A231626 -id:A231626 - cp's OEIS frontend

A343303 Numbers in A231626 but not in A343302; first of 5 consecutive deficient numbers in arithmetic progression with common difference > 1.

347, 1997, 2207, 2747, 2987, 2989, 3005, 3245, 3707, 3845, 4505, 4727, 4729, 5165, 6227, 7067, 7205, 7907, 8885, 9347, 9587, 9723, 9725, 11405, 13745, 14207, 14765, 17147, 17987, 18125, 18587, 18827, 18843, 18845, 19547, 20147, 20477, 21485, 22187, 22983, 22985

Offset: 1

Jianing Song, Apr 11 2021

nonn

Numbers k such that k, k+d, k+2*d, k+3*d and k+4*d are consecutive deficient numbers with some d > 1. Such k with d = 1 are listed in A343302.
All known terms have d = 2. If some k is the start of 5 consecutive deficient numbers in arithmetic progression with common difference 3, then k+1, k+4, k+7 and k+10 must be 4 consecutive terms in A096399. This may happen, but each of such k has to be extremely large.
If k is an even term here, then none of k, k+d, k+2*d, k+3*d and k+4*d is divisible by 6, so d must be divisible by 3.
It seems that most terms are congruent to 5 modulo 6. The smallest term congruent to 1 modulo 6 is a(6) = 2989, and the smallest term congruent to 3 modulo 6 is a(22) = 9723.

			347 is here since it is the start of 5 consecutive deficient numbers in arithmetic progression with common difference 2, namely 347, 349, 351, 353 and 355. Indeed, all of 348, 350, 352 and 354 are abundant.

Jianing Song, Table of n, a(n) for n = 1..16607 (all terms <= 10^7).

Cf. A096399.

Set difference of A231626 by A343302.

DefQ[n_] := DivisorSigma[1, n] < 2 n; m = 2; z1 = 2; cd = 1; a = {}; Do[If[DefQ[n], If[n - z1 == cd, m = m + 1; If[m > 4 && cd != 1, AppendTo[a, n - 4*cd]], m = 2; cd = n - z1]; z1 = n], {n, 3, 50000}]; a (* after the Mathematica program of A231626 *)

A316099 Abundant numbers that differ from the next abundant number by 6.

Original entry on oeis.org

12, 24, 30, 42, 48, 60, 72, 90, 114, 120, 126, 132, 144, 150, 162, 168, 180, 186, 210, 228, 234, 240, 246, 252, 264, 282, 288, 294, 312, 324, 330, 342, 354, 372, 384, 402, 408, 420, 426, 432, 450, 468, 480, 492, 504, 510, 522, 534, 552, 564, 582, 588, 594, 600

Offset: 1

Muniru A Asiru, Jun 25 2018

nonn

From Amiram Eldar, Sep 02 2022: (Start)
All the terms are even, since all the multiples of 6 that are larger than 6 are abundant numbers.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 8, 85, 865, 8716, 87668, 875528, 8761027, 87606693, 875947187, ... . Apparently, the asymptotic density of this sequence exists and equals 0.087... . (End)

			12 is abundant, 13, 14, 15, 16 and 17 are deficient, 18 is abundant.
24 is abundant, 25, 26, 27, 28 and 29 are deficient, 30 is abundant.

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

Subsequence of A005101.
Cf. A316097, A343302.
Cf. A231626 (which has many common terms when 1 is subtracted).

A:=Filtered([1..800],n->Sigma(n)>2*n);;  a:=List(Filtered([1..Length(A)-1],i->A[i+1]-A[i]=6),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]=6,[$1..nops(A)-1]));

q[n_] := DivisorSigma[1, n] > 2 n; Select[Range[600], q[#] && SelectFirst[# + Range[6], q] == # + 6 &] (* Giovanni Resta, Jul 01 2018 *)

list(lim) = {my(k = 1, k2); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 6, print1(k1, ", ")); k1 = k2));} \\ Amiram Eldar, Mar 01 2025

a(n) = A005101(A316097(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)

A231628 Smallest sets of 6 consecutive deficient numbers in arithmetic progression. The initial deficient number is listed.

Original entry on oeis.org

2987, 4727, 9723, 18843, 22983, 30543, 35147, 39947, 45047, 50463, 55787, 56807, 58055, 58779, 69183, 78047, 81947, 85743, 101147, 106143, 108255, 109247, 117123, 134087, 139743, 139803, 152567, 171287, 174347, 175907, 182643, 189767, 197027, 199803, 202127

Offset: 1

Shyam Sunder Gupta, Nov 11 2013

nonn

			2987, 2989, 2991, 2993, 2995, 2997 is the smallest set of 6 consecutive deficient numbers in arithmetic progression so 2987 is in the list.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..5000

Cf. A005100, A231623, A228963, A231624, A231625, A231626.

DefQ[n_] := DivisorSigma[1, n] < 2 n; m = 2; z1 = 2; cd = 1; a = {}; Do[If[DefQ[n], If[n - z1 == cd, m = m + 1; If[m > 5, AppendTo[a, n - 5*cd]], m = 2; cd = n - z1]; z1 = n], {n, 3, 1000000}]; a

A231629 First of 7 consecutive deficient numbers in arithmetic progression.

Original entry on oeis.org

801339, 962649, 7353339, 21964299, 41642139, 48049689, 55455939, 89034939, 89851449, 92253849, 105259539, 107948379, 109455339, 114295449, 116754939, 122349369, 135575979, 156009849, 159521049, 173645439, 188586699, 192674169, 193137849, 220301769, 221355125

Offset: 1

Shyam Sunder Gupta, Nov 11 2013

nonn

			801339, 801341, 801343, 801345, 801347, 801349, 801351 is the smallest set of 7 consecutive deficient numbers in arithmetic progression so 801339 is in the list.

Shyam Sunder Gupta, Table of n, a(n) for n = 1..126

Cf. A005100, A231623, A228964, A231624, A231625, A231626, A231628.

DefQ[n_] := DivisorSigma[1, n] < 2 n; m = 2; z1 = 2; cd = 1; a = {}; Do[If[DefQ[n], If[n - z1 == cd, m = m + 1; If[m > 6, AppendTo[a, n - 6*cd]], m = 2; cd = n - z1]; z1 = n], {n, 3, 1000000000}]; a

A231630 Smallest sets of 8 consecutive deficient numbers in arithmetic progression. The initial deficient number is listed.

Original entry on oeis.org

221355125, 221355127, 402640539, 402640541, 668862579, 668862581, 739577139, 739577141

Offset: 1

Shyam Sunder Gupta, Nov 11 2013

nonn

			221355125, 221355127, 221355129, 221355131, 221355133, 221355135, 221355137, 221355139 is the smallest set of 8 consecutive deficient numbers in arithmetic progression so 221355125 is in the list.

Cf. A005100, A231623, A228965, A231624, A231625, A231626, A231628, A231629.

DefQ[n_] := DivisorSigma[1, n] < 2 n; m = 2; z1 = 2; cd = 1; a = {}; Do[If[DefQ[n], If[n - z1 == cd, m = m + 1; If[m > 7, AppendTo[a, n - 7*cd]], m = 2; cd = n - z1]; z1 = n], {n, 3, 1000000000}]; a

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A343303 Numbers in A231626 but not in A343302; first of 5 consecutive deficient numbers in arithmetic progression with common difference > 1.

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A316099 Abundant numbers that differ from the next abundant number by 6.

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

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A231628 Smallest sets of 6 consecutive deficient numbers in arithmetic progression. The initial deficient number is listed.

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A231629 First of 7 consecutive deficient numbers in arithmetic progression.

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A231630 Smallest sets of 8 consecutive deficient numbers in arithmetic progression. The initial deficient number is listed.

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Mathematica