A101113 -id:A101113 - cp's OEIS frontend

A005101 Abundant numbers (sum of divisors of m exceeds 2m).

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270

Offset: 1

N. J. A. Sloane

A number m is abundant if sigma(m) > 2m (this sequence), perfect if sigma(m) = 2m (cf. A000396), or deficient if sigma(m) < 2m (cf. A005100), where sigma(m) is the sum of the divisors of m (A000203).
While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number!
It appears that for m abundant and > 23, 2*A001055(m) - A101113(m) is NOT 0. - Eric Desbiaux, Jun 01 2009
If m is a term so is every positive multiple of m. "Primitive" terms are in A091191.
If m=6k (k>=2), then sigma(m) >= 1 + k + 2*k + 3*k + 6*k > 12*k = 2*m. Thus all such m are in the sequence.
According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Thus the n-th abundant number is asymptotic to 4.0322*n < n/A(2) < 4.0421*n. - Daniel Forgues, Oct 11 2015
From Bob Selcoe, Mar 28 2017 (prompted by correspondence with Peter Seymour): (Start)
Applying similar logic as the proof that all multiples of 6 >= 12 appear in the sequence, for all odd primes p:
i) all numbers of the form j*p*2^k (j >= 1) appear in the sequence when p < 2^(k+1) - 1;
ii) no numbers appear when p > 2^(k+1) - 1 (i.e., are deficient and are in A005100);
iii) when p = 2^(k+1) - 1 (i.e., perfect numbers, A000396), j*p*2^k (j >= 2) appear.
Note that redundancies are eliminated when evaluating p only in the interval [2^k, 2^(k+1)].
The first few even terms not of the forms i or iii are {70, 350, 490, 550, 572, 650, 770, ...}. (End)

L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Pure Appl. Math., Vol. 44 (1913), pp. 264-296.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 59.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 128.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. Britton, Perfect Number Analyser.
C. K. Caldwell, The Prime Glossary, abundant number.
Marc Deléglise, Bounds for the density of abundant integers, Experiment. Math., Volume 7, Issue 2 (1998), pp. 137-143.
Jason Earls, On Smarandache repunit n numbers, in Smarandache Notions Journal, Vol. 14, No. 1 (2004), page 243.
S. Flora Jeba, Anirban Roy, and Manjil P. Saikia, On k-Facile Perfect Numbers, Algebra and Its Applications (ICAA-2023) Springer Proc. Math. Stat., Vol. 474, 111-121. See p. 112.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Walter Nissen, Abundancy : Some Resources .
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
Tyler Ross, A Perfect Number Generalization and Some Euclid-Euler Type Results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.7.5. See p. 10.
Eric Weisstein's World of Mathematics, Abundant Number.
Eric Weisstein's World of Mathematics, Abundance.
Wikipedia, Abundant number.
Index entries for "core" sequences.

Cf. A005835, A005100, A091194, A091196, A080224, A091191 (primitive).
Cf. A005231 and A006038 (odd abundant numbers).
Cf. A094268 (n consecutive abundant numbers).
Cf. A173490 (even abundant numbers).
Cf. A001065.
Cf. A000396 (perfect numbers).
Cf. A302991.

a005101 n = a005101_list !! (n-1)
a005101_list = filter (\x -> a001065 x > x) [1..]
-- Reinhard Zumkeller, Nov 01 2015, Jan 21 2013

with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d,`,n) fi: od:
isA005101 := proc(n)
    simplify(numtheory[sigma](n) > 2*n) ;
end proc: # R. J. Mathar, Jun 18 2015
A005101 := proc(n)
    option remember ;
    local a ;
    if n =1 then
        12 ;
    else
        a := procname(n-1)+1 ;
        while numtheory[sigma](a) <= 2*a do
            a := a+1 ;
        end do ;
        a ;
    end if ;
end proc: # R. J. Mathar, Oct 11 2017

abQ[n_] := DivisorSigma[1, n] > 2n; A005101 = Select[ Range[270], abQ[ # ] &] (* Robert G. Wilson v, Sep 15 2005 *)
Select[Range[300], DivisorSigma[1, #] > 2 # &] (* Vincenzo Librandi, Oct 12 2015 *)

isA005101(n) = (sigma(n) > 2*n) \\ Michael B. Porter, Nov 07 2009

from sympy import divisors
def ok(n): return sum(divisors(n)) > 2*n
print(list(filter(ok, range(1, 271)))) # Michael S. Branicky, Aug 29 2021

from sympy import divisor_sigma
from itertools import count, islice
def A005101_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) > 2*n, count(max(startvalue, 1))) # generator of terms >= startvalue
A005101_list = list(islice(A005101_gen(), 20)) # Chai Wah Wu, Jan 14 2022

a(n) is asymptotic to C*n with C=4.038... (Deléglise, 1998). - Benoit Cloitre, Sep 04 2002
A005101 = { n | A033880(n) > 0 }. - M. F. Hasler, Apr 19 2012
A001065(a(n)) > a(n). - Reinhard Zumkeller, Nov 01 2015

A191323 Increasing sequence generated by these rules: a(1)=1, and if x is in a then [3x/2]+1 and 3x+1 are in a, where [ ]=floor.

Original entry on oeis.org

1, 2, 4, 7, 11, 13, 17, 20, 22, 26, 31, 34, 40, 47, 52, 61, 67, 71, 79, 92, 94, 101, 103, 107, 119, 121, 139, 142, 152, 155, 157, 161, 179, 182, 184, 202, 209, 214, 229, 233, 236, 238, 242, 269, 274, 277, 283, 304, 310, 314, 322, 344, 350, 355, 358, 364, 404, 412, 416, 418, 425, 427, 457, 466, 472, 484, 517, 526, 533, 538, 547, 553

Offset: 1

Clark Kimberling, May 30 2011

nonn

This sequence represents a class of sequences generated by rules of the form "a(1)=1, and if x is in a then floor(hx+i) and floor(jx+k) are in a, where h and j are rational numbers and i and k are positive integers."  In the following examples, the floor function is denoted by [ ].
A191323:  [3x/2]+1, 3x+1
A191324:  [3x/2]+1, 3x+2
A191325:  [3x/2], [5x/2]
A191326:  [3x/2], [7x/2]
A191327:  [5x/2], [7x/2]
A191328:  [5x/3], [7x/3]
Other families of sequences generated by "rules" are listed at A191803, A191106, A101113 and A191203.

			1 -> 2,4 -> 6,7,13 -> 10,11,19,20,22,40 -> ...

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A190503, A191100, A191113, A191203.

h = 3; i = 1; j = 3; k = 1; f = 1; g = 12;
a=Union[Flatten[NestList[{Floor[h#/2]+i,j#+k}&,f,g]]]
(* A191323 *)

A191134 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x+1 and 4x-1 are in a.

Original entry on oeis.org

1, 3, 4, 10, 11, 13, 15, 31, 34, 39, 40, 43, 46, 51, 59, 94, 103, 118, 121, 123, 130, 135, 139, 154, 155, 159, 171, 178, 183, 203, 235, 283, 310, 355, 364, 370, 375, 391, 406, 411, 418, 463, 466, 471, 478, 483, 491, 514, 519, 535, 539, 550, 555, 610, 615, 619, 635, 683, 706, 711, 731, 811, 850, 931, 939, 1066, 1093, 1111, 1126

Offset: 1

Clark Kimberling, May 28 2011

nonn

See A101113/

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A191113.

import Data.Set (singleton, deleteFindMin, insert)
a191134 n = a191134_list !! (n-1)
a191134_list = f $ singleton 1
   where f s = m : (f $ insert (3*m+1) $ insert (4*m-1) s')
             where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Jun 01 2011

h = 3; i = 1; j = 4; k = -1; f = 1; g = 9;
a = Union[Flatten[NestList[{h # + i, j # + k} &, f, g]]]   (* A191134 *)
b = (a - 1)/3; c = (a + 1)/4; r = Range[1, 1500];
d = Intersection[b, r] (* A191192 *)
e = Intersection[c, r] (* A191193 *)

A101114 Let t(G) = number of unitary factors of the Abelian group G. Then a(n) = sum t(G) over all Abelian groups G of order <= n.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 23, 27, 31, 33, 41, 43, 47, 51, 61, 63, 71, 73, 81, 85, 89, 91, 103, 107, 111, 117, 125, 127, 135, 137, 151, 155, 159, 163, 179, 181, 185, 189, 201, 203, 211, 213, 221, 229, 233, 235, 255, 259, 267, 271, 279, 281, 293, 297, 309, 313, 317

Offset: 1

Russ Cox, Dec 01 2004

From Schmidt paper: Let A denote the set of all Abelian groups. Under the operation of direct product, A is a semigroup with identity element E, the group with one element. G_1 and G_2 are relatively prime if the only common direct factor of G_1 and G_2 is E. We say that G_1 and G_2 are unitary factors of G if G=G_1 X G_2 and G_1, G_2 are relatively prime. Let t(G) denote the number of unitary factors of G. Sequence gives T(n) = sum_{G in A, |G| <= n} t(G).

			A101113 begins 1, 2, 2, 4, 2. So a(5) = 11.

Schmidt, Peter Georg, Zur Anzahl unitaerer Faktoren abelscher Gruppen. [On the number of unitary factors in Abelian groups] Acta Arith., 64 (1993), 237-248.
Wu, J., On the average number of unitary factors of finite Abelian groups, Acta Arith. 84 (1998), 17-29.

Cf. A101113.

Sum[Apply[Times, 2*Map[PartitionsP, Map[Last, FactorInteger[i]]]], {i, n}]

a(n) = partial sums of A101113

cp's OEIS Frontend

A005101 Abundant numbers (sum of divisors of m exceeds 2m).

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A191323 Increasing sequence generated by these rules: a(1)=1, and if x is in a then [3x/2]+1 and 3x+1 are in a, where [ ]=floor.

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A191134 Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x+1 and 4x-1 are in a.

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A101114 Let t(G) = number of unitary factors of the Abelian group G. Then a(n) = sum t(G) over all Abelian groups G of order <= n.

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