A096399 -id:A096399 - cp's OEIS frontend

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

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

A383390 Numbers k such that k^2 and (k+1)^2 are both abundant numbers.

Original entry on oeis.org

104, 495, 584, 735, 944, 1155, 1364, 1484, 2144, 2204, 2415, 2624, 2924, 2925, 3135, 3255, 3794, 3795, 4304, 4484, 4784, 4844, 5264, 5355, 5445, 5564, 5565, 5655, 5775, 5984, 6104, 6764, 7424, 7455, 7664, 7755, 7875, 8084, 8294, 8295, 8414, 8415, 8924, 9009, 9344, 9944, 9975

Offset: 1

Amiram Eldar, Apr 25 2025

nonn

The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 5, 47, 459, 4655, 46733, 460693, 4612685, 46177602, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00461... .

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

Subsequence of A381738.
A383391 and A096399 are subsequences.
Cf. A005101, A063734.

Select[Range[10000], DivisorSigma[-1, #^2] > 2 && DivisorSigma[-1, (#+1)^2] > 2 &]

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); for(k = 2, lim, q2 = is1(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A295042 Numbers k such that both k and (k+1) are abundant, and neither is divisible by 3.

Original entry on oeis.org

55959128224, 68972878975, 91653987424, 171967420624, 350441716624, 372944997424, 386136575824, 711480344575, 769856312224, 789255692224, 818564922175, 997039218175, 1071710665024, 1216042052224, 1340586071824, 1925671372624, 1954925637664, 2045947528624

Offset: 1

Amiram Eldar, Nov 13 2017

nonn

Subsequence of A096399.

All terms are of the form 3j+1, with j = 18653042741, 22990959658, 30551329141, 57322473541, 116813905541, 124314999141, 128712191941, 237160114858, 256618770741, 263085230741, 272854974058, 332346406058, ...

			k = 55959128224 is in the sequence as sigma(k) > 2*k and sigma(k + 1) > 2*(k + 1). - _David A. Corneth_, Apr 11 2021

David A. Corneth, Table of n, a(n) for n = 1..988 (first 87 terms from Giovanni Resta)

Cf. A005101, A096399, A115414.

abQ[n_] := Mod[n, 3] > 0 && DivisorSigma[1, n] > 2 n; abQ1[n_] := abQ[n - 1]; abQ2[n_] := abQ[n + 1]; s = Import["b115414.txt", "Data"][[All, -1]]; s1 = Select[s, abQ1] - 1; s2 = Select[s, abQ2]; seq = Union[s1, s2] (* using the b-File from A115414 *)

isoka(n) = (n%3) && (sigma(n) > 2*n);
isok(n) = isoka(n) && isoka(n+1); \\ Michel Marcus, Nov 13 2017

a(13)-a(18) from Giovanni Resta, Aug 22 2018

A333054 Numbers m such that r(m) > r(k) for all k < m, where r(m) = min(sigma(m)/m, sigma(m+1)/(m+1)), and sigma(m) is the sum of divisors of m (A000203).

Original entry on oeis.org

1, 2, 8, 14, 44, 104, 495, 944, 4095, 5775, 5984, 21735, 98175, 862784, 4096575, 7194824, 14753024, 879207615, 1969789184, 2275962975, 3968862975, 12567844575, 39566665215, 44803620225, 77510285775, 125617830975, 162902829375

Offset: 1

Amiram Eldar, Mar 06 2020

The corresponding values of r(a(n)) are 1, 1.333..., 1.444..., 1.6, 1.733..., 1.828..., 1.890..., 1.970..., 1.999..., 2.044..., 2.085..., 2.120..., 2.181..., 2.243..., 2.248..., 2.252..., 2.360..., 2.397..., 2.407..., 2.408..., 2.411...
The least number m such that both m and m+1 are k-abundant (i.e., their abundancy indices sigma(m)/m > k and sigma(m+1)/(m+1) > k) is a term in this sequence. E.g., a(10) = 5775 = A096399(1).
a(28) > 5*10^11. - Amiram Eldar, Jan 02 2021

			The values of min(sigma(k)/k, sigma(k+1)/(k+1)) for k = 1, 2, ... 8 are 1, 4/3, 4/3, 6/5, 6/5, 8/7, 8/7, 13/9. The record values in this range, 1, 4/3 and 13/9, are obtained at k = 1, 2, and 8.

Cf. A000203, A004394, A068403, A096399, A333053.

seq={}; rminmax = 0; r1 = 1; Do[r2 = DivisorSigma[1, n]/n; rmin = Min[r1, r2]; If[rmin > rminmax, rminmax = rmin; AppendTo[seq, n-1]]; r1 = r2, {n, 2, 10^6}]; seq

a(22)-a(27) from Amiram Eldar, Jan 02 2021

A339937 Numbers k such that k and k+1 are both coreful abundant numbers (A308053).

Original entry on oeis.org

2282175, 33350624, 46734975, 86424975, 87152624, 105674624, 126114975, 169707824, 179762624, 214491375, 221370975, 235857824, 266022224, 270586575, 278524575, 297774224, 360021375, 372683024, 380858624, 395715375, 425840624, 470624175, 489873824, 503963775

Offset: 1

Amiram Eldar, Dec 23 2020

nonn

			2282175 is a term since 2282175 and 2282176 are both coreful abundant numbers.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Subsequence of A308053.

Similar sequences: A096399, A283418, A318167, A327635, A327942, A330872, A331412, A333951.

f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); abQ[n_] := s[n] > 2*n; q1 = False; seq = {}; Do[q2 = abQ[n]; If[q1 && q2, AppendTo[seq, n - 1]]; q1 = q2, {n, 2, 10^8}]; seq

A347937 Numbers k such that k and k+1 are both terms of A347935.

Original entry on oeis.org

2282175, 16769024, 18356624, 27252224, 32493824, 35820224, 46577024, 50968575, 51962624, 53992575, 55130624, 61854975, 63101024, 63140175, 69980624, 72525375, 73378304, 74376224, 80791424, 82389824, 98834175, 102650624, 105674624, 107769375, 109001024, 110238975

Offset: 1

Amiram Eldar, Sep 20 2021

nonn

			2282175 is a term since A187795(2282175) = 4801650 > 2*2282175 = 4564350 and A187795(2282176) = 4630080 > 2*2282176 = 4564352.

David A. Corneth, Table of n, a(n) for n = 1..2372 (first 103 terms from Amiram Eldar)

Subsequence of A005101, A096399 and A347935.

Cf. A187795, A347936, A347939.

abQ[n_] := DivisorSigma[1, n] > 2*n; s[n_] := DivisorSum[n, # &, abQ[#] &]; q[n_] := s[n] > 2*n; seq = {}; q1 = q[1]; Do[q2 = q[n]; If[q1 && q2, AppendTo[seq, n-1]]; q1 = q2, {n, 2, 2*10^7}]; seq

isok1(k) = sumdiv(k, d, if (sigma(d)>2*d, d)) > 2*k; \\ A347935
isok(k) = isok1(k) && isok1(k+1); \\ Michel Marcus, Sep 20 2021

A348276 Numbers k such that k and k+1 are both noninfinitary abundant numbers (A348274).

Original entry on oeis.org

64198575, 84909824, 86424975, 110238975, 113223824, 191206575, 211266224, 224722575, 231058575, 231800624, 240069375, 240584175, 245383424, 262648575, 262911824, 279597824, 293893424, 297774224, 333773055, 338676975, 340250624, 340829775, 347244975, 372683024

Offset: 1

Amiram Eldar, Oct 09 2021

nonn

			64198575 is a term since A348271(64198575) = 69470136 > 64198575 and A348271(64198576) = 65363424 > 64198576.

Cf. A348271.
Subsequence of A096399 and A348274.
Similar sequences: A318167, A327635, A327942, A331412.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ f @@@ FactorInteger[n]; q[n_] := DivisorSigma[1,n] - isigma[n] > n; seq = {}; q1 = q[1]; Do[q2 = q[n]; If[q1 && q2, AppendTo[seq, n-1]]; q1=q2 ,{n,2,10^8}]; seq

A348606 Numbers k such that k and k+1 are both nonexponential abundant numbers.

Original entry on oeis.org

21735, 76544, 170624, 301664, 345344, 348704, 382304, 739935, 862784, 1218944, 1262624, 1272704, 1314495, 1370655, 1376864, 1539615, 1558304, 1707615, 1718144, 1761375, 1845375, 1890944, 1926015, 2100735, 2132864, 2223584, 2415104, 2463615, 2581215, 2675295, 2747744

Offset: 1

Amiram Eldar, Oct 25 2021

nonn

			21735 is a term since A160135(21735) = 21930 > 21735 and A160135(21736) = 23230 > 21736.

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

Cf. A160135.
Subsequence of A096399 and A348604.
Similar sequences: A318167, A327635, A327942, A331412, A348276.

esigma[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; q[n_] := DivisorSigma[1, n] - esigma[n] > n; Select[Range[1, 3*10^6], q[#] && q[#+1] &]

A380933 Numbers k such that k and k+1 are both in A380929.

Original entry on oeis.org

121643775, 157390064, 161019495, 275734304, 584899875, 1493214975, 1614323655, 2043708975, 3081783375, 3118599224, 3426851295, 3902652495, 3947893424, 5849043375, 11731509855, 12138531615, 13008843224, 14598032624, 17588484584, 19782621495, 20191564575, 20759209064

Offset: 1

Amiram Eldar, Feb 08 2025

Numbers k such that A380845(k) > 2*k and A380845(k+1) > 2*(k+1).

			121643775 is a term since A380845(121643775) = 244722015 > 2 * 121643775 = 243287550, and A380845(121643776) = 256456081 > 2 * 121643776 = 243287552.

Amiram Eldar, Table of n, a(n) for n = 1..32

Subsequence of A096399 and A380929.
Cf. A380845, A380932.
Similar sequences: A283418, A318167, A327635, A327942, A331412, A333951, A357608, A364727, A364861.

q[k_] := Module[{h = DigitCount[k, 2, 1]}, DivisorSum[k, # &, DigitCount[#, 2, 1] == h &] > 2*k];
seq[lim_] := Module[{s = {}}, Do[If[q[k], If[q[k-1], AppendTo[s, k-1]]; If[q[k+1], AppendTo[s, k]]], {k, 3, lim, 2}]; s];
seq[3*10^8]

isab(k) = {my(h = hammingweight(k)); sumdiv(k, d, d*(hammingweight(d) == h)) > 2*k;}
list(lim) = forstep(k = 3, lim, 2, if(isab(k), if(isab(k-1), print1(k-1, ", ")); if(isab(k+1), print1(k, ", "))));

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).

cp's OEIS Frontend

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

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A383390 Numbers k such that k^2 and (k+1)^2 are both abundant numbers.

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A295042 Numbers k such that both k and (k+1) are abundant, and neither is divisible by 3.

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Extensions

A333054 Numbers m such that r(m) > r(k) for all k < m, where r(m) = min(sigma(m)/m, sigma(m+1)/(m+1)), and sigma(m) is the sum of divisors of m (A000203).

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A339937 Numbers k such that k and k+1 are both coreful abundant numbers (A308053).

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A347937 Numbers k such that k and k+1 are both terms of A347935.

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A348276 Numbers k such that k and k+1 are both noninfinitary abundant numbers (A348274).

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A348606 Numbers k such that k and k+1 are both nonexponential abundant numbers.

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A380933 Numbers k such that k and k+1 are both in A380929.