A316099 -id:A316099 - cp's OEIS frontend

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

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

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

Muniru A Asiru, Jun 25 2018

nonn

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

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

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.

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]);

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]));

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 *)

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

a(n) = A005101(A316096(n)). - Amiram Eldar, Mar 01 2025

A343302 Numbers k such that k through k+4 are all deficient (in A005100).

Original entry on oeis.org

1, 7, 13, 31, 43, 49, 61, 73, 91, 115, 121, 127, 133, 145, 151, 163, 169, 181, 187, 211, 229, 235, 241, 247, 253, 265, 283, 289, 295, 313, 325, 331, 343, 355, 373, 385, 403, 409, 421, 427, 433, 451, 469, 481, 505, 511, 523, 535, 553, 565, 583, 589, 595

Offset: 1

Jianing Song, Apr 11 2021

nonn

Since every multiple of 6 (other than 6 itself) is an abundant number, the maximum length of consecutive runs of deficient numbers is 5.
All terms are congruent to 1 modulo 6.
This is a proper subset of A231626, with the smallest missing term being 347: here only the first members of 5 consecutive deficient numbers in arithmetic progression with common difference 1 are allowed. Terms of A231626 that are not here are listed in A343303.

			115 is a term since all of 115, 116, 117, 118 and 119 are deficient.
2989 is not a term since 2989 + 3 = 2992 is an abundant number.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A005100 (deficient numbers), A316099, A343301.

Set difference of A231626 by A343303.

Position[Partition[DivisorSigma[-1, Range[600]], 5, 1], ?(Max[#] < 2 &), 1] // Flatten (* _Amiram Eldar, Mar 21 2024 *)
SequencePosition[Table[If[DivisorSigma[1,n]<2n,1,0],{n,600}],{1,1,1,1,1}][[;;,1]] (* Harvey P. Dale, Jul 16 2025 *)

isA343302(k) = if(k%6!=1, 0, for(i=0, 4, if( sigma(k+i) >= 2*(k+i), return(0) )); 1)

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

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

Jaroslav Krizek, Jan 16 2020

Sequence is finite; any multiple of 6 is abundant.

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).

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

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A343302 Numbers k such that k through k+4 are all deficient (in A005100).

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

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

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