A125115 -id:A125115 - cp's OEIS frontend

A125238 Differences between consecutive deficient numbers.

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1

Offset: 1

Jason G. Wurtzel, Nov 25 2006

The first 3 occurs at n=4340. - R. J. Mathar, Jul 08 2015

			a(1) = 1 because 2-1 = 1.
a(5) = 2 because 7-5 = 2.

R. J. Mathar, Table of n, a(n) for n = 1..1000

Cf. A005100, A094268, A125115, A318172.

A:=Filtered([1..100],n->Sigma(n)<2*n);; a:=List([1..Length(A)-1],i->A[i+1]-A[i]); # Muniru A Asiru, Jun 26 2018

A125238 := proc(n)
    A005100(n+1)-A005100(n) ;
end proc: # R. J. Mathar, Jul 08 2015

Differences@ Select[Range@ 105, DivisorSigma[1, #] < 2 # &] (* Michael De Vlieger, Jun 29 2018 *)

a(n) = A005100(n+1) - A005100(n).

Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 1/A318172 = 1.3291... - Amiram Eldar, Oct 21 2020

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

A306497 Abundant numbers that differ from the next abundant number by 5.

Original entry on oeis.org

5391411025, 26957055120, 28816162375, 33426748350, 34393484125, 37739877175, 40342627320, 48150877770, 50866790970, 53356378075, 59305521270, 60711976320, 61164628525, 63395557225, 64899009175, 67275433225, 70088343325, 74922022170, 75665665075, 76781129425

Offset: 1

Sergio Pimentel, Feb 19 2019

nonn

Since all multiples of 6 are abundant, numbers in this sequence have to be abundant numbers of the form 6n or 6n + 1. Most common difference between abundant numbers is 6, followed by 2, 4, 3, 1. 5 is the least common.

			a(1) = 5391411025 is in the sequence since it is abundant and the next abundant number is 5391411030 which is a(1)+5 and all the numbers in between are deficient.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A005101, A096399, A231086, A228382, A316098, A316099, A115414, A125115.

isok(n) = for(k=1, 4, if(sigma(n+k) > 2*(n+k), return(0))); (sigma(n) > 2*n) && (sigma(n+5) > 2*(n+5)); \\ Daniel Suteu, Jul 24 2019

Either a(n) or a(n)+5 are in A115414. - Amiram Eldar, Jul 16 2019

More terms from Amiram Eldar, Jul 16 2019

cp's OEIS Frontend

A125238 Differences between consecutive deficient 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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A306497 Abundant numbers that differ from the next abundant number by 5.

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