) do
%p if isidiv(d, n) then
%p a := a union {d} ;
%p end if;
%p end do:
%p a ;
%p end proc:
%p A049417 := proc(n)
%p local d;
%p add(d, d=idivisors(n)) ;
%p end proc:
%p seq(A049417(n),n=1..100) ; # _R. J. Mathar_, Feb 19 2011
%t 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}]
%t (* Second program: *)
%t 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;
%t Array[a, 100] (* _Jean-François Alcover_, Mar 23 2020, after Paul Abbott in A077609 *)
%o (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
%o (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
%o (Haskell)
%o a049417 1 = 1
%o a049417 n = product $ zipWith f (a027748_row n) (a124010_row n) where
%o f p e = product $ zipWith div
%o (map (subtract 1 . (p ^)) $
%o zipWith (*) a000079_list $ map (+ 1) $ a030308_row e)
%o (map (subtract 1 . (p ^)) a000079_list)
%o -- _Reinhard Zumkeller_, Sep 18 2015
%o (Python)
%o from math import prod
%o from sympy import factorint
%o def A049417(n): return prod(p**(1<<i)+1 for p, e in factorint(n).items() for i, j in enumerate(bin(e)[-1:1:-1]) if j=='1') # _Chai Wah Wu_, Jul 11 2024
%Y Cf. A037445, A004607, A091732, A127661, A293355.
%Y Cf. A049418 (3-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).
%Y Cf. A000079, A030308, A027748, A124010.
%K nonn,mult
%O 1,2
%A _Yasutoshi Kohmoto_, Dec 11 1999
%E More terms from _Wouter Meeussen_, Sep 02 2001