cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Muniru A Asiru, Jun 25 2018

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    A:=Filtered([1..20000],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+3);
  • Maple
    with(numtheory): A:=select(n->sigma(n)>2*n,[$1..20000]):  a:=select(j->A[j+1]=A[j]+3,[$1..nops(A)-1]);
  • Mathematica
    Position[Map[{#1, #2 - 3} & @@ # &, Partition[Select[Range[12000], DivisorSigma[1, #] > 2 # &], 2, 1]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, Jun 29 2018 *)
  • PARI
    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

Formula

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

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

Views

Author

Muniru A Asiru, Jun 25 2018

Keywords

Links

Crossrefs

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.

Programs

  • GAP
    A:=Filtered([1..700],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+6);
  • Maple
    with(numtheory): A:=select(n->sigma(n)>2*n,[$1..700]):  a:=select(j->A[j+1]=A[j]+6,[$1..nops(A)-1]);
  • Mathematica
    Position[Map[{#1, #2 - 6} & @@ # &, Partition[Select[Range[10^3], DivisorSigma[1, #] > 2 # &], 2, 1]], ?(SameQ @@ # &)][[All, 1]] (* _Michael De Vlieger, Jun 29 2018 *)
  • PARI
    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

Formula

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

A316098 Abundant numbers that differ from the next abundant number by 4.

Original entry on oeis.org

20, 36, 56, 66, 80, 84, 96, 104, 108, 140, 156, 176, 192, 200, 204, 216, 224, 260, 272, 276, 300, 308, 320, 336, 360, 368, 380, 392, 396, 416, 440, 444, 456, 464, 476, 486, 500, 516, 528, 540, 546, 560, 572, 576, 608, 612, 620, 636, 644, 650, 680, 696, 704
Offset: 1

Views

Author

Muniru A Asiru, Jun 25 2018

Keywords

Examples

			20 is abundant, 21, 22 and 23 are deficient, 24 is abundant.
36 is abundant, 37, 38 and 39 are deficient, 40 is abundant.

Links

Crossrefs

Subsequence of A005101.
Abundant numbers that differ from the next abundant number by k: A096399 (k=1), A228382 (k=3), this sequence (k=4), A306497 (k=5), A316099 (k=6).
Cf. A316096.

Programs

  • GAP
    A:=Filtered([1..800],n->Sigma(n)>2*n);;  a:=List(Filtered([1..Length(A)-1],i->A[i+1]-A[i]=4),j->A[j]);
  • Maple
    with(numtheory):  A:=select(n->sigma(n)>2*n,[$1..800]): a:=seq(A[i],i in select(n->A[n+1]-A[n]=4,[$1..nops(A)-1]));
  • Mathematica
    q[n_] := DivisorSigma[1,n] > 2 n; Select[Range[704], q[#] && q[# + 4] && ! q[# + 1] && ! q[# + 2] && ! q[# + 3] &] (* Giovanni Resta, Jul 01 2018 *)
SequencePosition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,750}],{1,0,0,0,1}][[;;,1]] (* Harvey P. Dale, Mar 02 2023 *)
  • PARI
    list(lim) = {my(k = 1, k2); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 4, print1(k1, ", ")); k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Formula

a(n) = A005101(A316096(n)). - Amiram Eldar, Mar 01 2025
Showing 1-3 of 3 results.

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