A109729 -id:A109729 - cp's OEIS frontend

A111592 Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.

12, 20, 24, 30, 40, 42, 54, 56, 66, 70, 78, 84, 88, 102, 104, 114, 120, 138, 140, 174, 186, 222, 224, 234, 246, 258, 270, 282, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 650, 654, 672, 678, 762, 786, 812

Offset: 1

Jason Earls, Aug 09 2005

All admirable numbers are abundant.
If 2^n-2^k-1 is an odd prime then m=2^(n-1)*(2^n-2^k-1) is in the sequence because 2^k is one of the proper divisors of m and sigma(m)-2m=(2^n-1)*(2^n-2^k)-2^n*(2^n-2^k-1)=2^k hence m=(sigma(m)-m)-2^k, namely m is an Admirable number. This is one of the results of the following theorem that I have found. Theorem: If 2^n-j-1 is an odd prime and m=2^(n-1)*(2^n-j-1) then sigma(m)-2m=j. The case j=0 is well known. - Farideh Firoozbakht, Jan 28 2006
In particular, these numbers have abundancy 2 to 3: 2 < sigma(n)/n <= 3. - Charles R Greathouse IV, Jan 30 2014
Subsequence of A083207. - Ivan N. Ianakiev, Mar 20 2017
The concept of admirable numbers was developed by educator Jerome Michael Sachs (1914-2012) for a television in-service training course in mathematics for elementary school teachers. - Amiram Eldar, Aug 22 2018
Odd terms are listed in A109729. For abundant nonsquares, it is equivalent to say sigma(n)/2 - n divides n. For squares, sigma(n)/2 - n is half-integer, but n could still be an integer multiple. This first occurs for n = m^2 with even m = 2^k*(2^(2*k+1)-1), k = 1, 2, 3, 6, ... (A146768), and odd m = 13167. - M. F. Hasler, Jan 26 2020

			12 = 1+3+4+6-2, 20 = 2+4+5+10-1, etc.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
F. Firoozbakht, M. F. Hasler, Variations on Euclid's formula for Perfect Numbers, JIS 13 (2010) #10.3.1.
Giovanni Resta, admirable numbers
J. M. Sachs, Admirable Numbers and Compatible Pairs, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
T. Trotter, Admirable Numbers

Subsequence of A005101 (abundant numbers).
Cf. A000396 (perfect numbers), A005100 (deficient numbers), A000203 (sigma), A061645.
Cf. A109729 (odd admirable numbers).

with(numtheory); isadmirable := proc(n) local b, d, S; b:=false; S:=divisors(n) minus {n}; for d in S do if sigma(n)-2*d=2*n then b:=true; break fi od; return b; end: select(proc(z) isadmirable(z) end, [$1..1000]); # Walter Kehowski, Aug 12 2005

fQ[n_] := Block[{d = Most[Divisors[n]], k = 1}, l = Length[d]; s = Plus @@ d; While[k < l && s - 2d[[k]] > n, k++ ]; If[k > l || s != n + 2d[[k]], False, True]]; Select[ Range[821], fQ[ # ] &] (* Robert G. Wilson v, Aug 13 2005 *)
Select[Range[812],MemberQ[Most[Divisors[#]],(DivisorSigma[1,#]-2*#)/2]&] (* Ivan N. Ianakiev, Mar 23 2017 *)

for(n=1,10^3,ap=sigma(n)-2*n;if(ap>0 && (ap%2)==0,d=ap/2;if(d!=n && (n%d)==0, print1(n",")))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

is(n)=if(issquare(n)||issquare(n/2),0,my(d=sigma(n)/2-n); d>0 && d!=n && n%d==0) \\ Charles R Greathouse IV, Jun 21 2011

Better definition from Walter Kehowski, Aug 12 2005

A329188 Odd unitary admirable numbers: the odd terms of A328328.

Original entry on oeis.org

80535, 354585, 403095, 430815, 437745, 442365, 5388495, 6126645, 9338595, 36340395, 130689195, 747242265, 1335049485, 2224695165, 4085490255, 9665740455, 10394173335, 11534750535, 13837748925, 33378237165, 73088757105, 94849396005, 109544822205, 216654032595

Offset: 1

Amiram Eldar, Nov 07 2019

nonn

Of the first 10^4 unitary admirable numbers only 6 are odd.

a(21) > 6*10^10.

Giovanni Resta, Table of n, a(n) for n = 1..43 (terms < 10^13)

The unitary version of A109729.
Intersection of A005408 and A328328.
Subsequence of A129485.

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); uadmQ[n_] := (ab = usigma[n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && CoprimeQ[2*n/ab, ab/2]; s = {}; Do[If[uadmQ[n], AppendTo[s, n]], {n, 1, 10^6, 2}]; s

Data corrected by Amiram Eldar, May 12 2020

Terms a(21) and beyond from Giovanni Resta, May 12 2020

A334973 Odd bi-unitary admirable numbers: the odd terms of A334972.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, May 18 2020

nonn

Of the first 10^4 bi-unitary admirable numbers only 11 are odd.

The bi-unitary version of A109729.
Intersection of A005408 and A334972.
Subsequence of A293186.
Cf. A329188, A334975.

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]

A334975 Odd infinitary admirable numbers: the odd terms of A334974.

Original entry on oeis.org

945, 43065, 46035, 80535, 354585, 403095, 430815, 437745, 442365, 2305875, 3525795, 4404105, 4891887, 5388495, 5927985, 6126645, 6220665, 6375105, 6537375, 7853625, 8109585, 8731125, 9071865, 9338595, 9784125, 13241745, 23760555, 33381855, 34592805, 35642295

Offset: 1

Amiram Eldar, May 18 2020

nonn

Of the first 10^4 infinitary admirable numbers only 9 are odd.

The infinitary version of A109729.
Intersection of A005408 and A334974.
Subsequence of A127666.
Cf. A329188, A334973.

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[1, 5*10^5, 2], infAdmQ]

A364727 Numbers k such that k and k+1 are both admirable numbers (A111592).

Original entry on oeis.org

29691198404, 478012798575, 2789405835075, 22929723392715

Offset: 1

Amiram Eldar, Aug 05 2023

a(1)-a(2) were found by Giovanni Resta.

Giovanni Resta, admirable numbers, Numbers Aplenty.

Subsequence of A096399 and A111592.

Cf. A109729, A109730.

a(4) from Martin Ehrenstein, Aug 06 2023

A336681 Odd exponential admirable numbers: the odd terms of A336680.

Original entry on oeis.org

6485886225, 71344748475, 110260065825, 123231838275, 125730522225, 149175383175, 162485579025, 185601564225, 188090700525, 191620685025, 195686793225, 201062472975, 239977790325, 265921335225, 278893107675, 304836652575, 343751969925, 395639059725, 434554377075

Offset: 1

Amiram Eldar, Jul 30 2020

nonn

Exponential admirable numbers that are odd are relatively rare: there are 5742336 even exponential admirable numbers that are smaller than the first odd term, i.e., a(1) = A336680(5742337).

			6485886225 is a term since 6485886225 = 80535 + 241605 + ... + (-8456175) + ... + 2161962075 is the sum of its proper exponential divisors with one of them, 8456175, taken with a minus sign.

Amiram Eldar, Table of n, a(n) for n = 1..65

The exponential version of A109729.
Intersection of A005408 and A336680.
Subsequence of A321147.
Similar sequences: A329188, A334973, A334975.

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[1, 10^9, 2], expAdmQ]

cp's OEIS Frontend

A111592 Admirable numbers. A number n is admirable if there exists a proper divisor d' of n such that sigma(n)-2d'=2n, where sigma(n) is the sum of all divisors of n.

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A329188 Odd unitary admirable numbers: the odd terms of A328328.

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A334973 Odd bi-unitary admirable numbers: the odd terms of A334972.

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A334975 Odd infinitary admirable numbers: the odd terms of A334974.

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A364727 Numbers k such that k and k+1 are both admirable numbers (A111592).

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A336681 Odd exponential admirable numbers: the odd terms of A336680.

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Mathematica