A096399 -id:A096399 - cp's OEIS frontend

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

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

A328327 Numbers k such that both k and k+1 are Zumkeller numbers (A083207).

Original entry on oeis.org

5984, 7424, 21735, 21944, 26144, 27404, 39375, 43064, 49664, 56924, 58695, 61424, 69615, 70784, 76544, 77175, 79695, 81080, 81675, 82004, 84524, 84644, 89775, 91664, 98175, 103455, 104895, 106784, 109395, 111824, 116655, 116864, 120015, 121904, 122264, 126224

Offset: 1

Amiram Eldar, Oct 12 2019

nonn

Terms k such that both k and k+1 are primitive Zumkeller numbers (A180332) are 82004, 84524, 158235, 516704, 2921535, 5801984, ... (A361934).

There are infinitely many such k as proven by Somu et al. (2023). - Duc Van Khanh Tran, Dec 07 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Giovanni Resta)
Sai Teja Somu, Andrzej Kukla, and Duc Van Khanh Tran, Some results on Zumkeller numbers, arXiv:2310.14149 [math.NT], 2023.

Cf. A083207, A180332, A361934.

Subsequence of A096399.

zumkellerQ[n_] := Module[{d = Divisors[n], t, ds, x}, ds = Plus @@ d; If[Mod[ds, 2] > 0, False, t = CoefficientList[Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; zq1 = False; s = {}; Do[zq2 = zumkellerQ[n]; If[zq1 && zq2, AppendTo[s, n - 1]]; zq1 = zq2, {n, 2, 10^5}]; s (* after T. D. Noe at A083207 *)

from itertools import count, islice
from sympy import divisors
def A328327_gen(startvalue=1): # generator of terms >= startvalue
    m = -1
    for n in count(max(startvalue,1)):
        d = divisors(n)
        s = sum(d)
        if s&1^1 and n<<1<=s:
            d = d[:-1]
            s2, ld = (s>>1)-n, len(d)
            z = [[0 for  in range(s2+1)] for  in range(ld+1)]
            for i in range(1, ld+1):
                y = min(d[i-1], s2+1)
                z[i][:y] = z[i-1][:y]
                for j in range(y,s2+1):
                    z[i][j] = max(z[i-1][j],z[i-1][j-y]+y)
                if z[i][s2] == s2:
                    if m == n-1:
                        yield m
                    m = n
                    break
A328327_list = list(islice(A328327_gen(),5)) # Chai Wah Wu, Feb 13 2023

A333951 Numbers k such that both k and k+1 are recursive abundant numbers (A333928).

Original entry on oeis.org

56924, 82004, 84524, 109395, 158235, 241604, 261260, 266475, 285075, 361844, 442035, 445004, 469755, 611324, 666315, 694484, 712844, 922635, 968715, 971684, 1102724, 1172115, 1190475, 1199835, 1239524, 1304324, 1338435, 1430715, 1442924, 1486275, 1523115, 1550835

Offset: 1

Amiram Eldar, Apr 11 2020

nonn

			56924 is a term since A333926(56924) = 120960 > 2 * 56924, and A333926(56925) = 116064 > 2 * 56925.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A333928.
Cf. A333926, A333950.
Analogous sequences: A096399, A283418 (primitive), A318167 (bi-unitary), A327635 (infinitary), A327942 (nonunitary), A331412 (unitary).

recDivQ[n_, 1] = True; recDivQ[n_, d_] := recDivQ[n, d] = Divisible[n, d] && AllTrue[FactorInteger[d], recDivQ[IntegerExponent[n, First[#]], Last[#]] &]; recDivs[n_] := Select[Divisors[n], recDivQ[n, #] &]; f[p_, e_] := 1 + Total[p^recDivs[e]]; recDivSum[1] = 1; recDivSum[n_] := Times @@ (f @@@ FactorInteger[n]); recAbQ[n_] := recDivSum[n] > 2*n; Select[Range[2*10^5], recAbQ[#] && recAbQ[# + 1] &]

A381548 Numbers k such that k and k+1 both have an odd number of abundant divisors.

Original entry on oeis.org

5984, 7424, 21735, 27404, 43064, 56924, 76544, 82004, 89775, 109395, 144584, 158235, 164835, 165375, 174824, 222704, 266475, 271215, 300104, 311024, 322335, 326655, 326864, 334304, 347984, 350175, 371924, 387584, 393855, 414315, 442035, 445004, 447524, 477224

Offset: 1

Amiram Eldar, Feb 26 2025

nonn

Numbers k such that k and k+1 are both in A381546.

			5984 is a term since it has 9 abundant divisors (88, 176, 272, 352, 544, 748, 1496, 2992, 5984) and 5984 + 1 = 5985 has one abundant divisor (5985 itself).
21735 is a term since it has 3 abundant divisors (945, 7245, 21735) and 21735 + 1 = 21736 has 9 abundant divisors (88, 104, 572, 836, 1144, 1672, 1976, 10868, 21736).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A096399 and A381546.

Subsequences: A381549.

q[n_] := q[n] = OddQ[DivisorSum[n, 1 &, DivisorSigma[-1, #] > 2 &]]; Select[Range[500000], q[#] && q[#+1] &]

is1(k) = sumdiv(k, d, (sigma(d, -1) > 2)) % 2;
list(lim) = forstep(k = 3, lim, 2, if(is1(k), if(is1(k-1), print1(k-1, ", ")); if(is1(k+1), print1(k, ", "))));

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

A357608 Numbers k such that k and k+1 are both in A357605.

Original entry on oeis.org

76544, 104895, 126224, 165375, 170624, 174824, 201824, 245024, 257984, 271215, 273104, 316575, 338624, 387855, 447615, 469664, 477224, 540224, 618975, 633555, 641024, 659295, 705375, 752895, 770175, 842624, 843975, 862784, 870975, 893024, 913275, 957824, 1047375

Offset: 1

Amiram Eldar, Oct 06 2022

nonn

Numbers k such that A162296(k) > 2*k and A162296(k+1) > 2*(k+1).

			76544 is a term since 76544 and 76545 are both in A357605: A162296(76544) = 170688 > 2*76544 and A162296(76545) = 157248 > 2*76545.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A162296.
Subsequence of A013929, A096399 and A357605.
Similar sequences: A096399, A283418, A318167, A327635, A327942, A331412, A333951.

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; Times @@ ((p^(e + 1) - 1)/(p - 1)) - Times @@ (p + 1) > 2*n]; Select[Range[2, 10^6], q[#] && q[#+1] &]

A329525 a(n) is the smallest positive number k such that k and k+n are both abundant.

Original entry on oeis.org

5775, 18, 942, 20, 940, 12, 945, 12, 936, 20, 4725, 12, 4712, 40, 930, 20, 928, 12, 2816, 20, 924, 18, 945, 12, 920, 30, 918, 12, 2176, 12, 3465, 24, 912, 20, 910, 12, 7208, 18, 906, 20, 4095, 12, 5312, 12, 900, 20, 945, 12, 896, 20, 894, 18, 4672, 12, 945, 24

Offset: 1

Jaroslav Krizek, Nov 15 2019

nonn

Sequences of numbers k such that k and k+n are both abundant for any n: A096399 (n = 1), A231086 (n = 2).

			Number 5775 is the smallest abundant number k such that k+1 = 5576 is also abundant.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A005101, A096399, A231086, A294937.

[Min([m: m in[1..10^4] | SumOfDivisors(m) gt 2*m and SumOfDivisors(m+n) gt 2*(m+n)]): n in [1..60]];

A329525(n) = for(k=1, oo, if((sigma(k) > (k+k)) && (sigma(n+k) > 2*(n+k)), return(k))); \\ Antti Karttunen, Nov 15 2019

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

Original entry on oeis.org

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 *)

A349870 Numbers k such that k and k+1 are both nobly abundant numbers (A349758).

Original entry on oeis.org

735, 819, 1035, 1196, 1274, 1275, 1449, 1665, 1700, 1924, 1925, 1952, 1988, 2204, 2324, 2331, 2540, 2655, 2960, 2975, 3068, 3195, 3267, 3324, 3339, 3380, 3549, 3555, 3626, 3717, 4004, 4059, 4164, 4220, 4235, 4256, 4556, 4563, 4598, 4599, 4635, 4655, 4675, 4719

Offset: 1

Amiram Eldar, Dec 03 2021

nonn

			735 is a term since 735 and 736 are both nobly abundant numbers: A000005(735) = A000005(736) = 12, A000203(735) = 1368 and A000203(736) = 1512 are all abundant numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A005101, A096399.

Subsequence of A349758.

abQ[n_] := DivisorSigma[1, n] > 2*n; nobAbQ[n_] := And @@ abQ /@ DivisorSigma[{0, 1}, n]; Select[Range[50000], nobAbQ[#] && nobAbQ[# + 1] &]

A364727 Numbers k such that k and k+1 are both admirable numbers (A111592).

Original entry on oeis.org

29691198404, 478012798575, 2789405835075, 22929723392715

Offset: 1

Amiram Eldar, Aug 05 2023

a(1)-a(2) were found by Giovanni Resta.

Giovanni Resta, admirable numbers, Numbers Aplenty.

Subsequence of A096399 and A111592.

Cf. A109729, A109730.

a(4) from Martin Ehrenstein, Aug 06 2023

cp's OEIS Frontend

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

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A328327 Numbers k such that both k and k+1 are Zumkeller numbers (A083207).

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A333951 Numbers k such that both k and k+1 are recursive abundant numbers (A333928).

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A381548 Numbers k such that k and k+1 both have an odd number of abundant divisors.

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

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A357608 Numbers k such that k and k+1 are both in A357605.

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A329525 a(n) is the smallest positive number k such that k and k+n are both abundant.

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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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