A111592 -id:A111592 - cp's OEIS frontend

A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2d = 2k, where isigma is the sum of infinitary divisors function (A049417).

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

Offset: 1

Amiram Eldar, May 18 2020

nonn

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, ...

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The infinitary version of A111592.
Subsequence of A129656.
Cf. A036537, A049417, A077609, A328328, A334972.

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]

A329189 3-admirable numbers: 3-abundant numbers (A068403) k such that exists a proper divisor d of k such that sigma(k) - 2d = 3k, where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

180, 240, 360, 420, 504, 540, 600, 780, 1080, 1344, 1872, 1890, 2016, 2184, 2352, 2376, 2688, 3192, 3276, 3744, 4284, 4320, 4680, 5292, 5376, 5796, 6048, 6552, 7128, 7344, 7440, 8190, 9504, 10296, 10416, 13776, 14850, 18600, 19824, 19872, 20496, 21528, 22932

Offset: 1

Amiram Eldar, Nov 07 2019

nonn

Analogous to admirable numbers (A111592) as 3-perfect numbers (A005820) are analogous to perfect numbers (A000396).

The proper divisors of each term k can be added to a sum of 2*k with one divisor taken with a minus sign.

			180 is a term since its proper divisors can be added to 1 + 2 - 3 + 4 + 5 + 6 + 9 + 10 + 12 + 15 + 18 + 20 + 30 + 36 + 45 + 60 + 90 = 360 = 2 * 180, with one divisor, 3, taken with a minus sign.

Amiram Eldar, Table of n, a(n) for n = 1..810 (terms below 10^11)

Cf. A000203, A000396, A068403, A111592.

aQ[n_] := (ab = DivisorSigma[1, n] - 3 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2]; Select[Range[23000], aQ]

A109730 Lesser of twin admirable numbers: k such that k and k+2 are both admirable numbers.

Original entry on oeis.org

40, 54, 102, 138, 222, 364, 366, 474, 532, 642, 834, 1036, 1146, 1372, 1504, 1876, 1986, 2994, 3052, 3556, 4396, 4564, 5514, 5572, 5622, 6198, 6412, 6522, 7026, 7912, 7924, 8202, 8596, 8706, 9424, 9714, 10444, 10722, 11226, 11406, 12066, 12964

Offset: 1

Jason Earls, Aug 09 2005

nonn

Conjecture: Sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A111592.

Select[Range[13000], MemberQ[Most[Divisors[#]], (DivisorSigma[1,#]-2*#)/2] && MemberQ[Most[Divisors[#+2]], (DivisorSigma[1,#+2]-2*(#+2))/2]&] (* James C. McMahon, Mar 29 2024 *)

A111667 Abundance of admirable numbers.

Original entry on oeis.org

4, 2, 12, 12, 10, 12, 12, 8, 12, 4, 12, 56, 4, 12, 2, 12, 120, 12, 56, 12, 12, 12, 56, 78, 12, 12, 180, 12, 56, 12, 12, 56, 12, 8, 12, 12, 12, 2, 12, 56, 12, 56, 12, 12, 12, 12, 12, 56, 2, 12, 672, 12, 12, 12, 56, 12, 12, 8, 56, 12, 12, 12, 30, 12, 32, 12, 56, 12, 12, 12, 12, 56

Offset: 1

Jason Earls, Aug 14 2005

			a(1)=4 because the first admirable number is 12 and sigma(12)-24 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A033880, A111592.

fn[n_]:=DivisorSigma[1,n]-2*n;fn/@Select[Range[1155], MemberQ[Most[Divisors[#]], (DivisorSigma[1, #]-2*#)/2]&] (* James C. McMahon, Jun 02 2024 *)

A111947 Admirable Harshad numbers.

Original entry on oeis.org

12, 20, 24, 30, 40, 42, 54, 70, 84, 102, 114, 120, 140, 222, 224, 234, 270, 308, 364, 402, 476, 644, 1002, 1148, 1204, 1638, 1652, 2022, 2202, 2212, 3164, 3250, 3472, 4172, 6200, 7588, 8204, 8432, 8596, 9424, 10002, 10014, 10724, 10792, 11202, 11228

Offset: 1

Jason Earls, Aug 22 2005

			a(3)=24 because 1+2+3+4+8+12-6 = 24 and 24/6 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005349 and A111592.

Cf. A111948.

harsQ[n_] := Divisible[n, Plus @@  IntegerDigits[n]]; admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2]; Select[Range[12000], harsQ[#] && admQ[#] &] (* Amiram Eldar, Oct 27 2019 *)

A111948 Admirable Harshad numbers n such that the subtracted divisor is equal to the digital sum of n.

Original entry on oeis.org

24, 42, 114, 222, 402, 2022, 2202, 7588, 8596, 10014, 11202, 12102, 17668, 21102, 27748, 29764, 31002, 32788, 39844, 42868, 43876, 45388, 46396, 48916, 49924, 55972, 56476, 57484, 58492, 65548, 66556, 69076, 70588, 71596, 78148, 81676

Offset: 1

Jason Earls, Aug 22 2005

			a(2)=42 because 1+2+3+7+14+21-6 = 42 and 42/6 = 7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A111592, A005349, A111947.

harsQ[n_] := Divisible[n, Plus @@  IntegerDigits[n]]; admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] &&  Plus @@ IntegerDigits[n] == ab/2 && ab/2 < n && Divisible[n, ab/2]; Select[Range[100000], harsQ[#] && admQ[#] &] (* Amiram Eldar, Oct 27 2019 *)

A165772 Numbers d*p where d is a perfect number and p

Original entry on oeis.org

30, 84, 140, 308, 364, 476, 532, 644, 1488, 2480, 3472, 5456, 6448, 8432, 9424, 11408, 14384, 18352, 20336, 21328, 23312, 24384, 26288, 29264, 30256, 33232, 35216, 36208, 39184, 40640, 41168, 44144, 48112, 50096, 51088, 53072, 54064, 56048

Offset: 1

M. F. Hasler, Oct 11 2009

nonn

A subsequence of A109321, and thus admirable numbers (A111592, solutions to sigma(x)-2x = 2d with d being a proper divisor of x): If d is a perfect number (A000396), then for any prime pA111592) and d > sqrt(dp).

			For d = 6 = 2*3, we must omit 3*d (because 3 | d) and get a(1) = 5*d = 30.
For d = 28 = 4*7, we get a(2) = 3*d = 84, a(3) = 5*d = 140, we omit 7*d,
  a(4) = 11*d = 308, a(5) = 13*d = 364, a(6) = 17*d = 476, a(7) = 19*d = 532,
  a(8) = 23*d = 644. So far all terms are in order of increasing size.
For d = 496 = 16*31, we get a(9) = 3*d = 1488 through a(21) = 47*d = 23312 (omitting 31*d), but the next larger term a(22) comes from the next perfect number, see below. Then we get a(23) = 53*d = 26288 through a(29) = 39184, a(31) = 41168 through a(38) = 56048, and a(40) = 62992.
For d = 8128 = 64*127, we get a(22) = 3*d = 24384, a(30) = 5*d = 40640, a(39) = 56896, a(41) = 89408, and all following terms up to 3*4096*8191.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000396, A109321, A111592.

f[p_] := (2^p - 1)*2^(p - 1); evenPerf[n_] := f[MersennePrimeExponent[n]]; sp[p_, max_] := With[{pn = f[p]}, pn * Select[Complement[Range[3, Min[pn - 1, max/pn]], {2^p - 1}], PrimeQ]];
seq[max_] := Module[{s = {}, k = 1}, While[(pn = evenPerf[k]) < max/3, s = Join[s, sp[MersennePrimeExponent[k], max]]; k++]; Union[s]]; seq[60000] (* Amiram Eldar, Aug 05 2023, assuming that there are no odd perfect numbers below max *)

forprime(q=1,9, isprime(2^q-1)||next; print("\n/* q="q", d=",d=(2^q-1)<<(q-1)," */"); forprime(p=3,d-1, d%p || next; print1(d*p,", "))) /* Note: This prints the terms in order of increasingly large perfect numbers, not in order of increasing terms: e.g., 243536, the last value for d = 496 = (2^5-1)*2^4, is printed before 24384, first term for d = 8128 = (2^7-1)*2^6. */

A165772_upto(N=10^5)=select({
  is_A165772(n)=my(v=valuation(n, 2), P); isprime(v+1) && (n=divrem(n>>v, P=2^(v+1)-1))[2]==0 && n[1] < P<M. F. Hasler, Jul 30 2024

A291457 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 3.

Original entry on oeis.org

180, 240, 360, 420, 480, 540, 600, 660, 780, 840, 1080, 1320, 1560, 1890, 1920, 2016, 2040, 2184, 2280, 2352, 2376, 2688, 2760, 2856, 3000, 3192, 3360, 3480, 3720, 3744, 4284, 4320, 4440, 4680, 4704, 4896, 4920, 5160, 5292, 5640, 5796, 6048, 6360, 6552, 7080, 7128

Offset: 1

Paolo P. Lava, Aug 24 2017

Case k=2 are the admirable numbers (A111592).

Subset of A023197, A068403, A068545, A204828, A204830.

			One of the proper divisors of 1080 is 120 and sigma(1080) - 3*120 = 3600 - 360 = 3240 = 3*1080.
One of the proper divisors of 17850 is 6 and sigma(17850) - 3*6 = 53568 - 18 = 53550 = 3*17850.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1500 from Paolo P. Lava)

Cf. A000203, A023197, A068403, A068545, A111592, A204828, A204830, A291458, A291459.

with(numtheory): P:=proc(q,h) local a,b,c,k; c:=0; a:=sort([op(divisors(q))]); for k from 1 to nops(a)-1 do if sigma(q)-h*a[k]=h*q then c:=1; break; fi; od; if c=1 then q; fi; end: seq(P(i,3),i=1..7200);

k=3; Select[Range[7128], (t = DivisorSigma[1, #]/k - #; # > t > 0 && IntegerQ[t] && Mod[#, t] == 0) &] (* Giovanni Resta, Aug 25 2017 *)

A291458 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 4.

Original entry on oeis.org

27720, 60480, 65520, 90720, 98280, 105840, 115920, 120120, 120960, 128520, 131040, 143640, 151200, 163800, 180180, 191520, 205920, 207900, 211680, 218400, 229320, 235620, 241920, 249480, 264600, 272160, 289800, 292320, 312480, 332640, 360360, 372960, 393120, 414960

Offset: 1

Paolo P. Lava, Aug 24 2017

Case k=2 are the admirable numbers (A111592).

Subset of A023198, A068404, A204831, A230608.

			One of the proper divisors of 27720 is 360 and sigma(27720) - 4*360 = 112320 - 1440 = 110880 = 4*27720.
One of the proper divisors of 115920 is 144 and sigma(115920) - 4*144 = 464256 - 576 = 463680 = 4*115920.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A023198, A068404, A111592, A204831, A230608, A291457, A291459.

with(numtheory): P:=proc(q,h) local a,k,n; for n from 1 to q do a:=sort([op(divisors(n))]);
for k from 1 to nops(a)-1 do if sigma(n)-h*a[k]=h*n then print(n); break; fi; od; od; end: P(10^9,4);

With[{k = 4}, Select[Range[5 * 10^5], Function[n, AnyTrue[Most@ Divisors@ n, DivisorSigma[1, n] - k # == k n &]]]] (* Michael De Vlieger, Aug 24 2017 *)
(* or *)
k=4; Select[Range[5*^5], (t = DivisorSigma[1, #]/k - #; #>t>0 && IntegerQ[t] && Mod[#, t] == 0) &] (* much faster, Giovanni Resta, Aug 25 2017 *)

A109321 Admirable numbers n such that the subtracted divisor is > sqrt(n).

Original entry on oeis.org

24, 30, 84, 120, 140, 224, 234, 270, 308, 364, 476, 532, 644, 672, 1488, 1638, 2480, 3472, 3724, 4095, 5456, 5624, 6200, 6435, 6448, 8432, 9424, 11408, 14384, 15872, 18352, 20336, 21328, 23312, 24384, 26288, 29264, 29450, 30256, 33232, 35150

Offset: 1

Jason Earls, Aug 20 2005

nonn

Solutions to sigma(x)=2(x+d) with d > sqrt(x) being a proper divisor of x. The subsequence A165772 contains most of the terms. - M. F. Hasler, Oct 11 2009

			a(2)=30 because 1+2+3+5+10+15-6 = 30 and 6 > sqrt(30) = 5.477...

Amiram Eldar, Table of n, a(n) for n = 1..1200

Cf. A111592.

aQ[n_] := (d = DivisorSigma[1, n] - 2n) > 0 && EvenQ[d] && Mod[n, d/2] == 0 && d < 2n && d^2 > 4n; Select[Range[35150], aQ]  (* Amiram Eldar, Sep 22 2019 *)

is_A109321(n)= my(d=sigma(n)-2*n); d>0 && bittest(d,0)==0 && d<2*n && d*d>4*n && 2*n%d==0 \\ M. F. Hasler, Oct 11 2009

cp's OEIS Frontend

A334974 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).

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A329189 3-admirable numbers: 3-abundant numbers (A068403) k such that exists a proper divisor d of k such that sigma(k) - 2*d = 3*k, where sigma(k) is the sum of divisors of k (A000203).

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A109730 Lesser of twin admirable numbers: k such that k and k+2 are both admirable numbers.

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A111667 Abundance of admirable numbers.

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A111947 Admirable Harshad numbers.

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A111948 Admirable Harshad numbers n such that the subtracted divisor is equal to the digital sum of n.

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A165772 Numbers d*p where d is a perfect number and p

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A291457 Numbers n having a proper divisor d such that sigma(n) - k*d = k*n. Case k = 3.

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A334974 Infinitary admirable numbers: numbers k such that there is a proper infinitary divisor d of k such that isigma(k) - 2d = 2k, where isigma is the sum of infinitary divisors function (A049417).

A329189 3-admirable numbers: 3-abundant numbers (A068403) k such that exists a proper divisor d of k such that sigma(k) - 2d = 3k, where sigma(k) is the sum of divisors of k (A000203).

A291457 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 3.

A291458 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 4.