A048242 -id:A048242 - cp's OEIS frontend

A048260 The sum of 2 (not necessarily distinct) abundant numbers.

24, 30, 32, 36, 38, 40, 42, 44, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150

Offset: 1

Jud McCranie

20161 is the largest number not in this sequence.

			12 is abundant, so 12+12=24 is in the sequence.
957 = 12 + 945 is in the sequence because 12 and 945 are abundant.

The Penguin Dictionary of Curious and Interesting Numbers, David Wells, entry 20161.

Cf. A005101. Complement of A048242.

a(n) = n + 1456 if n > 18705. - Charles R Greathouse IV, Feb 21 2017

A283550 Numbers that are not the sum of abundant numbers (not necessarily distinct).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 41, 43, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107

Offset: 1

Amiram Eldar, Mar 10 2017

a(496) = 991 is the last term.
Subsequence of A048242 (Numbers that are not the sum of two abundant numbers).
Giovanni Resta noticed that 991 is the largest number that is not a sum of abundant numbers.

			20161 = 12 + 304 + 19845 is the sum of 3 abundant numbers and thus not in the sequence (although it is in A048242, since it is not the sum of 2 abundant numbers).

Amiram Eldar, Table of n, a(n) for n = 1..496
Giovanni Resta, abundant numbers

Cf. A271113, A048242, A005101.

AbundantQ[n_] := DivisorSigma[1, n] > 2 n; a = Select[Range[1000], AbundantQ[#] &]; nn = Dimensions[a][[1]]; t = Rest[CoefficientList[ Series[Product[(1 + x^a[[k]]), {k, nn}], {x, 0, a[[-1]]}], x]]; f = Flatten[Position[t, 0]]

A270660 Numbers in the range of the sum of abundant divisors function.

Original entry on oeis.org

12, 18, 20, 24, 30, 32, 36, 38, 40, 42, 44, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168, 170, 172, 174, 176, 178, 180

Offset: 1

Timothy L. Tiffin, Jul 12 2016

nonn

Possible values for the sum of abundant divisors of the positive integers, written in ascending order.
Distinct nonzero values found in A187795.
The smallest odd integer in this sequence is 945, and the first run of three consecutive integers is 944, 945, 946.
This sequence contains every even integer greater than 46 and every odd integer greater than 20161.

Sin Hitotumatu, On the Limit for the Representation by the Sum of Two Abundant Numbers, Publications of the Research Institute for Mathematical Sciences of Kyoto University, 8 (1972/1973), 111-116.

Cf. A048242, supersequence of A005101 and A048260, A187795.

A271113 Numbers not in the range of the sum of abundant divisors function.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 39, 41, 43, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131, 133, 135, 137, 139

Offset: 1

Timothy L. Tiffin, Jul 13 2016

Numbers which do not appear in A187795 or in A270660; that is, there is no integer N whose sum of abundant divisors is equal to a(n) for any n.
This is a finite sequence that contains every odd positive integer less than 945, twelve even integers with 46 being the largest, and has the prime number 20161 as its last term.
A048242 contains the first three primitive abundant numbers: 12, 18, 20.

Sin Hitotumatu, On the Limit for the Representation by the Sum of Two Abundant Numbers, Publications of the Research Institute for Mathematical Sciences of Kyoto University, 8 (1972/1973), 111-116.

Cf. A005101, subsequence of A048242 and A263837, A048260, A187795, A270660 (complement).

A306720 Even numbers that are not the sum of two unitary abundant numbers (not necessarily distinct).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 86, 88, 90, 92, 94, 98, 102, 104, 106, 110, 114, 116, 118, 122, 124, 126, 128, 130, 134, 138, 142, 146, 150

Offset: 1

Amiram Eldar, Mar 06 2019

The unitary version of A048242.

a(6066) = 530086 is the last term. te Riele proved that every even number larger than 530086 is the sum of two unitary abundant numbers (not necessarily distinct). The corresponding sequence of odd numbers is also finite, but he did not calculate the last term, and only showed that it is below 2004452254833.

			Since the unitary abundant numbers begin with 30, 42, 66, 70, ... the first integers which are missing from this sequence are 60 = 30 + 30, 72 = 30 +42, 84 = 42 + 42, 96 = 30 + 66, 100 = 30 + 70, ...

Amiram Eldar, Table of n, a(n) for n = 1..6066
Herman J. J. te Riele, On the representation of the positive integers as the sum of two unitary abundant numbers, Stichting Mathematisch Centrum, Numerieke Wiskunde NW 19/75 (1975).

Cf. A034683, A048242.

A306776 Numbers that are the sum of two abundant numbers (not necessarily distinct) in a record number of ways.

Original entry on oeis.org

24, 36, 48, 60, 84, 90, 96, 108, 120, 144, 168, 180, 216, 240, 264, 288, 300, 336, 360, 408, 420, 480, 540, 576, 588, 600, 660, 720, 780, 840, 924, 960, 1008, 1080, 1140, 1200, 1260, 1320, 1380, 1428, 1440, 1500, 1560, 1620, 1680, 1920, 1980, 2040, 2100, 2280

Offset: 1

Amiram Eldar, Mar 09 2019

nonn

The record values of number of ways are 1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 15, 17, ... (see link for more values).

According to Andree, Mr. James Jones of Moore, Oklahoma, has shown that 371280 can be expressed as a sum of two abundant numbers in more than 43000 different ways and that record-breaking values are likely to be multiples of 60. Indeed, except for the 19 terms 24, 36, 48, 84, 90, 96, 108, 144, 168, 216, 264, 288, 336, 408, 576, 588, 924, 1008, and 1428, apparently all the others are divisible by 60.

			a(1) = 24 = 12 + 12 (one way);
a(2) = 36 = 12 + 24 = 18 + 18 (2 ways);
a(3) = 48 = 12 + 36 = 18 + 30 = 24 + 24 (3 ways);
a(4) = 60 = 12 + 48 = 18 + 42 = 20 + 40 = 24 + 36 = 30 + 30 (5 ways).

Eric A. Weiss, ed., A Computer Science Reader: Selections from ABACUS, Springer Science & Business Media, New York, 1988, p. 336.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Amiram Eldar, Table of n, a(n), record values for n = 1..1000
Richard V. Andree, Computer-Assisted Problem Solving, ABACUS, Vol. 2, No. 3 (Spring 1985), pp. 61-71 (reprinted in Weiss's book).

Cf. A005101, A048242, A283550, A300794.

nm=1000; ab=Select[Range[nm], DivisorSigma[1,#] > 2# &]; f[n_] := Length[ IntegerPartitions[n, {2}, ab]]; s={}; fm=0; Do[f1 = f[n]; If[f1>fm, fm=f1; AppendTo[s,n]], {n, 1, nm}]; s

A375389 a(n) is the smallest abundant number k such that n - k is abundant, or -1 if there is no such k.

Original entry on oeis.org

-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 12, -1, -1, -1, -1, -1, 12, -1, 12, -1, -1, -1, 12, -1, 18, -1, 20, -1, 12, -1, 20, -1, -1, -1, 12, -1, 20, -1, 12, -1, 12, -1, 20, -1, 18, -1, 12, -1, 20, -1, 24, -1, 12, -1, 12, -1, 30, -1, 12, -1, 18

Offset: 1

Robert Israel, Aug 13 2024

a(n) >= 12 for n >= 20162.
a(n) = 12 if n >= 24 and n == 0 (mod 6).
12 <= a(n) <= 20 if n >= 26 and n == 2 (mod 6).
12 <= a(n) <= 40 if n >= 52 and n == 4 (mod 6).
If a(n) > 0 then 0 < a(k n - (k-1) a(n)) <= a(n) for all positive integers k.

			a(30) = 12 because 30 = 12 + 18 where 12 and 18 are abundant numbers.

Robert Israel, Table of n, a(n) for n = 1..21000

Cf. A005101, A048242.

Ab:= select(t -> numtheory:-sigma(t) > 2*t, [$1..10^4]):
f:= proc(n) local i,x;
  for i from 1 do
    x:= Ab[i];
    if 2*x > n then return -1 fi;
    if ListTools:-BinarySearch(Ab, n-x) <> 0 then return x fi
  od;
end proc:
map(f, [$1..100]);

A337933 Numbers that are the sum of two abundant numbers in exactly one way.

Original entry on oeis.org

24, 30, 32, 38, 40, 44, 50, 52, 56, 58, 62, 64, 70, 957, 963, 965, 969, 975, 981, 985, 987, 993, 999, 1001, 1005, 1011, 1015, 1017, 1023, 1025, 1029, 1033, 1035, 1041, 1045, 1047, 1049, 1053, 1057, 1059, 1065, 1071, 1077, 1083, 1085, 1089, 1095, 1101, 1105, 1107, 1113

Offset: 1

Wesley Ivan Hurt, Oct 01 2020

An easy to calculate upper bound for terms is 12*(A047802(2)+1) = 64696932312. This and all larger numbers can be expressed as the sum of an abundant multiple of 6 and a multiple of A047802(2) in at least two ways. - Peter Munn, Feb 09 2021

			24 is in the sequence since it is the sum of two abundant numbers in exactly one way as 24 = 12 + 12.
30 is in the sequence since it is the sum of two abundant numbers in exactly one way as 30 = 12 + 18.

Cf. A005101, A047802, A048242.

Table[If[Sum[(1 - Sign[Floor[(2 (n - i))/DivisorSigma[1, n - i]]])*(1 - Sign[Floor[(2 i)/DivisorSigma[1, i]]]), {i, Floor[n/2]}] == 1, n, {}], {n, 1200}] // Flatten

cp's OEIS Frontend

A048260 The sum of 2 (not necessarily distinct) abundant numbers.

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A283550 Numbers that are not the sum of abundant numbers (not necessarily distinct).

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A270660 Numbers in the range of the sum of abundant divisors function.

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A271113 Numbers not in the range of the sum of abundant divisors function.

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A306720 Even numbers that are not the sum of two unitary abundant numbers (not necessarily distinct).

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A306776 Numbers that are the sum of two abundant numbers (not necessarily distinct) in a record number of ways.

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A375389 a(n) is the smallest abundant number k such that n - k is abundant, or -1 if there is no such k.

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A337933 Numbers that are the sum of two abundant numbers in exactly one way.

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