A275082 -id:A275082 - cp's OEIS frontend

A091191 Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor.

12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114, 138, 174, 186, 196, 222, 246, 258, 272, 282, 304, 308, 318, 354, 364, 366, 368, 402, 426, 438, 464, 474, 476, 498, 532, 534, 550, 572, 582, 606, 618, 642, 644, 650, 654, 678, 748, 762, 786, 812, 822

Offset: 1

Reinhard Zumkeller, Dec 27 2003

nonn

A080224(a(n)) = 1.
This is a supersequence of the primitive abundant number sequence A071395, since many of these numbers will be positive integer multiples of the perfect numbers (A000396). - Timothy L. Tiffin, Jul 15 2016
If the terms of A071395 are removed from this sequence, then the resulting sequence will be A275082. - Timothy L. Tiffin, Jul 16 2016

			12 is a term since 1, 2, 3, 4, and 6 (the proper divisors of 12) are either deficient or perfect numbers, and thus not abundant. - _Timothy L. Tiffin_, Jul 15 2016

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On the density of the abundant numbers, J. London Math. Soc. 9 (1934), pp. 278-282.
Eric Weisstein's World of Mathematics, Abundant Number

Cf. A006038 (odd terms), A005101 (abundant numbers), A091192.
Cf. A027751, A071395 (subsequence), supersequence of A275082.
Cf. A294930 (characteristic function), A294890.

a091191 n = a091191_list !! (n-1)
a091191_list = filter f [1..] where
   f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
         where pdivs = a027751_row x
-- Reinhard Zumkeller, Jan 31 2014

isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
isA091191 := proc(n) local d; if isA005101(n) then for d in numtheory[divisors](n) minus {1,n} do if isA005101(d) then return false; end if; end do: return true; else false; end if; end proc:
for n from 1 to 200 do if isA091191(n) then printf("%d\n",n) ; end if;end do: # R. J. Mathar, Mar 28 2011

t = {}; n = 1; While[Length[t] < 100, n++; If[DivisorSigma[1, n] > 2*n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)
Select[Range@ 840, DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] <= 2 # &, Most@ Divisors@ #] == 1 &] (* Michael De Vlieger, Jul 16 2016 *)

is(n)=sumdiv(n,d,sigma(d,-1)>2)==1 \\ Charles R Greathouse IV, Dec 05 2012

Erdős shows that a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 05 2012

A071395 Primitive abundant numbers (abundant numbers all of whose proper divisors are deficient numbers).

Original entry on oeis.org

20, 70, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030, 4070, 4095, 4216, 4288

Offset: 1

Joe McCauley (mccauley(AT)davesworld.net), Jun 12 2002

nonn

This is a subsequence of the primitive abundant number sequence A091191, since none of these numbers are a positive integer multiple of a perfect number (A000396). - Timothy L. Tiffin, Jul 15 2016
If the terms of this sequence are removed from A091191, then the resulting sequence will be A275082. - Timothy L. Tiffin, Jul 16 2016
Numbers n such that A294927(n) = 0 and A294937(n) = 1. - Antti Karttunen, Nov 14 2017

			20 is a term since 1, 2, 4, 5, and 10 (the proper divisors of 20) are all deficient numbers. - _Timothy L. Tiffin_, Jul 15 2016

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 46, also section B2, 1994.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Primitive Abundant Number

Cf. A006038, A000396, A005100, A005101, subsequence of A091191, A275082.

Cf. A294927, A294937.

abundance:= proc(n) option remember;  numtheory:-sigma(n)-2*n end proc:
select(n -> abundance(n) > 0 and andmap(t -> abundance(t) < 0, numtheory:-divisors(n) minus {n}), [$1..10000]); # Robert Israel, Nov 15 2017

Select[Range@ 5000, DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &] (* Michael De Vlieger, Jul 16 2016 *)

isA071395(v) = {if (sigma(v) <= 2*v, return (0)); fordiv (v, d, if ((d != v) && (sigma(d) >= 2*d), return (0));); return (1);} \\ Michel Marcus, Mar 10 2013

Offset corrected by Donovan Johnson, Aug 28 2011

A274338 The period 10 sequence of the iterated sum of deficient divisors function (A187793) starting at 52.

Original entry on oeis.org

52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78, 84, 52, 98, 171, 260, 308, 336, 76, 140, 78

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065).  It appears to be the only one of period (order, length) 10 that A187793 generates under iteration.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.

			a(1) = 52;
a(2) = sigma(52) = 98;
a(3) = sigma(98) = 171;
a(4) = sigma(171) = 260;
a(5) = sigma(260) - 260 - 20 = 308;
a(6) = sigma(308) - 308 - 28 = 336;
a(7) = 1 + 2 + 3 + 4 + 7 + 8 + 14 + 16 + 21 = 76 [since 336 has more abundant divisors than deficient ones];
a(8) = sigma(76) = 140;
a(9) = sigma(140) - 140 - 70 - 28 - 20 = 78;
a(10) = sigma(78) - 78 - 6 = 84;
a(11) = sigma(84) - 84 - 42 - 28 - 12 - 6 = 52 = a(1).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274339, A274340, A274380, A274549, A275082.

a(n)=n=n%10; if(n>0, sumdiv(a(n-1),d,if(sigma(d,-1)<2,d,0)), 84) \\ Charles R Greathouse IV, Jun 23 2016

Vec(x*(52 + 98*x + 171*x^2 + 260*x^3 + 308*x^4 + 336*x^5 + 76*x^6 + 140*x^7 + 78*x^8 + 84*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)) + O(x^50)) \\ Colin Barker, Jan 30 2020

a(n+10) = a(n).

G.f.: x*(52 + 98*x + 171*x^2 + 260*x^3 + 308*x^4 + 336*x^5 + 76*x^6 + 140*x^7 + 78*x^8 + 84*x^9) / ((1 - x)*(1 + x)*(1 - x + x^2 - x^3 + x^4)*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Jan 30 2020

A274339 The period 3 sequence of the iterated sum of deficient divisors function (A187793) starting at 15.

Original entry on oeis.org

15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18, 15, 24, 18

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be the only one of period (order, length) 3 that A187793 generates under iteration.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.
A284326 also generates this sequence under iteration. - Timothy L. Tiffin, Feb 22 2022

			a(1) = 15;
a(2) = sigma(15) = 24;
a(3) = sigma(24) - 24 - 12 - 6 = 18;
a(4) = sigma(18) - 18 - 6 = 15 = a(1).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274340, A274380, A274549, A275082, A284326.

LinearRecurrence[{0,0,1},{15,24,18},90] (* or *) PadRight[{},90,{15,24,18}] (* Harvey P. Dale, Aug 06 2023 *)

Vec(3*x*(5 + 8*x + 6*x^2) / ((1 - x)*(1 + x + x^2)) + O(x^40)) \\ Colin Barker, Jan 30 2020

a(n+3) = a(n).

G.f.: 3*x*(5 + 8*x + 6*x^2) / ((1 - x)*(1 + x + x^2)). - Colin Barker, Jan 30 2020

A274340 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 19.

Original entry on oeis.org

19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36, 19, 20, 22, 36

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be one of two sequences of period (order, length) 4 that A187793 generates under iteration. The other one is A274380.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.

			a(1) = 19;
a(2) = sigma(19) = 20;
a(3) = sigma(20) - 20 = 22;
a(4) = sigma(22) = 36;
a(5) = sigma(36) - 36 - 18 - 12 - 6 = 19 = a(1).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274339, A274380, A274549, A275082.

PadRight[{},100,{19,20,22,36}] (* Paolo Xausa, Oct 16 2023 *)

Vec(x*(19 + 20*x + 22*x^2 + 36*x^3) / (1 - x^4) + O(x^80)) \\ Colin Barker, Jan 30 2020

a(n+4) = a(n).
a(n) = A187793(a(n-1)).
G.f.: x*(19 + 20*x + 22*x^2 + 36*x^3) / (1 - x^4). - Colin Barker, Jan 30 2020

A274380 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 34.

Original entry on oeis.org

34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48, 34, 54, 42, 48

Offset: 1

Timothy L. Tiffin, Jun 22 2016

This sequence is generated in a similar way to aliquot sequences or sociable chains, which are generated by iterating the sum of proper divisors function (A001065). It appears to be one of two sequences of period (order, length) 4 that A187793 generates under iteration. The other one is A274340.
If sigma(N) is the sum of positive divisors of N, then:
  a(n+1) = sigma(a(n)) if a(n) is a deficient number (A005100),
  a(n+1) = sigma(a(n))-a(n) if a(n) is a primitive abundant number (A071395),
  a(n+1) = sigma(a(n))-a(n)-m if a(n) is an abundant number with one proper divisor m that is either perfect (A275082) or abundant, and so forth.
This is used in the example below.
A284326 also generates this sequence under iteration. - Timothy L. Tiffin, Feb 22 2022

			a(1) = 34;
a(2) = sigma(34) = 54;
a(3) = sigma(54) - 18 - 6 = 42;
a(4) = sigma(42) - 42 - 6 = 48;
a(5) = sigma(48) - 48 - 24 - 12 - 6 = 34 = a(1);
  :
  :

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001065, A005100, A005101, A071395, A125310, A187793, A274338, A274339, A274340, A274549, A275082, A284326.

Vec(2*x*(17 + 27*x + 21*x^2 + 24*x^3) / ((1 - x)*(1 + x)*(1 + x^2)) + O(x^80)) \\ Colin Barker, Jan 30 2020

a(n+4) = a(n).

G.f.: 2*x*(17 + 27*x + 21*x^2 + 24*x^3) / ((1 - x)*(1 + x)*(1 + x^2)). - Colin Barker, Jan 30 2020

cp's OEIS Frontend

A091191 Primitive abundant numbers: abundant numbers (A005101) having no abundant proper divisor.

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A071395 Primitive abundant numbers (abundant numbers all of whose proper divisors are deficient numbers).

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A274338 The period 10 sequence of the iterated sum of deficient divisors function (A187793) starting at 52.

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A274339 The period 3 sequence of the iterated sum of deficient divisors function (A187793) starting at 15.

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A274340 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 19.

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A274380 The period 4 sequence of the iterated sum of deficient divisors function (A187793) starting at 34.

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