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.

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A126167 Number of primitive exponential amicable pairs (i,j) with i

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 5, 8, 8, 12
Offset: 1

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Author

Ant King, Dec 21 2006

Keywords

Comments

There are infinitely many exponential amicable pairs, for multiplying an exponential amicable pair by a squarefree integer coprime to each of its members will generate another exponential amicable pair. Accordingly, we refer to pairs like (90972,100548) as primitive exponential amicable pairs and to pairs like (454860,502740) that can be obtained from them as nonprimitive. This sequence counts the primitive pairs only.

Examples

			a(7)=3 because there are 3 primitive exponential pairs (m,n) with m<n and m<=10^7

References

  • Hagis, Peter Jr.; Some Results Concerning Exponential Divisors, International Journal of Mathematics and Mathematical Sciences, Vol. 11, No. 2, (1988), pp. 343-350.

Links

Crossrefs

Cf. A051377, A054979, A049419, A054980, A126164, A126165, A126166.

Extensions

Link corrected and reference added by Andrew Lelechenko, Dec 04 2011

A324278 Exponential untouchable numbers: numbers that are not the sum of aliquot exponential divisors of any number.

Original entry on oeis.org

1, 4, 8, 9, 16, 20, 25, 27, 28, 32, 40, 44, 45, 49, 52, 54, 63, 64, 68, 75, 76, 81, 88, 92, 96, 99, 104, 108, 116, 117, 121, 124, 125, 128, 136, 144, 147, 148, 152, 153, 160, 164, 169, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208, 212, 216, 224
Offset: 1

Views

Author

Amiram Eldar, Feb 20 2019

Keywords

Comments

The terms are conjectural and based on a search for solutions to esigma(x) - x = k for k in the range of the data section and x < 10^12 (esigma(x) - x = A051377(x) - x = A126164(x) is the sum of aliquot exponential divisors of x). - Amiram Eldar, Jan 22 2020

Crossrefs

Cf. A051377, A126164, A005114, A063948 (unitary), A324276 (bi-unitary), A324277 (infinitary).

Programs

  • Mathematica
    fun[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := esigma[n] = Times @@ fun @@@ FactorInteger[n]; untouchableQ[n_] := Catch[ Do[ If[n == esigma[k]-k, Throw[True]], {k, 0, (n+1)^2}]] === Null; Reap[ Table[ If[ untouchableQ[n], Sow[n]], {n, 1, 130}]][[2, 1]] (* after Jean-François Alcover at A005114 *)

Extensions

Data corrected by Amiram Eldar, Jan 22 2020

A323753 Lesser member of primitive exponential amicable pairs.

Original entry on oeis.org

90972, 937692, 4548600, 44030448, 46884600, 453842928, 712931184, 906494400, 20907057600, 34793179200, 47646797328, 53469838800, 240707724300
Offset: 1

Views

Author

Amiram Eldar, Jan 26 2019

Keywords

Comments

Exponential amicable pair (m,n) is primitive if there is no prime number that is a unitary divisor of both m and n. All the other amicable pairs can be generated from primitive pairs by multiplying them with a squarefree integer coprime to each of the members of the pair. Hagis found the first 6 terms in 1988. Pedersen found the next 7 terms in 1999.
a(14) <= 588330137304.
The larger counterparts are in A323754.

Examples

			(90972 = 2^2*3^2*7*19^2, 100548 = 2^2*3^3*7^2*19) are a primitive pair since they are an exponential amicable pair (A126165, A126166) and they do not have a common prime divisor with multiplicity 1 in both.
(454860, 502740) = 5 * (90972, 100548) are not a primitive pair since 5 divides both of them only once.

Links

Crossrefs

Cf. A051377, A054979, A049419, A054980, A126164, A126165, A126166, A126167, A323754.

Programs

  • Mathematica
    rad[n_] := Times @@ First /@ FactorInteger[n]; pf[n_] := Denominator[n/rad[n]^2]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; es[n_] := esigma[n] - n; s = {}; Do[m = es[n]; If[m > n && es[m] == n && CoprimeQ[pf[n], pf[m]], AppendTo[s, n]], {n, 1, 10^7}]; s (* after Jean-François Alcover at A055231 and A051377 *)

A323754 Larger member of primitive exponential amicable pairs.

Original entry on oeis.org

100548, 968436, 5027400, 48665232, 48421800, 468723024, 845775504, 938024640, 26989110720, 40792003200, 48200025744, 63433162800, 303008547060
Offset: 1

Views

Author

Amiram Eldar, Jan 26 2019

Keywords

Comments

The lesser counterparts are in A323753.
a(14) <= 647935817256.

Examples

			(90972 = 2^2*3^2*7*19^2, 100548 = 2^2*3^3*7^2*19) are a primitive pair since they are an exponential amicable pair (A126165, A126166) and they do not have a common prime divisor with multiplicity 1 in both.
(454860, 502740) = 5 * (90972, 100548) are not a primitive pair since 5 divides both of them only once.

Links

Crossrefs

Cf. A051377, A054979, A049419, A054980, A126164, A126165, A126166, A126167, A323753.

Programs

  • Mathematica
    rad[n_] := Times @@ First /@ FactorInteger[n]; pf[n_] := Denominator[n/rad[n]^2]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; es[n_] := esigma[n] - n; s = {}; Do[m = es[n]; If[m > n && es[m] == n && CoprimeQ[pf[n], pf[m]], AppendTo[s, m]], {n, 1, 10^7}]; s (* after Jean-François Alcover at A055231 and A051377 *)
Previous Showing 11-14 of 14 results.

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