A049417 -id:A049417 - cp's OEIS frontend

A126170 Larger member of an infinitary amicable pair.

126, 846, 1260, 7920, 8460, 11760, 10856, 14595, 17700, 43632, 45888, 49308, 83142, 62700, 71145, 73962, 96576, 83904, 107550, 88730, 178800, 112672, 131100, 125856, 168730, 149952, 196650, 203432, 206752, 224928, 306612, 365700, 399592, 419256, 460640, 548550

Offset: 1

Ant King, Dec 21 2006

nonn

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.

			a(5)=8460 because the fifth infinitary amicable pair is (5940,8460) and 8460 is its largest member.

Amiram Eldar, Table of n, a(n) for n = 1..7916
Jan Munch Pedersen, Tables of Aliquot Cycles.

Cf. A126169, A049417, A126168, A037445.

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}]] ]] ] Null; properinfinitarydivisorsum[k] := Plus @@ InfinitaryDivisors[k] - k; InfinitaryAmicableNumberQ[k_] := If[Nest[properinfinitarydivisorsum, k, 2] == k && ! properinfinitarydivisorsum[k] == k, True, False]; data1 = Select[ Range[10^6], InfinitaryAmicableNumberQ[ # ] &]; data2 = properinfinitarydivisorsum[ # ] & /@ data1; data3 = Table[{data1[[k]], data2[[k]]}, {k, 1, Length[data1]}]; data4 = Select[data3, First[ # ] < Last[ # ] &]; Table[Last[data4[[k]]], {k, 1, Length[data4]}]
fun[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_] := Times @@ (fun @@@ FactorInteger[n]) - n; s = {}; Do[k = infs[n]; If[k > n && infs[k] == n, AppendTo[s, k]], {n, 2, 10^5}]; s (* Amiram Eldar, Jan 22 2019 *)

The values of n for which isigma(m)=isigma(n)=m+n and n>m.

a(33)-a(36) from Amiram Eldar, Jan 22 2019

A063947 Infinitary harmonic numbers: harmonic mean of infinitary divisors is an integer.

Original entry on oeis.org

1, 6, 45, 60, 90, 270, 420, 630, 2970, 5460, 8190, 9100, 15925, 27300, 36720, 40950, 46494, 54600, 81900, 95550, 136500, 163800, 172900, 204750, 232470, 245700, 257040, 409500, 464940, 491400, 646425, 716625, 790398, 791700, 819000, 900900

Offset: 1

Wouter Meeussen, Sep 03 2001

Amiram Eldar, Table of n, a(n) for n = 1..239 (terms below 10^10)
P. Hagis, Jr. and G. L. Cohen, Infinitary harmonic numbers, Bull. Australian math. Soc., 41 (1990), 151-158 (Math. Rev. 91d:11001) (asymptotics).
Eric Weisstein's World of Mathematics, Harmonic Mean
Wikipedia, Harmonic mean

Cf. A037445, A049417.

Cf. A077609.

import Data.Ratio (denominator)
import Data.List (genericLength)
a063947 n = a063947_list !! (n-1)
a063947_list = filter ((== 1) . denominator . hm . a077609_row) [1..]
   where hm xs = genericLength xs / sum (map (recip . fromIntegral) xs)
-- Reinhard Zumkeller, Jul 10 2013

bitty[ k_ ] := Union[ Flatten[ Outer[ Plus, Sequence @@ ({0, #} & /@ Union[ (2^Range[ 0, Floor[ Log[ 2, k ] ] ] ) Reverse[ IntegerDigits[ k, 2 ] ] ] ) ] ] ]; 1 + Flatten[ Position[ Table[ (Length[ # ] /(Plus @@ (1/#)) &)@ (Apply[ Times, (First[ it ] ^ (# /. z -> List)) ] & /@ Flatten[ Outer[ z, Sequence @@ (bitty /@ Last[ it = Transpose[ FactorInteger[ k ] ] ] ), 1 ] ]), {k, 2, 2^22 + 1} ], Integer ] ] (* _Robert G. Wilson v, Sep 04 2001 *)

More terms from David W. Wilson, Sep 04 2001

A348271 a(n) is the sum of noninfinitary divisors of n.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 8, 0, 0, 0, 14, 0, 9, 0, 12, 0, 0, 0, 0, 5, 0, 0, 16, 0, 0, 0, 12, 0, 0, 0, 41, 0, 0, 0, 0, 0, 0, 0, 24, 18, 0, 0, 56, 7, 15, 0, 28, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 24, 42, 0, 0, 0, 36, 0, 0, 0, 45, 0, 0, 20, 40, 0, 0, 0, 84, 39

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

			a(12) = 8 since 12 has 2 noninfinitary divisors, 2 and 6, and 2 + 6 = 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Infinitary Divisor.

Cf. A000203, A036537, A049417, A077609, A162643, A327633.

Similar sequences: A034448, A048146, A051377, A188999.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; a[n_]:= DivisorSigma[1,n] - isigma[n]; Array[a, 100]

a(n) = A000203(n) - A049417(n).
a(n) = 0 if and only if the number of divisors of n is a power of 2, (i.e., n is in A036537).
a(n) > 0 if and only if the number of divisors of n is not a power of 2, (i.e., n is in A162643).

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

Ant King, Jan 26 2007

nonn

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

			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.

R. J. Mathar, Table of n, a(n) for n = 1..839
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
Hans Havermann, Graphs of infinitary aliquot sequences for 840, 1152, 2442, 2658, 2982, 5766, 6216, 6870, 7560, 8670, 9030, 9570 (click to see full plots)
D. Moews, A database of aliquot cycles (2015)
J. O. M. Pedersen, Tables of Aliquot Cycles [Broken link]
J. O. M. Pedersen, Tables of Aliquot Cycles [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

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

# 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

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 *)

A244963 a(n) = sigma(n) - n * Product_{p|n, p prime} (1 + 1/p).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 4, 0, 0, 0, 7, 0, 3, 0, 6, 0, 0, 0, 12, 1, 0, 4, 8, 0, 0, 0, 15, 0, 0, 0, 19, 0, 0, 0, 18, 0, 0, 0, 12, 6, 0, 0, 28, 1, 3, 0, 14, 0, 12, 0, 24, 0, 0, 0, 24, 0, 0, 8, 31, 0, 0, 0, 18, 0, 0, 0, 51, 0, 0, 4, 20, 0, 0, 0, 42, 13

Offset: 1

Omar E. Pol, Jul 15 2014

a(n) = 0 if and only if n is a squarefree number (A005117), otherwise a(n) > 0.

If n is semiprime, then a(n) = 1+floor(sqrt(n))-ceiling(sqrt(n)). - Wesley Ivan Hurt, Dec 25 2016

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203 (sigma), A001615 (Dedekind psi), A005117 (positions of zeros), A013929, A049417.

A244963:= n -> numtheory:-sigma(n) - n*mul(1+1/t[1],t=ifactors(n)[2]):
seq(A244963(n),n=1..1000); # Robert Israel, Jul 15 2014

nn = 200;Table[Sum[d, {d, Divisors[n]}], {n, 1, nn}] -
Table[Sum[n/d Abs[MoebiusMu[d]], {d, Divisors[n]}], {n, 1, nn}] (* Geoffrey Critzer, Mar 18 2015 *)

A001615(n) = (n * sumdivmult(n, d, issquarefree(d)/d)); \\ This function from Charles R Greathouse IV, Sep 09 2014
A244963(n) = (sigma(n) - A001615(n)); \\ Antti Karttunen, Nov 22 2017

a(n) = A000203(n) - A001615(n).

Sum_{k=1..n} a(k) ~ c*n^2 + O(n*log(n)), where c = Pi^2/12 - 15/(2*Pi^2) = 0.062558... - Amiram Eldar, Mar 02 2021

A327635 Numbers k such that both k and k+1 are infinitary abundant numbers (A129656).

Original entry on oeis.org

21735, 21944, 43064, 58695, 188055, 262184, 414855, 520695, 567944, 611415, 687015, 764504, 792855, 809864, 812889, 833624, 874664, 911624, 945944, 976184, 991304, 1019655, 1026375, 1065015, 1073709, 1157624, 1201095, 1218944, 1248344, 1254015, 1272375, 1272704

Offset: 1

Amiram Eldar, Sep 20 2019

nonn

The least k such that k, k+1 and k+2 are all infinitary abundant numbers is a(75976) = 2666847104.

			21735 is in the sequence since both 21735 and 21736 are infinitary abundant: isigma(21735) = 46080 > 2 * 21735, and isigma(21736) = 50400 > 2 * 21736 (isigma is the sum of infinitary divisors, A049417).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A049417, A127666, A129656.

Cf. A096399, A292704, A318167.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten @ (f @@@ FactorInteger[n]) + 1); abQ[n_] := isigma[n] > 2n; s={}; ab1 = 0; Do[ab2 = abQ[n]; If[ab1 && ab2, AppendTo[s, n-1]]; ab1 = ab2, {n, 2, 10^5}]; s

A074847 Sum of 4-infinitary divisors of n: if n=Product p(i)^r(i) and d=Product p(i)^s(i), each s(i) has a digit a<=b in its 4-ary expansion everywhere that the corresponding r(i) has a digit b, then d is a 4-infinitary-divisor of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 17, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 51, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 68, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 119, 84, 144, 68, 126, 96

Offset: 1

Yasutoshi Kohmoto, Sep 10 2002

If we group the exponents e in the Bower-Harris formula into the sets with d_k=0, 1, 2 and 3, we see that every n has a unique representation of the form n=prod q_i *prod (r_j)^2 *prod (s_k)^3, where each of q_i, r_j, s_k is a prime power of the form p^(k^4), p prime, k>=0. Using this representation, a(n)=prod (q_i+1)prod ((r_j)^2+r_j+1)prod ((s_k)^3+(s_k)^2+s_k+1) by simple expansion of the quotient on the right hand side of the Bower-Harris formula. - Vladimir Shevelev, May 08 2013

			2^4*3 is a 4-infinitary-divisor of 2^5*3^2 because 2^4*3 = 2^10*3^1 and 2^5*3^2 = 2^11*3^2 in 4-ary expanded power. All corresponding digits satisfy the condition. 1<=1, 0<=1, 1<=2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A049417 (2-infinitary), A049418 (3-infinitary), A097863 (5-infinitary).

Cf. A000302, A030386, A027748, A074848, A124010.

following Bower and Harris, cf. A049418:
a074847 1 = 1
a074847 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = product $ zipWith div
           (map (subtract 1 . (p ^)) $
                zipWith (*) a000302_list $ map (+ 1) $ a030386_row e)
           (map (subtract 1 . (p ^)) a000302_list)
-- Reinhard Zumkeller, Sep 18 2015

A074847 := proc(n) option remember; ifa := ifactors(n)[2] ; a := 1 ; if nops(ifa) = 1 then p := op(1,op(1,ifa)) ; e := op(2,op(1,ifa)) ; d := convert(e,base,4) ; for k from 0 to nops(d)-1 do a := a*(p^((1+op(k+1,d))*4^k)-1)/(p^(4^k)-1) ; end do: else for d in ifa do a := a*procname( op(1,d)^op(2,d)) ; end do: return a; end if; end proc:
seq(A074847(n),n=1..100) ; # R. J. Mathar, Oct 06 2010

f[p_, e_] := Module[{d = IntegerDigits[e, 4]}, m = Length[d]; Product[(p^((d[[j]] + 1)*4^(m - j)) - 1)/(p^(4^(m - j)) - 1), {j, 1, m}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 09 2020 *)

Multiplicative. If e = sum_{k >= 0} d_k 4^k (base 4 representation), then a(p^e) = prod_{k >= 0} (p^(4^k*{d_k+1}) - 1)/(p^(4^k) - 1). - Christian G. Bower and Mitch Harris, May 20 2005

More terms from R. J. Mathar, Oct 06 2010

A306984 Infinitary weird numbers: infinitary abundant numbers (A129656) that are not infinitary pseudoperfect numbers (A306983).

Original entry on oeis.org

70, 4030, 5390, 5830, 7192, 7400, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 11830, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17920, 17990, 18410, 18830, 18970, 19390, 19670

Offset: 1

Amiram Eldar, Mar 18 2019

nonn

Differs from bi-unitary weird numbers from n >= 32 (a(32) = 17920 is not bi-unitary weird).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A006037, A049417, A064114, A126168, A129656, A292986, A321146, A306983.

idivs[x_] := If[x == 1, 1, Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[ x ] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m &])]] ;s = {}; Do[d = Most[idivs[n]]; If[Total[d]

A049418 3-i-sigma(n): sum of 3-infinitary divisors of n: if n=Product p(i)^r(i) and d=Product p(i)^s(i), each s(i) has a digit a<=b in its ternary expansion everywhere that the corresponding r(i) has a digit b, then d is a 3-i-divisor of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 9, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 36, 31, 42, 28, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 54, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 84, 72, 72, 80, 90, 60, 168, 62, 96, 104, 73, 84, 144, 68, 126, 96

Offset: 1

Yasutoshi Kohmoto

			Let n = 28 = 2^2*7. Then a(n) = (2^2 + 2 + 1)*(7 + 1) = 56. - _Vladimir Shevelev_, May 07 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. O. M. Pedersen, Tables of Aliquot Cycles, backup on web.archive.org of no more exiting page, as of May 2014
J. O. M. Pedersen, Tables of Aliquot Cycles [Cached copy, pdf file only]

Cf. A049417 (2-infinitary), A074847 (4-infinitary), A097863 (5-infinitary).

Cf. A000244, A030341, A027748, A124010.

following Bower and Harris:
a049418 1 = 1
a049418 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = product $ zipWith div
           (map (subtract 1 . (p ^)) $
                zipWith (*) a000244_list $ map (+ 1) $ a030341_row e)
           (map (subtract 1 . (p ^)) a000244_list)
-- Reinhard Zumkeller, Sep 18 2015

A049418 := proc(n) option remember; local ifa,a,p,e,d,k ; ifa := ifactors(n)[2] ; a := 1 ; if nops(ifa) = 1 then p := op(1,op(1,ifa)) ; e := op(2,op(1,ifa)) ; d := convert(e,base,3) ; for k from 0 to nops(d)-1 do a := a*(p^((1+op(k+1,d))*3^k)-1)/(p^(3^k)-1) ; end do: else for d in ifa do a := a*procname( op(1,d)^op(2,d)) ; end do: return a; end if; end proc:
seq(A049418(n),n=1..40) ; # R. J. Mathar, Oct 06 2010

A049418[n_] := Module[{ifa = FactorInteger[n], a = 1, p, e, d, k}, If[ Length[ifa] == 1, p = ifa[[1, 1]]; e = ifa[[1, 2]]; d = Reverse[ IntegerDigits[e, 3] ]; For[k = 1, k <= Length[d], k++, a = a*(p^((1 + d[[k]])*3^(k - 1)) - 1)/(p^(3^(k - 1)) - 1)], Do[ a = a*A049418[ d[[1]]^d[[2]] ], {d, ifa}]]; Return[a] ]; A049418[1] = 1; Table[ A049418[n] , {n, 1, 69}] (* Jean-François Alcover, Jan 03 2012, after R. J. Mathar *)

apply( {A049418(n)=vecprod([prod(k=1,#n=digits(f[2],3),(f[1]^(3^(#n-k)*(n[k]+1))-1)\(f[1]^3^(#n-k)-1))|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Sep 21 2022

Multiplicative with a(p^e) = prod_{k >= 0} (p^(3^k*{d_k+1}) - 1)/(p^(3^k) - 1), where e = sum_{k >= 0} d_k 3^k (base 3 representation). - Christian G. Bower and Mitch Harris, May 20 2005. [Edited by M. F. Hasler, Sep 21 2022]

Denote P_3 = {p^3^k}, k = 0, 1, ..., p runs primes. Then every n has a unique representation of the form n = prod q_i prod (r_j)^2, where q_i, r_j are distinct elements of P_3. Using this representation, we have a(n) = prod (q_i+1)*prod ((r_j)^2+r_j+1). - Vladimir Shevelev, May 07 2013

More terms from Naohiro Nomoto, Sep 10 2001

A097863 Sum of 5-infinitary divisors of n.

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 33, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 99, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 124

Offset: 1

Yasutoshi Kohmoto

If n=Product p_i^r_i and d=Product p_i^s_i, each s_i has a digit a<=b in its 5-ary expansion everywhere that the corresponding r_i has a digit b, then d is a 5-infinitary-divisor of n.

			a(32) = a(2^10) = 2^10 + 2^0 = 32 + 1 = 33, in 5-ary expansion. This is the first term which is different from sigma(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A049417, A049418, A074847, A097464.

Cf. A000351, A031235, A027748, A124010.

following Bower and Harris, cf. A049418:
a097863 1 = 1
a097863 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f p e = product $ zipWith div
           (map (subtract 1 . (p ^)) $
                zipWith (*) a000351_list $ map (+ 1) $ a031235_row e)
           (map (subtract 1 . (p ^)) a000351_list)
-- Reinhard Zumkeller, Sep 18 2015

A097863 := proc(n) option remember; local ifa, a, p, e, d, k ; ifa := ifactors(n)[2] ; a := 1 ; if nops(ifa) = 1 then p := op(1, op(1, ifa)) ; e := op(2, op(1, ifa)) ; d := convert(e, base, 5) ; for k from 0 to nops(d)-1 do a := a*(p^((1+op(k+1, d))*5^k)-1)/(p^(5^k)-1) ; end do: else for d in ifa do a := a*procname( op(1, d)^op(2, d)) ; end do: return a; end if; end proc:

f[p_, e_] := Module[{d = IntegerDigits[e, 5]}, m = Length[d]; Product[(p^((d[[j]] + 1)*5^(m - j)) - 1)/(p^(5^(m - j)) - 1), {j, 1, m}]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 09 2020 *)

Denote by P_5={p^5^k} the two-parameter set when k=0,1,... and p runs prime values. Then every n has a unique representation of the form n=prod q_i prod (r_j)^2 prod (s_k)^3 prod (t_m)^4, where q_i, r_j, s_k, t_m are distinct elements of P_5. Using this representation, we have a(n)=prod (q_i+1)prod ((r_j)^2+r_j+1)prod ((s_k)^3+(s_k)^2+s_k+1) prod ((t_m)^4+(t_m)^3+(t_m)^2+t_m+1). - Vladimir Shevelev, May 08 2013

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A126170 Larger member of an infinitary amicable pair.

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A063947 Infinitary harmonic numbers: harmonic mean of infinitary divisors is an integer.

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A348271 a(n) is the sum of noninfinitary divisors of n.

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A127661 Lengths of the infinitary aliquot sequences.

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A244963 a(n) = sigma(n) - n * Product_{p|n, p prime} (1 + 1/p).

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A327635 Numbers k such that both k and k+1 are infinitary abundant numbers (A129656).

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A074847 Sum of 4-infinitary divisors of n: if n=Product p(i)^r(i) and d=Product p(i)^s(i), each s(i) has a digit a<=b in its 4-ary expansion everywhere that the corresponding r(i) has a digit b, then d is a 4-infinitary-divisor of n.

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