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 1-7 of 7 results.

A007357 Infinitary perfect numbers.

Original entry on oeis.org

6, 60, 90, 36720, 12646368, 22276800, 126463680, 4201148160, 28770487200, 287704872000, 1446875426304, 2548696550400, 14468754263040, 590325173932032, 3291641594841600, 8854877608980480, 32916415948416000
Offset: 1

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Author

N. J. A. Sloane

Keywords

Comments

Numbers N whose sum of infinitary divisors equals 2*N: A049417(N)=2*N. - Joerg Arndt, Mar 20 2011
6 is the only infinitary perfect number which is also perfect number (A000396). 6 is also the only squarefree infinitary perfect number. - Vladimir Shevelev, Mar 02 2011

Examples

			Let n=90. Its unique expansion over distinct terms of A050376 is 90=2*5*9. Thus the infinitary divisors of 90 are 1,2,5,9,10,18,45,90. The number of such divisors is 2^3, i.e., the cardinality of the set of all subsets of the set {2,5,9}. The sum of such divisors is (2+1)*(5+1)*(9+1)=180 and the sum of proper such divisors is 90=n. Thus 90 is in the sequence. - _Vladimir Shevelev_, Mar 02 2011

References

  • G. L. Cohen, personal communication.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A129656 (infinitary abundant), A129657 (infinitary deficient).

Programs

  • Maple
    isA007357 := proc(n)
    A049417(n) = 2*n ;
    simplify(%) ;
end proc:
for n from 1 do
    if isA007357(n) then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, Oct 05 2017
  • Mathematica
    infiPerfQ[n_] := 2n == Total[If[n == 1, 1, Sort @ Flatten @ Outer[ Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m&])]]];
For[n = 6, True, n += 6, If[infiPerfQ[n], Print[n]]] (* Jean-François Alcover, Feb 08 2021 *)

Formula

{n: A049417(n) = 2*n}. - R. J. Mathar, Mar 18 2011
a(n)==0 (mod 6). Thus there are no odd infinitary perfect numbers. - Vladimir Shevelev, Mar 02 2011

Extensions

More terms from Eric W. Weisstein, Jan 27 2004

A129656 Infinitary abundant numbers: integers for which A126168 (n)>n, or equivalently for which A049417 (n)>2n.

Original entry on oeis.org

24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, 96, 102, 104, 114, 120, 138, 150, 168, 174, 186, 210, 216, 222, 246, 258, 264, 270, 280, 282, 294, 312, 318, 330, 354, 360, 366, 378, 384, 390, 402, 408, 420, 426, 438, 440, 456, 462, 474, 480, 486, 498
Offset: 1

Views

Author

Ant King, Apr 29 2007

Keywords

Comments

For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~7.95n. It follows that the density of the infinitary abundant numbers is 1/7.95, which is about 0.126.

Examples

			The third integer that is exceeded by its proper infinitary divisor sum is 40. Hence a(3)=40.

Links

Crossrefs

Cf. A126168, A049417, A127666, A129657, A007357.

Programs

  • Mathematica
    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;InfinitaryAbundantNumberQ[k_]:=If[properinfinitarydivisorsum[k]>k,True,False];Select[Range[500],InfinitaryAbundantNumberQ[ # ] &]
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}]]; isigma[1]=1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[1000], isigma[#]>2# &] (* Amiram Eldar, May 12 2019 *)

A300664 Infinitary 3-abundant numbers: numbers n such that isigma(n) >= 3n, where isigma is the sum of infinitary divisors of n (A049417).

Original entry on oeis.org

120, 840, 1080, 1320, 1512, 1560, 1848, 1890, 1920, 2040, 2184, 2280, 2376, 2688, 2760, 2856, 3000, 3192, 3480, 3720, 4440, 4920, 5160, 5640, 5880, 6360, 7080, 7320, 7560, 8040, 8520, 8760, 9240, 9480, 9720, 9960, 10680, 10920, 11640, 11880, 12120, 12360
Offset: 1

Views

Author

Amiram Eldar, Mar 10 2018

Keywords

Comments

Analogous to 3-abundant numbers (A023197) with isigma (A049417) instead of sigma (A000203).

Examples

			840 is in the sequence since isigma(840) = 2880 > 3 * 840.

Links

Crossrefs

Cf. A007357, A023197, A049417, A129656, A129657.

Programs

  • Mathematica
    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; Infinitary3AbundantNumberQ[k_] :=  If[properinfinitarydivisorsum[k] >= 2 k, True, False]; Select[Range[15000], Infinitary3AbundantNumberQ[#] &] (* after Ant King at A129656 *)

A372298 Primitive infinitary abundant numbers (definition 1): infinitary abundant numbers (A129656) whose all proper infinitary divisors are infinitary deficient numbers.

Original entry on oeis.org

40, 56, 70, 72, 88, 104, 756, 924, 945, 1092, 1188, 1344, 1386, 1428, 1430, 1596, 1638, 1760, 1870, 2002, 2016, 2080, 2090, 2142, 2176, 2210, 2394, 2432, 2470, 2530, 2584, 2720, 2750, 2944, 2990, 3040, 3128, 3190, 3200, 3230, 3250, 3400, 3410, 3496, 3712, 3770
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2024

Keywords

Examples

			40 is a term since it is an infinitary abundant number and all its proper infinitary divisors, {1, 2, 4, 5, 8, 10, 20}, are infinitary deficient numbers.
24 and 30, which are infinitary abundant numbers, are not primitive, because they are divisible by 6 which is an infinitary perfect number.

Links

Crossrefs

Subsequence of A129656 and A372299.
A372300 is a subsequence.
Cf. A007357, A077609, A129657.
Similar sequences: A071395, A298973, A302573, A307112, A307114, A307115.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, 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]; idefQ[n_] := isigma[n] < 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], idefQ]]; Select[Range[4000], q]
  • PARI
    isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) >= 2*x, idivs(n)) == [];

A372299 Primitive infinitary abundant numbers (definition 2): infinitary abundant numbers (A129656) having no proper infinitary divisors that are infinitary abundant numbers.

Original entry on oeis.org

24, 30, 40, 42, 54, 56, 66, 70, 72, 78, 88, 96, 102, 104, 114, 138, 150, 174, 186, 222, 246, 258, 282, 294, 318, 354, 366, 402, 420, 426, 438, 474, 486, 498, 534, 540, 582, 606, 618, 642, 654, 660, 678, 726, 756, 762, 780, 786, 822, 834, 894, 906, 924, 942, 945, 960, 978, 990
Offset: 1

Views

Author

Amiram Eldar, Apr 25 2024

Keywords

Examples

			24 is a term since it is an infinitary abundant number and none of its proper infinitary divisors, {1, 2, 3, 4, 6, 8, 12}, are infinitary abundant numbers.
The least infinitary abundant number that is not primitive is 120. It has 3 infinitary divisors, 24, 30, and 40, that are also infinitary abundant numbers.

Links

Crossrefs

Subsequence of A129656.
A372298 is a subsequence.
Cf. A077609, A129657.
Similar sequences: A091191, A302574, A339940.

Programs

  • Mathematica
    f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, 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]; iabQ[n_] := isigma[n] > 2*n; idivs[1] = {1};
idivs[n_] := Sort@ Flatten@ Outer[Times, Sequence @@ (FactorInteger[n] /. {p_, e_Integer} :> p^Select[Range[0, e], BitOr[e, #] == e &])];
q[n_] := Module[{d = idivs[n]}, Total[d] > 2*n && AllTrue[Most[d], !iabQ[#] &]]; Select[Range[1000], q]
  • PARI
    isidiv(d, f) = {if (d==1, return (1)); for (k=1, #f~, bne = binary(f[k, 2]); bde = binary(valuation(d, f[k, 1])); if (#bde < #bne, bde = concat(vector(#bne-#bde), bde)); for (j=1, #bne, if (! bne[j] && bde[j], return (0)); ); ); return (1); }
idivs(n) = {my(f = factor(n), d = divisors(f), idiv = []); for (k=1, #d, if (isidiv(d[k], f), idiv = concat(idiv, d[k])); ); idiv; } \\ Michel Marcus at A077609
isigma(n) = {my(f = factor(n), b); prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 1+f[i, 1]^(2^(#b-k)), 1)))} ;
is(n) = isigma(n) > 2*n && select(x -> x < n && isigma(x) > 2*x, idivs(n)) == [];

A336252 Infinitary barely deficient numbers: infinitary deficient numbers whose infinitary abundancy is closer to 2 than that of any smaller infinitary deficient number.

Original entry on oeis.org

1, 2, 8, 84, 110, 128, 1155, 3680, 6490, 8200, 8648, 12008, 18632, 32768, 724000, 1495688, 2095208, 3214090, 3477608, 3660008, 5076008, 12026888, 16102808, 26347688, 29322008, 33653888, 73995392, 615206030, 815634435, 2147483648, 42783299288, 80999455688
Offset: 1

Views

Author

Amiram Eldar, Jul 14 2020

Keywords

Comments

The infinitary abundancy of a number k is isigma(k)/k, where isigma is the sum of infinitary divisors of k (A049417).
The corresponding values of the infinitary abundancy are 1, 1.5, 1.875, 1.904..., 1.963..., ...

Examples

			8 is a term since it is infinitary deficient (A129657), and isigma(8)/8 = 15/8 is higher than isigma(k)/k for all the infinitary deficient numbers k < 8.

Crossrefs

Cf. A049417, A129657, A335054.
Similar sequences: A228450, A262228, A302572, A307122, A336253.

Programs

  • Mathematica
    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}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; seq = {}; r = 0; Do[s = isigma[n]/n; If[s < 2 && s > r, AppendTo[seq, n]; r = s], {n, 1, 10^6}]; seq

A186889 Oex perfect numbers: n such that A186644(n) = 2*n.

Original entry on oeis.org

6, 18, 20, 100, 1888, 2044928, 33099776, 35021696, 45335936, 533020672
Offset: 1

Views

Author

Vladimir Shevelev, Feb 28 2011

Keywords

Comments

There are no squarefree infinitary perfect numbers > 6 (cf. A007357). Therefore, the second and all further terms of the sequence are infinitary deficient (A129657).
No further term between 1888 and 1440000. - R. J. Mathar, Mar 18 2011
a(11) > 3*10^10. 1471763808896 is also a term. - Donovan Johnson, Jan 30 2013

Examples

			Let n = 100 with divisors 1, 2, 4, 5, 10, 20, 25, 50, and 100. By the definition in A186643, only 1, 4, 20, 25, 50, 100 among these are oex divisors. Since 1+4+20+25+50+100 = 2*100, 100 is in the sequence.

Crossrefs

Cf. A186443, A186444, A129657, A000396, A007357.

Programs

  • PARI
    for(n=4, 10^9, if(isprime(n), next); d=divisors(n); s=n+1; for(j=2, numdiv(n)-1, for(k=2, 30, if(n%d[j]^k<>0, if(k%2==0, s=s+d[j]); k=30))); if(s==2*n, print(n))) /* Donovan Johnson, Jan 28 2013 */

Extensions

a(6)-a(10) from Donovan Johnson, Jan 28 2013
Showing 1-7 of 7 results.

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