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 1-5 of 5 results.

A125115 Differences between consecutive abundant numbers.

Original entry on oeis.org

6, 2, 4, 6, 6, 4, 2, 6, 6, 2, 4, 6, 4, 2, 6, 2, 4, 4, 2, 6, 4, 2, 2, 4, 4, 2, 6, 6, 6, 6, 2, 4, 6, 6, 4, 2, 6, 6, 2, 4, 6, 6, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 4, 6, 6, 6, 6, 6, 2, 4, 6, 2, 4, 4, 2, 6, 6, 6, 4, 2, 2, 4, 6, 2, 4, 6, 6, 4, 2, 6, 2
Offset: 1

Views

Author

Jason G. Wurtzel, Nov 21 2006

Keywords

Comments

One may think that a(n) is always even and greater than 1. This is not the case as can be seen with A096399 or A228382. - Michel Marcus, Aug 21 2013

Examples

			a(1) = 6 because 18 - 12 = 6; a(4) = 6 because 30 - 24 = 6.

Links

Crossrefs

Cf. A005101, A096399, A228382, A302991.

Programs

  • GAP
    A:=Filtered([1..350],n->Sigma(n)>2*n);;  a:=List([1..Length(A)-1],i->A[i+1]-A[i]); # Muniru A Asiru, Jun 09 2018
  • Mathematica
    #[[2]] - #[[1]]&/@Partition[Select[Range[300], DivisorSigma[1, #] > 2# &], 2, 1] (* Harvey P. Dale, Dec 02 2006 *)
Differences[Select[Range[300], DivisorSigma[1, #] > 2# &]] (* Alonso del Arte, Apr 29 2019 *)
  • PARI
    lista(nn) = {lastab = 0; for (i=1, nn, if (sigma(i) > 2*i,if (lastab, print1(i - lastab, ", ")); lastab = i;););} \\ Michel Marcus, Aug 21 2013

Formula

From Amiram Eldar, Oct 21 2020: (Start)
a(n) = A005101(n+1) - A005101(n).
Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 1/A302991 = 4.0384... (End)

Extensions

More terms from Michel Marcus, Aug 21 2013

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

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

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

Views

Author

Sergio Pimentel, Feb 19 2019

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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

Programs

  • PARI
    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

Formula

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

Extensions

More terms from Amiram Eldar, Jul 16 2019

A331202 a(n) is the smallest abundant number that differs from the next abundant number by n.

Original entry on oeis.org

5775, 18, 942, 20, 5391411025, 12
Offset: 1

Views

Author

Jaroslav Krizek, Jan 16 2020

Keywords

Comments

Sequence is finite; any multiple of 6 is abundant.

Crossrefs

Cf. Sequences of abundant numbers that differ from the next abundant number by k for any k: A096399 (k = 1), A228382 (k = 3), A316098 (k = 4), A306497 (k = 5), A316099 (k = 6).
Cf. A005101 (abundant numbers), A094268.
Cf. A329525 (smallest abundant numbers k such that k and k+n are both abundant).
Showing 1-5 of 5 results.

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