A111592 -id:A111592 - cp's OEIS frontend

A109797 First of a pair of compatible numbers, where two numbers m and n are compatible if sigma(n)-2dn=sigma(m)-2dm=m+n, for some proper divisors dn and dm of m and n respectively.

24, 30, 40, 42, 48, 60, 80, 80, 96, 102, 126, 140, 140, 156, 156, 156, 174, 180, 180, 198, 216, 224, 224, 264, 276, 280, 294, 294, 300, 320, 340, 372, 380, 384, 440, 440, 468, 500, 504, 510, 528, 560, 582, 608, 616, 642, 680, 684, 690, 690, 696, 702, 736, 750

Offset: 1

Walter Kehowski, Aug 15 2005

nonn

Compatible numbers were introduced by Sachs in analogy to amicable numbers, as admirable numbers are analogous to perfect numbers. Some terms have more than one counterpart (A109798), like 80 (two counterparts: 102 and 104) or 156 (3 counterparts: 210, 230 and 234). - Amiram Eldar, Oct 26 2019

			sigma(42)-2(1)=96-2=94 and sigma(52)-2(2)=98-4=94 and 42+52=94 so a(4)=42.

Amiram Eldar, Table of n, a(n) for n = 1..1000
J. M. Sachs, Admirable Numbers and Compatible Pairs, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
T. Trotter, Admirable Numbers.

Cf. A109798, A111592.

L:=remove(proc(z) isprime(z) end, [$1..10000]): S:=proc(n) map(proc(z) sigma(n) -2*z end, divisors(n) minus {n}) end; CK:=map(proc(z) [z,S(z)] end, L): CL:=[]: for j from 1 to nops(CK)-1 do x:=CK[j,1]; sx:=sigma(x); Sx:=CK[j,2]; for k from j+1 to nops(CK) while CK[k,1]

seq = {}; Do[d = Most[Divisors[n]]; s = Total[d]; Do[m = s - 2*d[[k]]; If[m <= 0 || m <= n, Continue[]]; delta = DivisorSigma[1, m] - m - n; If[delta > 0 && EvenQ[delta] && delta/2 < m && Divisible[m, delta/2], AppendTo[seq, n]], {k, Length[d], 1, -1}], {n, 1, 750}]; seq (* Amiram Eldar, Oct 26 2019 *)

A109798 Second of a pair of compatible numbers, where two numbers m and n are compatible if sigma(n)-2dn=sigma(m)-2dm=m+n, for some proper divisors dn and dm of m and n respectively.

Original entry on oeis.org

28, 40, 42, 52, 60, 96, 102, 104, 124, 110, 182, 182, 188, 210, 230, 234, 184, 358, 362, 204, 312, 248, 252, 408, 372, 424, 306, 388, 418, 434, 376, 516, 384, 508, 530, 638, 782, 572, 888, 782, 828, 872, 592, 644, 820, 650, 938, 908, 1026, 1034, 1102, 976, 760

Offset: 1

Walter Kehowski, Aug 15 2005

nonn

The terms are arranged by the order of their lesser counterparts (A109797). - Amiram Eldar, Oct 26 2019

			sigma(42)-2(1)=96-2=94 and sigma(52)-2(2)=98-4=94 and 42+52=94 so a(4)=52.

Amiram Eldar, Table of n, a(n) for n = 1..1000
J. M. Sachs, Admirable Numbers and Compatible Pairs, The Arithmetic Teacher, Vol. 7, No. 6 (1960), pp. 293-295.
T. Trotter, Admirable Numbers.

Cf. A109797, A111592.

L:=remove(proc(z) isprime(z) end, [$1..10000]): S:=proc(n) map(proc(z) sigma(n) -2*z end, divisors(n) minus {n}) end; CK:=map(proc(z) [z,S(z)] end, L): CL:=[]: for j from 1 to nops(CK)-1 do x:=CK[j,1]; sx:=sigma(x); Sx:=CK[j,2]; for k from j+1 to nops(CK) while CK[k,1]

seq = {}; Do[d = Most[Divisors[n]]; s = Total[d]; Do[m = s - 2*d[[k]]; If[m <= 0 || m <= n, Continue[]]; delta = DivisorSigma[1, m] - m - n; If[delta > 0 && EvenQ[delta] && delta/2 < m && Divisible[m, delta/2], AppendTo[seq, m]], {k, Length[d], 1, -1}], {n, 1, 750}]; seq (* Amiram Eldar, Oct 26 2019 *)

A111646 Subtracted divisor in admirable numbers.

Original entry on oeis.org

2, 1, 6, 6, 5, 6, 6, 4, 6, 2, 6, 28, 2, 6, 1, 6, 60, 6, 28, 6, 6, 6, 28, 39, 6, 6, 90, 6, 28, 6, 6, 28, 6, 4, 6, 6, 6, 1, 6, 28, 6, 28, 6, 6, 6, 6, 6, 28, 1, 6, 336, 6, 6, 6, 28, 6, 6, 4, 28, 6, 6, 6, 15, 6, 16, 6, 28, 6, 6, 6, 6, 28, 6, 6, 6, 28, 6, 28, 6, 6, 28, 6, 6, 6, 6, 28, 496, 8, 6, 6, 6, 6

Offset: 1

Jason Earls, Aug 10 2005

			a(1)=2 because 12 = 1+3+4+6-2, the 2 is subtracted. a(6)=6 because 42 = 1+2+3+7+14+21-6, the 6 is subtracted.

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

Cf. A033880, A111592, A111667.

a(n) = A033880(A111592(n))/2 = A111667(n)/2. - Amiram Eldar, Jun 22 2019

A291459 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 5.

Original entry on oeis.org

294053760, 575134560, 739458720, 882161280, 1193512320, 1314593280, 1725403680, 2539555200, 2588105520, 2646483840, 2711348640, 3008396160, 3891888000, 4053329280, 4214770560, 4648644000, 4802878080, 5176211040, 5194949760, 5258373120, 6470263800, 6768891360, 7900532640

Offset: 1

Paolo P. Lava, Aug 24 2017

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

Subset of A215264.

			One of the proper divisors of 294053760 is 2056320 and sigma(294053760) - 5*2056320 = 1480550400 - 10281600 = 1470268800 = 5*294053760.
One of the proper divisors of 3891888000 is 314496 and sigma(3891888000) - 5*314496 = 19461012480 - 1572480 = 19459440000 = 5*3891888000.

Cf. A000203, A111592, A215264, A291457, A291458.

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(10^10,5);

A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2d = 2k, where esigma is the sum of exponential divisors function (A051377).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 30 2020

nonn

Equivalently, numbers that are equal to the sum of their proper exponential divisors, with one of them taken with a minus sign.

			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.

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

The exponential version of A111592.
Subsequence of A129575.
Similar sequences: A328328, A334972, A334974.
Cf. A051377, A322791.

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]

A109131 Admirable numbers such that the subtracted divisor is a triangular number.

Original entry on oeis.org

20, 24, 30, 42, 54, 66, 78, 84, 102, 104, 114, 138, 140, 174, 186, 222, 224, 246, 258, 282, 308, 318, 354, 364, 366, 402, 426, 438, 464, 474, 476, 498, 532, 534, 582, 606, 618, 642, 644, 650, 654, 678, 762, 786, 812, 822, 834, 868, 894, 906, 942, 945, 978

Offset: 1

Jason Earls, Aug 17 2005

			a(2)=24 because 1+2+3+4+8+12-6 = 24 and the subtracted divisor is triangular.

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

Cf. A000217 (triangular numbers), A111592 (admirable numbers).

triQ[n_] := IntegerQ @ Sqrt[8n + 1]; Select[Range[1000], MemberQ[Divisors[#], (d = (DivisorSigma[1, #] - 2#)/2)] && triQ[d] &] (* Amiram Eldar, Sep 21 2019 *)

A109396 Admirable Harshad numbers such that the subtracted divisor is also a Harshad number.

Original entry on oeis.org

12, 20, 24, 30, 40, 42, 54, 70, 102, 114, 120, 222, 270, 402, 1002, 2022, 2202, 10002, 10014, 10792, 11202, 12102, 21102, 31002, 32128, 45356, 103002, 110202, 111102, 128768, 740870, 1000002, 1000014, 1001202, 1002102, 1021002, 1111002, 1200102

Offset: 1

Jason Earls, Aug 26 2005

			12 is in the sequence since it is a Harshad number (1 + 2 = 3 is a divisor of 12), an admirable number: 1 - 2 + 3 + 4 + 6 = 12, and the subtracted divisor, 2, is also a Harshad number.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A111592, A005349, A111947, A111948.

hQ[n_] := Divisible[n, Plus @@ IntegerDigits@n]; aQ[n_] := hQ[n] && (d = DivisorSigma[1, n] - 2*n) > 0 && EvenQ[d] && d/2 < n && hQ[d/2] && Divisible[n, d/2]; Select[Range[50000], aQ] (* Amiram Eldar, Oct 27 2019 *)

A109745 Admirable numbers that set a new record for largest subtracted divisor.

Original entry on oeis.org

12, 24, 84, 120, 270, 672, 1488, 1638, 6200, 24384, 174592, 523776, 44736512, 91963648, 100651008, 459818240, 1476304896, 10200236032, 25769607168, 51001180160, 412316073984

Offset: 1

Jason Earls, Aug 10 2005

The records set are A000203(a(n))/2 - a(n) = 2, 6, 28, 60, 90, 336, 496, 546, 1240, 8128, 21824,... - R. J. Mathar, Feb 12 2008

Cf. A000203, A111592, A111646.

admDiv[n_] := Module[{ab = DivisorSigma[1, n] - 2*n}, If[ab > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2], ab/2, 0]];
seq[nmax_] := Module[{dmax = 0, s = {}, d}, Do[d = admDiv[n]; If[d > dmax, dmax = d; AppendTo[s, n]], {n, 1, nmax}]; s]; seq[6*10^5] (* Amiram Eldar, Aug 05 2023 *)

a(13)-a(20) from Donovan Johnson, Nov 11 2008

a(21) from Amiram Eldar, Aug 05 2023

A109759 Palindromic admirable numbers.

Original entry on oeis.org

66, 88, 222, 282, 464, 474, 606, 868, 2002, 20802, 24042, 24342, 24942, 29092, 41214, 44144, 45354, 46564, 47274, 60906, 64146, 66966, 67676, 80108, 81318, 83238, 85458, 85758, 87378, 89898, 2002002, 2008002, 2024202, 2027202, 2032302

Offset: 1

Jason Earls, Aug 12 2005

Amiram Eldar, Table of n, a(n) for n = 1..1000

Intersection of A002113 and A111592.

palQ[n_] := PalindromeQ @ IntegerDigits[n]; admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2]; Select[Range[2*10^6], palQ[#] && admQ[#] &] (* Amiram Eldar, Oct 27 2019 *)

A109788 Admirable numbers whose abundance is < 10.

Original entry on oeis.org

12, 20, 56, 70, 88, 104, 368, 464, 650, 836, 1888, 1952, 4030, 5830, 8925, 11096, 17816, 32128, 32445, 45356, 77744, 91388, 128768, 130304, 254012, 388076, 442365, 521728, 522752, 1848964, 2087936, 2291936, 8378368, 8382464, 13174976, 29465852, 35021696, 45335936

Offset: 1

Jason Earls, Aug 14 2005

nonn

Apparently all the abundant numbers with even abundance < 10 are admirable (checked for the first 60 terms). Of the 4 known numbers whose abundance is 10 (A223609), only the first, 40, is admirable. - Amiram Eldar, Nov 07 2019

Amiram Eldar, Table of n, a(n) for n = 1..60

Cf. A033880, A111592, A111667.

aQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab < 10 && ab/2 < n && Divisible[n, ab/2]; Select[Range[10^4], aQ] (* Amiram Eldar, Nov 07 2019 *)

for(n=1,10^9,my(ap=sigma(n)-2*n); if(ap>0 && ap<10 && ap%2==0, my(d=ap/2); if(d!=n && n%d==0, print1(n, ", ")))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 30 2008

a(36)-a(38) from Amiram Eldar, Nov 07 2019

cp's OEIS Frontend

A109797 First of a pair of compatible numbers, where two numbers m and n are compatible if sigma(n)-2dn=sigma(m)-2dm=m+n, for some proper divisors dn and dm of m and n respectively.

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A109798 Second of a pair of compatible numbers, where two numbers m and n are compatible if sigma(n)-2dn=sigma(m)-2dm=m+n, for some proper divisors dn and dm of m and n respectively.

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A111646 Subtracted divisor in admirable numbers.

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A291459 Numbers n having a proper divisor d such that sigma(n) - k*d = k*n. Case k = 5.

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A336680 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).

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Mathematica

A109131 Admirable numbers such that the subtracted divisor is a triangular number.

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Mathematica

A109396 Admirable Harshad numbers such that the subtracted divisor is also a Harshad number.

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Mathematica

A109745 Admirable numbers that set a new record for largest subtracted divisor.

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Extensions

A109759 Palindromic admirable numbers.

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A291459 Numbers n having a proper divisor d such that sigma(n) - kd = kn. Case k = 5.

A336680 Exponential admirable numbers: numbers k such that there is a proper exponential divisor d of k such that esigma(k) - 2d = 2k, where esigma is the sum of exponential divisors function (A051377).