A038182 -id:A038182 - cp's OEIS frontend

A038148 Number 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-infinitary-divisor of n.

1, 2, 2, 3, 2, 4, 2, 2, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 4, 3, 4, 2, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 4, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 4, 4, 4, 4, 4, 2, 12, 2, 4, 6, 3, 4, 8, 2, 6, 4, 8, 2, 6, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4, 4, 2, 12, 4, 6, 4, 4, 4, 12, 2, 6, 6, 9, 2, 8, 2, 4, 8

Offset: 1

Yasutoshi Kohmoto

Multiplicative: If e = sum d_k 3^k, then a(p^e) = prod (d_k+1). - Christian G. Bower, May 19 2005

			2^3*3 is a 3-infinitary-divisor of 2^5*3^2 because 2^3*3 = 2^10*3^1 and 2^5*3^2 = 2^12*3^2 in ternary expanded power. All corresponding digits satisfy the condition. 1 <= 1, 0 <= 2, 1 <= 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Frédéric Chyzak, Ivan Gutman, and Peter Paule, Predicting the number of hexagonal systems with 24 and 25 hexagons, Communications in Mathematical and Computer Chemistry (1999) No. 40, 139-151. See p. 141.
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]
Index entries for sequences computed from exponents in factorization of n

Cf. A006047, A037445, A038182, A074848.

A038148 := proc(n) if n= 1 then 1; else ifa := ifactors(n)[2] ;
a := 1; for f in ifa do e := convert(op(2,f),base,3) ; a := a*mul(d+1,d=e) ; end do: end if; end proc:
seq(A038148(n),n=1..50) ; # R. J. Mathar, Feb 08 2011

a[1] = 1; a[n_] := (k = 1; Do[k = k * Times @@ (IntegerDigits[f, 3] + 1), {f, FactorInteger[n][[All, 2]]}]; k); Table[a[n], {n, 1, 102}](* Jean-François Alcover, Feb 03 2012, after R. J. Mathar *)

A006047(n) = { my(m=1, d); while(n, d = (n%3); m *= (1+d); n = (n-d)/3); m; };
A038148(n) = factorback(apply(e -> A006047(e), factorint(n)[, 2])); \\ (After A037445) - Antti Karttunen, May 28 2017

(define (A038148 n) (if (= 1 n) n (* (A006047 (A067029 n)) (A038148 (A028234 n))))) ;; Antti Karttunen, May 28 2017

a(1) = 1; for n > 1, a(n) = A006047(A067029(n)) * a(A028234(n)). [After Christian G. Bower's 2005 comment.] - Antti Karttunen, May 28 2017

More terms from Naohiro Nomoto, Jun 21 2001

Data section further extended to 105 terms by Antti Karttunen, May 28 2017

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

Yasutoshi Kohmoto

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

			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.

Cf. A007357, A038182, A074849, A097863.

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 *)

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

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

A038148 Number 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-infinitary-divisor of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Scheme

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI