A169822 -id:A169822 - cp's OEIS frontend

A096399 Numbers k such that both k and k+1 are abundant.

5775, 5984, 7424, 11024, 21735, 21944, 26144, 27404, 39375, 43064, 49664, 56924, 58695, 61424, 69615, 70784, 76544, 77175, 79695, 81080, 81675, 82004, 84524, 84644, 89775, 91664, 98175, 103455, 104895, 106784, 109395, 111824, 116655, 116864, 120015, 121904, 122264

Offset: 1

John L. Drost, Aug 06 2004

nonn

Numbers k such that both sigma(k) > 2k and sigma(k+1) > 2*(k+1).
Numbers k such that both k and k+1 are in A005101.
Set difference of sequences A103289 and {2^m-1} for m in A103291.
The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 3, 27, 357, 3723, 36640, 365421, 3665799, 36646071, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000366... . - Amiram Eldar, Sep 02 2022

			sigma(5775) = sigma(3*5*5*7*11) = 11904 > 2*5775.
sigma(5776) = sigma(2*2*2*2*19*19) = 11811 > 2*5776.

T. D. Noe, Table of n, a(n) for n = 1..10000
Yong-Gao Chen and Hui Lv, On consecutive abundant numbers, arXiv:1603.06176 [math.NT], 2016.
Paul Erdős, Note on consecutive abundant numbers, J. London Math. Soc., 10 (1935), 128-131.
Carlos Rivera, Puzzle 878. Consecutive abundant integers, The Prime Puzzles and Problems Connection.

Cf. A000203, A005101, A023196, A103289, A103291, A169822.

fQ[n_] := DivisorSigma[1, n] > 2 n; Select[ Range@ 117000, fQ[ # ] && fQ[ # + 1] &] (* Robert G. Wilson v, Jun 11 2010 *)
Select[Partition[Select[Range[120000], DivisorSigma[1, #] > 2 # &], 2, 1], Differences@ # == {1} &][[All, 1]] (* Michael De Vlieger, May 20 2017 *)

for(i=1,1000000,if(sigma(i)>2*i && sigma(i+1)>2*(i+1),print(i))); \\ Max Alekseyev, Jan 28 2005

a(n) = A005101(A169822(n)). - Amiram Eldar, Mar 01 2025

Two further terms from Max Alekseyev, Jan 28 2005
Entry revised by N. J. A. Sloane, Dec 03 2006
Edited by T. D. Noe, Nov 15 2010

A303741 Numbers k such that A(k+1) = A(k) + 2, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

2, 7, 10, 14, 16, 19, 22, 23, 26, 31, 36, 39, 44, 45, 48, 51, 52, 59, 62, 65, 70, 71, 74, 79, 81, 82, 83, 86, 87, 90, 93, 96, 99, 104, 107, 110, 111, 114, 118, 120, 125, 128, 131, 133, 135, 138, 141, 146, 149, 150, 155, 156, 158, 164, 169, 170, 175, 178, 179

Offset: 1

Muniru A Asiru, Jun 22 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Muniru A Asiru)

A231086 is the main entry for this sequence.

Cf. A005101, A091194, A096399, A169822.

A:=Filtered([1..1000],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+2);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..1000]):  a:=select(j->A[j+1]=A[j]+2,[$1..nops(A)-1]);

Position[Differences[Select[Range[750], DivisorSigma[1, #] > 2*# &]], 2] // Flatten (* Amiram Eldar, Mar 15 2024 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 2, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Sequence is { k | A005101(k+1) = A005101(k) + 2 }.

a(n) = A091194(A231086(n)). - Amiram Eldar, Mar 01 2025

A316095 Numbers m such that A(m+1) = A(m) + 3, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

231, 232, 385, 386, 544, 545, 699, 700, 858, 859, 1014, 1015, 1172, 1173, 1326, 1327, 1431, 1488, 1600, 1601, 1645, 1646, 1699, 1700, 1806, 1807, 1850, 1959, 1960, 2015, 2016, 2093, 2094, 2119, 2120, 2221, 2222, 2272, 2273, 2378, 2379, 2433, 2434, 2583, 2584

Offset: 1

Muniru A Asiru, Jun 25 2018

nonn

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

A228382 is the main sequence for this entry.
Numbers m such that A(m+1) = A(m) + k, where A() = A005101() are the abundant numbers: A169822 (k=1), A303741 (k=2), this sequence (k=3), A316096 (k=4), A316097 (k=6).
Cf. A005101, A091194.

A:=Filtered([1..20000],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+3);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..20000]):  a:=select(j->A[j+1]=A[j]+3,[$1..nops(A)-1]);

Position[Map[{#1, #2 - 3} & @@ # &, Partition[Select[Range[12000], DivisorSigma[1, #] > 2 # &], 2, 1]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, Jun 29 2018 *)

lista(nn) = {my(va = select(x->(sigma(x) > 2*x), [1..nn]), dva = vector(#va-1, k, va[k+1] - va[k])); select(x->(x==3), dva, 1);} \\ Michel Marcus, Jul 03 2018

Sequence is { m | A005101(m+1) = A005101(m) + 3 }.
Sequence is { m | A125115(m) = 3 }.
a(n) = A091194(A228382(n)). - Amiram Eldar, Mar 01 2025

A316096 Numbers m such that A(m+1) = A(m) + 4, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

3, 6, 11, 13, 17, 18, 21, 24, 25, 32, 35, 40, 43, 46, 47, 50, 53, 60, 63, 64, 69, 72, 75, 78, 85, 88, 91, 94, 95, 100, 105, 106, 109, 112, 115, 117, 121, 124, 127, 130, 132, 136, 139, 140, 147, 148, 151, 154, 157, 159, 165, 168, 171, 176, 177, 180, 181, 184

Offset: 1

Muniru A Asiru, Jun 25 2018

nonn

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

A316098 is the main sequence for this entry.
Numbers m such that A(m+1) = A(m) + k, where A() = A005101() are the abundant numbers: A169822 (k=1), A303741 (k=2), A316095 (k=3), this sequence (k=4), A316097 (k=6).
Cf. A005101, A091194.

A:=Filtered([1..1000],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+4);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..1000]):  a:=select(j->A[j+1]=A[j]+4,[$1..nops(A)-1]);

Position[Map[{#1, #2 - 4} & @@ # &, Partition[Select[Range[10^3], DivisorSigma[1, #] > 2 # &], 2, 1]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, Jun 29 2018 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 4, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Sequence is { m | A005101(m+1) = A005101(m) + 4 }.
Sequence is { m | A125115(m) = 4 }.
a(n) = A091194(A316098(n)). - Amiram Eldar, Mar 01 2025

A316097 Numbers m such that A(m+1) = A(m) + 6, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

1, 4, 5, 8, 9, 12, 15, 20, 27, 28, 29, 30, 33, 34, 37, 38, 41, 42, 49, 54, 55, 56, 57, 58, 61, 66, 67, 68, 73, 76, 77, 80, 84, 89, 92, 97, 98, 101, 102, 103, 108, 113, 116, 119, 122, 123, 126, 129, 134, 137, 142, 143, 144, 145, 152, 153, 160, 161, 162, 163

Offset: 1

Muniru A Asiru, Jun 25 2018

nonn

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

A316099 is the main sequence for this entry.
Numbers m such that A(m+1) = A(m) + k, where A() = A005101() are the abundant numbers: A169822 (k=1), A303741 (k=2), A316095 (k=3), A316096 (k=4), this sequence (k=6).
Cf. A005101, A091194.

A:=Filtered([1..700],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+6);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..700]):  a:=select(j->A[j+1]=A[j]+6,[$1..nops(A)-1]);

Position[Map[{#1, #2 - 6} & @@ # &, Partition[Select[Range[10^3], DivisorSigma[1, #] > 2 # &], 2, 1]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, Jun 29 2018 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 6, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Sequence is { m | A005101(m+1) = A005101(m) + 6 }.
Sequence is { m | A125115(m) = 6 }.
a(n) = A091194(A316099(n)). - Amiram Eldar, Mar 01 2025

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A096399 Numbers k such that both k and k+1 are abundant.

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A303741 Numbers k such that A(k+1) = A(k) + 2, where A() = A005101() are the abundant numbers.

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A316095 Numbers m such that A(m+1) = A(m) + 3, where A() = A005101() are the abundant numbers.

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A316096 Numbers m such that A(m+1) = A(m) + 4, where A() = A005101() are the abundant numbers.

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A316097 Numbers m such that A(m+1) = A(m) + 6, where A() = A005101() are the abundant numbers.

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