A343302 -id:A343302 - 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

A343301 Numbers k such that 6k+1 through 6k+5 are all deficient (in A005100).

Original entry on oeis.org

0, 1, 2, 5, 7, 8, 10, 12, 15, 19, 20, 21, 22, 24, 25, 27, 28, 30, 31, 35, 38, 39, 40, 41, 42, 44, 47, 48, 49, 52, 54, 55, 57, 59, 62, 64, 67, 68, 70, 71, 72, 75, 78, 80, 84, 85, 87, 89, 92, 94, 97, 98, 99, 100, 104, 105, 109, 110, 111, 112, 114, 115, 118, 119

Offset: 1

Jianing Song, Apr 11 2021

nonn

Numbers k such that 6*k+1 is in A343302.

Note that no deficient number can be a multiple of 6.

			8 is a term since every one of 49, 50, 51, 52 and 53 is deficient.
157 is not a term since 943, 944, 946 and 947 are all deficient while 945 is not.

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

Cf. A005100 (deficient numbers), A343302, A343306.

q[n_] := AllTrue[Range[5], DivisorSigma[-1, 6*n + #] < 2 &]; Select[Range[0, 120], q] (* Amiram Eldar, Mar 21 2024 *)

isA343301(k) = for(i=1, 5, if( sigma(6*k+i) >= 2*(6*k+i), return(0) )); 1

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.

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

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A343301 Numbers k such that 6k+1 through 6k+5 are all deficient (in A005100).

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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.

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Author

Keywords

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Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

A343301 Numbers k such that 6*k+1 through 6*k+5 are all deficient (in A005100).

Views

Author

Keywords

Comments

Examples

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Programs

Mathematica

PARI

A343301 Numbers k such that 6k+1 through 6k+5 are all deficient (in A005100).