A096536 -id:A096536 - cp's OEIS frontend

A353539 Starts of runs of 3 consecutive numbers whose abundancy index is larger than Pi^2/6.

188, 230, 284, 374, 404, 434, 440, 494, 824, 854, 944, 1034, 1070, 1274, 1448, 1664, 1826, 1910, 1988, 2114, 2330, 2534, 2708, 2924, 3014, 3374, 3548, 3590, 3674, 3716, 3794, 3968, 4184, 4388, 4598, 4634, 4850, 5024, 5048, 5228, 5444, 5474, 5564, 5642, 5864, 5894

Offset: 1

Amiram Eldar, Apr 25 2022

nonn

The least odd term is a(64391) = 8110375.

			188 is a term since sigma(188)/188 = 336/188 = 1.787..., sigma(189)/189 = 320/189 = 1.693... and sigma(190)/190 = 360/190 = 1.894... are all larger than Pi^2/6 = 1.644... .

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

Cf. A000203, A013661.
Subsequence of A353537 and A353538.
Subsequences: A096536, A353540, A353541.

q[n_] := DivisorSigma[-1, n] > Pi^2/6; seq[count_, nConsec_] := Module[{tri = q /@ Range[nConsec], s = {}, c = 0, k = nConsec + 1}, While[c < count, If[And @@ tri, c++; AppendTo[s, k - nConsec]]; tri = Join[Rest[tri], {q[k]}]; k++]; s]; seq[46, 3]

isok(k) = ((sigma(k)/k) > Pi^2/6) && ((sigma(k+1)/(k+1)) > Pi^2/6) && ((sigma(k+2)/(k+2)) > Pi^2/6); \\ Michel Marcus, Apr 25 2022

A381549 Numbers k such that k, k+1 and k+2 all have an odd number of abundant divisors.

Original entry on oeis.org

96236031968, 229687160624, 274957745984, 331240852304, 363015363248, 386136575824, 407374391150, 623810538350, 734609097584, 745885389248, 1080953007848

Offset: 1

Amiram Eldar, Feb 26 2025

a(11) > 10^12, if it exists.

			96236031968 is a term since it has 7 abundant divisors (992, 512368, 1024736, 46580848, 93161696, 48118015984, 96236031968), 96236031968 + 1 = 96236031969 has 9 abundant divisors (7857927, 10025631, 12641013, 290743299, 2600973837, 3318483861, 4184175303, 5660943057, 96236031969), and 96236031968 + 2 = 96236031970 has one abundant divisor (96236031970 itself).

Subsequence of A096536, A381546 and A381548.

q[n_] := q[n] = OddQ[DivisorSum[n, 1 &, DivisorSigma[-1, #] > 2 &]]; With[{v = Import["https://oeis.org/A096536/b096536.txt", "Table"][[;; , 2]]}, Select[v, q[#] && q[# + 1] && q[# + 2] &]]

a(11) from Jinyuan Wang, Mar 12 2025

A364861 Numbers k such that k and k+1 are both S-abundant numbers (A181487).

Original entry on oeis.org

5984, 7424, 21944, 39375, 56924, 77175, 82004, 84524, 89775, 109395, 116655, 158235, 174824, 180495, 185535, 188055, 193544, 200024, 209055, 235935, 238095, 240344, 245024, 250964, 256095, 261260, 262184, 263024, 266475, 279279, 282975, 283815, 294975, 297296

Offset: 1

Amiram Eldar, Aug 11 2023

nonn

De Koninck and Ivić found that the least number k such that k, k+1, and k+2 are 3 consecutive integers that are S-abundant numbers is 171078830 (which is also the first term of A096536).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Aleksandar Ivić, On a sum of divisors problem, Publications de l'Institut Mathématique (Beograd), New Series, Vol. 64 (78) (1998), pp. 9-20.
Wikipedia, Granville number.

Subsequence of A181487.

Cf. A096399, A096536, A118372.

seq[kmax_] := Module[{s = {1}, a = {}, sum, q1 = False, q2}, Do[sum = Total[Select[Divisors[k], MemberQ[s, #] &]]; q2 = sum > k; If[!q2, AppendTo[s, k]]; If[q1 && q2, AppendTo[a, k-1]]; q1 = q2, {k, 2, kmax}]; a]; seq[40000]

lista(nmax) = {my(c = 0, s, q1 = 0, q2); for(n=2, nmax, if(sumdiv(n, d, !bittest(c, d)*d) > 2*n, c+=1<M. F. Hasler at A181487

A383391 Numbers k such that k^2, (k+1)^2 and (k+2)^2 are all abundant numbers.

Original entry on oeis.org

2924, 3794, 5564, 8294, 8414, 10064, 13454, 19304, 22154, 22814, 35684, 39974, 40544, 40754, 41768, 46214, 49994, 52064, 56264, 60884, 63854, 65624, 68354, 68474, 69068, 70244, 78974, 84824, 88604, 92168, 93224, 95354, 100694, 102464, 106028, 107084, 111110, 111824

Offset: 1

Amiram Eldar, Apr 25 2025

nonn

Are there numbers k such that k^2, (k+1)^2, (k+2)^2 and (k+3)^2 are all abundant numbers? There are none below 2.5*10^10.
Are there odd terms in this sequence? There are none below 2.5*10^10.
From David A. Corneth, Apr 26 2025: (Start)
If it exists then it is at least sqrt(A002110(24)/2 * 155925 + 1) - 1 ~= 4.3*10^19.
Proof: Exactly two of k, k+1, k+2 and k+3 are odd. Those two are coprime and differ by 2. Let them be m and m+2. Then sigma(m) > 2*m and sigma(m+2) > 2*(m+2). As they are coprime we have sigma(m*(m+2)) > 2*m*2*(m+2) so for a lower bound we look for the smallest odd t that sigma(t) > 4*t. The partial product of p / (p-1) for odd primes p first exceeds 4 when odd primes <= 79 are multiplied so t is divisible by 3 * 5 * 7 * ... * 79. A small search of multiples of this number gives A002110(24)/2 * 155925.
k * (k + 2) >= A002110(24)/2 * 155925 so k * (k + 2) + 1 = (k + 1)^2 >= A002110(24)/2 * 155925 + 1. Taking square roots on both sides and keeping the positive root gives the desired lower bound. (End)
From Yifan Xie, Apr 30 2025: (Start)
Both types of numbers exist, but the constructed ones are too large to be displayed here. For numbers k such that k^2, (k+1)^2, (k+2)^2 and (k+3)^2 are all abundant numbers, choose 4 disjoint subsets of the primes P_1, P_2, P_3 and P_4, and let the product of elements in P_i divide k+i-1. This is achievable because of the Chinese remainder theorem. If P_i contains p_1, ..., p_k, then sigma((k+i-1)^2)/(k+i-1)^2 >= Product_{i=1..k} (p_i+1)/p_i.
We are able to make the right hand side larger than 2 for each i because the infinite product Product_{p is prime} (p+1)/p = Product_{p is prime} (1+1/p) = Sum_{k is squarefree} 1/k diverges, since the squarefree numbers have asymptotic density 6/Pi^2.
For odd terms in this sequence, we can use only the odd primes to construct 3 prime subsets instead, and add a constraint that k == 1 (mod 2) after which the Chinese remainder theorem still applies. (End)

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

Subsequence of A381738 and A383390.

Cf. A002110, A005101, A063734, A096536.

q[n_] := q[n] = DivisorSigma[-1, n^2] > 2; Select[Range[120000], AllTrue[# + {0, 1, 2}, q] &]

is1(k) = {my(f = factor(k)); prod(i = 1, #f~, f[i,2] *= 2); sigma(f, -1) > 2;}
list(lim) = {my(q1 = is1(1), q2 = is1(2), q3); for(k = 3, lim, q3 = is1(k); if(q1 && q2 && q3, print1(k-2, ", ")); q1 = q2; q2 = q3);}

A357609 Numbers k such that k, k+1, and k+2 are all in A357605.

Original entry on oeis.org

10667829248, 14322877568, 25929352448, 26967189248, 31315096448, 32186016224, 35337613310, 36312573374, 41326711424, 53162460350, 69405075584, 71840196350, 72806666750, 73217757248, 83103523424, 106184935934, 109302242048, 111640866974, 115294917374, 116768901248

Offset: 1

Amiram Eldar, Oct 06 2022

nonn

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

			10667829248 is a term since 10667829248, 10667829249 and 10667829250 are all in A357605: A162296(10667829248) = 21342038784 > 2*10667829248, A162296(10667829249) = 21798236160 > 2*10667829249 and A162296(10667829250) = 21810824640 > 2*10667829250.

Amiram Eldar, Table of n, a(n) for n = 1..209 (terms below 10^12)

Cf. A162296.

Subsequence of A013929, A096536, A357605 and A357608.

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]; v = Cases[Import["https://oeis.org/A096536/b096536.txt", "Table"], {, }][[;; , 2]]; s = {}; Do[n = v[[k]]; If[q[n] && q[n+1] && q[n+2], AppendTo[s, n]], {k, 1, Length[v]}]; s

A358483 Numbers k such that k, k+1 and k+2 are all infinitary abundant numbers (A129656).

Original entry on oeis.org

2666847104, 2695309694, 8207456894, 8967010688, 12147283070, 12491149670, 13911605630, 14126720894, 17238119624, 17238704768, 18420223094, 20922243110, 21786026624, 25118874494, 26079705728, 26979164288, 27257009624, 30000503168, 30478990904, 30832299134, 32892108248

Offset: 1

Amiram Eldar, Nov 18 2022

nonn

			2666847104 is in the sequence since 2666847104, 2666847105 and 2666847106 are all infinitary abundant numbers (A129656): isigma(2666847104) = 5401952640 > 2 * 2666847104, isigma(2666847105) = 5374656000 > 2 * 2666847105, and isigma(2666847106) = 5419376640 > 2 * 2666847106 (isigma is the sum of infinitary divisors, A049417).

Amiram Eldar, Table of n, a(n) for n = 1..456 (terms below 10^12)

Subsequence of A096536, A129656 and A327635.

Cf. A049417.

f[p_, e_] := p^(2^(-1 + Position[Reverse @ IntegerDigits[e, 2], ?(# == 1 &)])); isigma[1] = 1; isigma[n] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); abQ[n_] := isigma[n] > 2*n; v = Cases[Import["https://oeis.org/A096536/b096536.txt", "Table"], {, }][[;; , 2]]; Select[v, And @@ abQ /@ (# + {0, 1, 2}) &]

A358484 Numbers k such that k, k+1 and k+2 are all bi-unitary abundant numbers (A292982).

Original entry on oeis.org

268005374, 600350750, 2666847104, 2683146464, 2695309694, 2849458688, 3904592768, 4112553248, 5368737374, 6554410784, 6955574624, 8207456894, 8967010688, 9220179968, 9868366430, 10529171288, 12147283070, 12411630944, 12491149670, 13911605630, 14126720894, 14396391008

Offset: 1

Amiram Eldar, Nov 18 2022

nonn

			268005374 is in the sequence since 268005374, 268005375 and 268005376 are all bi-unitary abundant numbers (A292982): bsigma(268005374) = 568995840 > 2 * 268005374, bsigma(268005375) = 540633600 > 2 * 268005375, and bsigma(268005376) = 541258200 > 2 * 268005376 (bsigma is the sum of bi-unitary divisors, A188999).

Amiram Eldar, Table of n, a(n) for n = 1..1546 (terms below 10^12)

Subsequence of A096536, A292982 and A318167.

Cf. A188999.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - If[OddQ[e], 0, p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ f @@@ FactorInteger[n]; abQ[n_] := bsigma[n] > 2*n; v = Cases[Import["https://oeis.org/A096536/b096536.txt", "Table"], {, }][[;; , 2]]; Select[v, And @@ abQ /@ (# + {0, 1, 2}) &]

A332575 Least start of a run of exactly n consecutive numbers that are norm-abundant in Gaussian integers (A332570).

Original entry on oeis.org

2, 9, 4, 12, 24, 185, 114, 1649, 692, 4977, 1412, 416345, 22624, 72233, 199892, 25262152, 1351880, 130824185, 16305324, 1688906313, 9412730, 10393378914, 721753400

Offset: 1

Amiram Eldar, Feb 16 2020

			a(2) = 9 since 9 and 10 are the least pair of 2 consecutive numbers that are norm-abundant in Gaussian integers, and 8 and 11 are not norm-abundant.

Cf. A005101, A096399, A096536, A103228, A103229, A103230, A332570.

normAbQ[z_] := Abs[DivisorSigma[1, z, GaussianIntegers -> True]]^2 > 2*Abs[z]^2; n = 1; count = 0; max = 15; seq = Table[0, {max}]; While[count < max, n1 = n; If[normAbQ[n], While[normAbQ[++n1]]; d = n1 - n; If[d <= max && seq[[d]] == 0, count++; seq[[d]] = n]]; n = n1 + 1]; seq

cp's OEIS Frontend

A353539 Starts of runs of 3 consecutive numbers whose abundancy index is larger than Pi^2/6.

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A381549 Numbers k such that k, k+1 and k+2 all have an odd number of abundant divisors.

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A364861 Numbers k such that k and k+1 are both S-abundant numbers (A181487).

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A383391 Numbers k such that k^2, (k+1)^2 and (k+2)^2 are all abundant numbers.

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A357609 Numbers k such that k, k+1, and k+2 are all in A357605.

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A358483 Numbers k such that k, k+1 and k+2 are all infinitary abundant numbers (A129656).

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A358484 Numbers k such that k, k+1 and k+2 are all bi-unitary abundant numbers (A292982).

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A332575 Least start of a run of exactly n consecutive numbers that are norm-abundant in Gaussian integers (A332570).

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