A097464 -id:A097464 - cp's OEIS frontend

A097863 Sum of 5-infinitary divisors of n.

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

A038182 3-infinitary perfect numbers k: 3-i-sigma(k) = 2*k, where 3-i-sigma = A049418.

Original entry on oeis.org

6, 28, 3024, 6552, 27578880, 49266240, 49095705098695680

Offset: 1

Yasutoshi Kohmoto

Similarly, we have 3-i-sigma(x)/x = r for the following numbers: r = 3 for x = 672, 13104, 4021920, 55157760, 98532480, 459818240, 372667889664, 7267023848448, 1178296922368696320, 5498718971053916160, ...; r = 4 for x = 2178540; r = 3/2 for x = 2, 24, 9192960, 196382820394782720. (Values above 10^7 from Yasutoshi Kohmoto, some terms may be missing.) - M. F. Hasler, Sep 21 2022

			Factorizations: 2*3, 2^2*7, 2^4*3^3*7, 2^3*3^2*7*13, 2^9*3^4*5*7*19, 2^6*3*5*19*37*73, 2^10*3^6*5*19^2*127*379*757.

J. O. M. Pedersen, Tables of Aliquot Cycles: backup on web.archive.org of no more existing web page, as of May 2014.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]

Cf. A007357, A037445, A038148, A049418, A074849, A097464.

f[p_, e_] := Module[{d = IntegerDigits[e, 3]}, m = Length[d]; Product[(p^((d[[j]] + 1)*3^(m - j)) - 1)/(p^(3^(m - j)) - 1), {j, 1, m}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[7000], s[#] == 2*# &] (* Amiram Eldar, Oct 24 2024 *)

is_A038182(n)=A049418(n)==2*n \\ M. F. Hasler, Sep 21 2022

Definition shortened by R. J. Mathar, Oct 06 2010

A074849 4-infinitary perfect numbers: numbers k such that 4-infinitary-sigma(k) = 2*k.

Original entry on oeis.org

6, 28, 36720, 222768, 12646368, 5154170112, 34725010231296

Offset: 1

Yasutoshi Kohmoto, Sep 10 2002

Here 4-infinitary-sigma(k) means sum of 4-infinitary-divisors of k. If k = 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 k.

			Factorizations: 2*3, 2^2*7, 2^4*3^3*5*17, 2^4*3^2*7*13*17, 2^5*3^4*7*17*41, 2^8*3^2*7*13^2*31*61, 2^12*3^5*7*11*41*43*257.

Cf. A007357, A038182, A074847, A097464.

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}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[300000], s[#] == 2*# &] (* Amiram Eldar, Oct 24 2024 *)

{k: A074847(k) = 2*k}. - R. J. Mathar, Mar 13 2024

A331108 Zeckendorf-infinitary perfect numbers: numbers k such that A331107(k) = 2*k.

Original entry on oeis.org

6, 60, 90, 3024, 133056, 1330560, 6879600, 28828800, 302702400, 698544000, 11763214848

Offset: 1

Amiram Eldar, Jan 09 2020

No more terms below 4*10^10.

			6 is a term since A331107(6) = 12 = 2*6.

Cf. A007357, A038182, A074849, A097464, A331107.

fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; Fibonacci[1 + Position[Reverse@fr, ?(# == 1 &)]]]; f[p, e_] := p^fb[e]; zsigma[1] = 1; zsigma[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); zPerfectQ[n_] := zsigma[n] == 2 n; Select[Range[10^4], zPerfectQ] (* after Robert G. Wilson v at A014417 *)

A331111 Dual-Zeckendorf-infinitary perfect numbers: numbers k such that A331110(k) = 2*k.

Original entry on oeis.org

6, 60, 90, 655200, 28828800, 238140000, 10478160000

Offset: 1

Amiram Eldar, Jan 09 2020

No more terms below 2.8*10^10.

			6 is a term since A331110(6) = 12 = 2*6.

Cf. A007357, A038182, A074849, A097464, A331108, A331110.

fibTerms[n_] := Module[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; fr];
dualZeck[n_] := Module[{v = fibTerms[n]}, nv = Length[v]; i = 1; While[i <= nv - 2, If[v[[i]] == 1 && v[[i + 1]] == 0 && v[[i + 2]] == 0, v[[i]] = 0; v[[i + 1]] = 1; v[[i + 2]] = 1; If[i > 2, i -= 3]]; i++]; i = Position[v, _?(# > 0 &)]; If[i == {}, {}, v[[i[[1, 1]] ;; -1]]]];
f[p_, e_] := p^Fibonacci[1 + Position[Reverse @ dualZeck[e], _?(# == 1 &)]];
dzsigma[1] = 1; dzsigma[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); seqQ[n_] := dzsigma[n] == 2n; Select[Range[10^6], seqQ]

A376889 Numbers k such that A376888(k) = 2*k.

Original entry on oeis.org

6, 60, 90, 336, 5040, 87360, 764400, 11466000, 620568000, 9478560000, 14217840000, 22805874000

Offset: 1

Amiram Eldar, Oct 08 2024

a(12) > 7*10^10, if it exists.
28279283760000, 282792837600000 and 1583639890560000 are also terms.
k! is a term for k = 3 and 7, and for no other factorial of k < 10^4.

Cf. A376888.
Subsequence of A023196.
Similar sequences: A007357, A038182, A074849, A097464, A331108, A331111.

ff[q_, s_] := (q^(s + 1) - 1)/(q - 1); f[p_, e_] := Module[{k = e, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 0, AppendTo[s, {p^(m - 1)!, r}];]; m++]; Times @@ ff @@@ s]; fsigma[1] = 1; fsigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^6], fsigma[#] == 2*# &]

fdigits(n) = {my(k = n, m = 2, r, s = []); while([k, r] = divrem(k, m); k != 0 || r != 0, s = concat(s, r); m++); s;}
fsigma(n) = {my(f = factor(n), p = f[, 1], e = f[, 2], d); prod(i = 1, #p, prod(j = 1, #d=fdigits(e[i]), (p[i]^(j!*(d[j]+1)) - 1)/(p[i]^j! - 1)));}
is(k) = fsigma(k) == 2*k;

cp's OEIS Frontend

A097863 Sum of 5-infinitary divisors of n.

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A038182 3-infinitary perfect numbers k: 3-i-sigma(k) = 2*k, where 3-i-sigma = A049418.

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A074849 4-infinitary perfect numbers: numbers k such that 4-infinitary-sigma(k) = 2*k.

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A331108 Zeckendorf-infinitary perfect numbers: numbers k such that A331107(k) = 2*k.

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A331111 Dual-Zeckendorf-infinitary perfect numbers: numbers k such that A331110(k) = 2*k.

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A376889 Numbers k such that A376888(k) = 2*k.

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