A316095 -id:A316095 - cp's OEIS frontend

A228382 Abundant numbers that differ from the next abundant number by 3.

942, 945, 1572, 1575, 2202, 2205, 2832, 2835, 3462, 3465, 4092, 4095, 4722, 4725, 5352, 5355, 5772, 5985, 6432, 6435, 6612, 6615, 6822, 6825, 7242, 7245, 7425, 7872, 7875, 8082, 8085, 8412, 8415, 8502, 8505, 8922, 8925, 9132, 9135, 9552, 9555, 9762, 9765

Offset: 1

Michel Marcus, Aug 21 2013

nonn

Apparently these numbers come up mostly by pairs m, m+3 with m even; the odd terms being a subsequence of A005231. But this is not always the case (e.g., note the term 7425).

The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 2, 43, 393, 3635, 37599, 374092, 3731903, 37338208, 373256850, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00373... . - Amiram Eldar, May 30 2023

			942 is abundant, 943 and 944 are deficient, 945 is abundant.
945 is abundant, 946 and 947 are deficient, 948 is abundant.

Muniru A Asiru, Table of n, a(n) for n = 1..10000

Subsequence of A005101.

Cf. A005231, A096399, A316095.

a:=Filtered([1..130000],n->Sigma(n)>2*n and Sigma(n+1)<2*(n+1) and Sigma(n+2)<2*(n+2) and Sigma(n+3)>2*(n+3)); # Muniru A Asiru, Jun 09 2018

with(numtheory): select(n->sigma(n)>2*n and sigma(n+1)<2*(n+1) and sigma(n+2)<2*(n+2) and sigma(n+3)>2*(n+3),[$1..12000]); # Muniru A Asiru, Jun 09 2018

With[{abs = Select[Range[10000], DivisorSigma[-1, #] > 2 &]}, abs[[Position[Differences[abs], 3] // Flatten]]] (* Amiram Eldar, May 30 2023 *)
SequencePosition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,10000}],{1,0,0,1}][[;;,1]] (* Harvey P. Dale, Aug 31 2024 *)

isok(n) = (sigma(n)> 2*n) && (sigma(n+1)< 2*(n+1)) && (sigma(n+2) < 2*(n+2)) && (sigma(n+3) > 2*(n+3)); \\ Michel Marcus, Aug 21 2013

a(n) = A005101(A316095(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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A228382 Abundant numbers that differ from the next abundant number by 3.

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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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GAP

Maple

Mathematica

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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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Author

Keywords

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula