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Odd bi-unitary admirable numbers: the odd terms of A334972.
+20
2
945, 43065, 46035, 48195, 80535, 354585, 403095, 430815, 437745, 442365, 458055, 2305875, 3525795, 4404105, 4891887, 5388495, 5803245, 6126645, 6220665, 6375105, 6537375, 7853625, 7981875, 8109585, 8731125, 9071865, 9338595, 9784125, 13241745, 13351635, 23760555
COMMENTS
Of the first 10^4 bi-unitary admirable numbers only 11 are odd.
MATHEMATICA
fun[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), (p^(e + 1) - 1)/(p - 1) - p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); buDivQ[n_, 1] = True; buDivQ[n_, div_] := If[Mod[#2, #1] == 0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; buAdmQ[n_] := (ab = bsigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && buDivQ[n, ab/2]; Select[Range[1, 5*10^5, 2], buAdmQ]
Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2*d = 2*k, where isigma is the sum of infinitary divisors function ( A049417).
+10
5
24, 30, 40, 42, 54, 56, 66, 70, 78, 88, 96, 102, 104, 114, 120, 138, 150, 174, 186, 222, 246, 258, 270, 282, 294, 318, 354, 360, 366, 402, 420, 426, 438, 474, 486, 498, 534, 540, 582, 606, 618, 630, 642, 654, 660, 678, 726, 762, 780, 786, 822, 834, 894, 906, 942
COMMENTS
Equivalently, numbers that are equal to the sum of their proper infinitary divisors, with one of them taken with a minus sign.
Admirable numbers ( A111592) whose number of divisors is a power of 2 ( A036537) are also infinitary admirable numbers, since all of their divisors are infinitary. Terms with number of divisors that is not a power of 2 are 96, 150, 294, 360, 420, 486, 540, 630, 660, 726, 780, 960, 990, ...
EXAMPLE
150 is in the sequence since 150 = 1 + 2 + 3 - 6 + 25 + 50 + 75 is the sum of its proper infinitary divisors with one of them, 6, taken with a minus sign.
MATHEMATICA
fun[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, 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]; infDivQ[n_, 1] = True; infDivQ[n_, d_] := BitAnd[IntegerExponent[n, First /@ (f = FactorInteger[d])], (e = Last /@ f)] == e; infAdmQ[n_] := (ab = isigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && infDivQ[n, ab/2]; Select[Range[1000], infAdmQ]
Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2*d = 2*k, where esigma is the sum of exponential divisors function ( A051377).
+10
2
900, 1764, 4356, 4500, 4900, 6084, 6300, 7056, 8820, 9900, 10404, 11700, 12348, 12996, 14700, 15300, 17100, 19044, 19404, 20700, 21780, 22932, 26100, 27900, 29988, 30276, 30420, 30492, 31500, 33300, 33516, 34596, 35280, 36900, 38700, 40572, 42300, 42588, 47700
COMMENTS
Equivalently, numbers that are equal to the sum of their proper exponential divisors, with one of them taken with a minus sign.
EXAMPLE
900 is a term since 900 = 30 + 60 + 90 + 150 - 180 + 300 + 450 is the sum of its proper exponential divisors with one of them, 180, taken with a minus sign.
MATHEMATICA
dQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m]); expDivQ[n_, d_] := Module[{ft = FactorInteger[n]}, And @@ MapThread[dQ, {ft[[;; , 2]], IntegerExponent[d, ft[[;; , 1]]]}]]; esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; expAdmQ[n_] := (ab = esigma[n] - 2*n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && expDivQ[n, ab/2]; Select[Range[50000], expAdmQ]
CROSSREFS
The exponential version of A111592.
Nonunitary admirable numbers: numbers k such that there is a nonunitary divisor d of k such that nusigma(k) - 2*d = k, where nusigma is the sum of nonunitary divisors function ( A048146).
+10
1
48, 80, 96, 108, 120, 160, 168, 180, 192, 216, 224, 252, 264, 280, 300, 312, 320, 336, 352, 360, 384, 396, 408, 416, 432, 448, 456, 468, 480, 504, 528, 540, 552, 560, 600, 612, 624, 640, 672, 684, 696, 704, 720, 744, 756, 768, 792, 816, 828, 832, 840, 864, 880
COMMENTS
Equivalently, numbers that are equal to the sum of their nonunitary divisors, with one of them taken with a minus sign.
EXAMPLE
48 is a term since 48 = 2 - 4 + 6 + 8 + 12 + 24 is the sum of its nonunitary divisors with one of them, 4, taken with a minus sign.
MATHEMATICA
usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); nusigma[n_] := DivisorSigma[1, n] - usigma[n]; nuAdmQ[n_] := (ab = nusigma[n] - n) > 0 && EvenQ[ab] && ab/2 < n && !CoprimeQ[ab/2, 2*n/ab]; Select[Range[1000], nuAdmQ]
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