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 1-5 of 5 results.

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

Views

Author

Yasutoshi Kohmoto

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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

Programs

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

Extensions

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

A097464 5-infinitary perfect numbers: numbers k such that 5-infinitary-sigma(k) = 2*k.

Original entry on oeis.org

6, 28, 496, 47520, 288288, 308474880
Offset: 1

Views

Author

Yasutoshi Kohmoto

Keywords

Comments

Here 5-infinitary-sigma(k) means sum of 5-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 5-ary expansion everywhere that the corresponding r_i has a digit b, then d is a 5-infinitary-divisor of k.
Is it certain that 308474880 is the 6th term? M. F. Hasler, Nov 20 2010
Data is verified. a(7) > 10^11, if it exists. - Amiram Eldar, Oct 24 2024

Examples

			Factorizations: 2*3, 2^2*7, 2^4*31, 2^5*3^3*5*11, 2^5*3^2*7*11*13, 2^10*3*5*7*19*151.

Crossrefs

Cf. A007357, A038182, A074849, A097863.

Programs

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

Formula

{k: A097863(k) = 2*k}.

Extensions

Missing a(4) inserted by R. J. Mathar, Nov 20 2010

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

Views

Author

Amiram Eldar, Jan 09 2020

Keywords

Comments

No more terms below 4*10^10.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Jan 09 2020

Keywords

Comments

No more terms below 2.8*10^10.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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

Views

Author

Amiram Eldar, Oct 08 2024

Keywords

Comments

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.

Crossrefs

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

Programs

  • Mathematica
    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*# &]
  • PARI
    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;
Showing 1-5 of 5 results.

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