A096399 -id:A096399 - cp's OEIS frontend

A330872 Numbers k such that k and k+1 are both primitive abundant numbers (A071395).

82004, 158235, 516704, 2921535, 5801984, 10846016, 12374144, 12603824, 18738224, 24252074, 32409530, 33696975, 35356544, 36149295, 41078114, 42541190, 43485584, 65090864, 88304475, 90725775, 181480695, 183872535, 213261795, 233762528, 242301344, 254502495, 254630144

Offset: 1

Amiram Eldar, Apr 29 2020

nonn

Not to be confused with A283418 in which the primitive abundant numbers can have perfect numbers as divisors (as defined in A091191).

			82004 is a term since both 82004 and 82005 are abundant, and all of their proper divisors are deficient numbers.

Amiram Eldar, Table of n, a(n) for n = 1..100

Subsequence of A005101, A071395, A096399 and A283418.

primAbQ[n_] := DivisorSigma[1, n] > 2 n && AllTrue[Most @ Rest @ Divisors[n], DivisorSigma[1, #] < 2*# &]; q1 = False; seq = {}; Do[q2 = primAbQ[n]; If[q1 && q2, AppendTo[seq, n - 1]]; q1 = q2, {n, 2, 6*10^6}]; seq

A125115 Differences between consecutive abundant numbers.

Original entry on oeis.org

6, 2, 4, 6, 6, 4, 2, 6, 6, 2, 4, 6, 4, 2, 6, 2, 4, 4, 2, 6, 4, 2, 2, 4, 4, 2, 6, 6, 6, 6, 2, 4, 6, 6, 4, 2, 6, 6, 2, 4, 6, 6, 4, 2, 2, 4, 4, 2, 6, 4, 2, 2, 4, 6, 6, 6, 6, 6, 2, 4, 6, 2, 4, 4, 2, 6, 6, 6, 4, 2, 2, 4, 6, 2, 4, 6, 6, 4, 2, 6, 2

Offset: 1

Jason G. Wurtzel, Nov 21 2006

One may think that a(n) is always even and greater than 1. This is not the case as can be seen with A096399 or A228382. - Michel Marcus, Aug 21 2013

			a(1) = 6 because 18 - 12 = 6; a(4) = 6 because 30 - 24 = 6.

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

Cf. A005101, A096399, A228382, A302991.

A:=Filtered([1..350],n->Sigma(n)>2*n);;  a:=List([1..Length(A)-1],i->A[i+1]-A[i]); # Muniru A Asiru, Jun 09 2018

#[[2]] - #[[1]]&/@Partition[Select[Range[300], DivisorSigma[1, #] > 2# &], 2, 1] (* Harvey P. Dale, Dec 02 2006 *)
Differences[Select[Range[300], DivisorSigma[1, #] > 2# &]] (* Alonso del Arte, Apr 29 2019 *)

lista(nn) = {lastab = 0; for (i=1, nn, if (sigma(i) > 2*i,if (lastab, print1(i - lastab, ", ")); lastab = i;););} \\ Michel Marcus, Aug 21 2013

From Amiram Eldar, Oct 21 2020: (Start)
a(n) = A005101(n+1) - A005101(n).
Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = 1/A302991 = 4.0384... (End)

More terms from Michel Marcus, Aug 21 2013

A169822 Numbers k such that A(k+1) = A(k) + 1, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

1432, 1487, 1849, 2742, 5380, 5434, 6474, 6786, 9752, 10674, 12311, 14115, 14557, 15237, 17266, 17558, 18987, 19138, 19761, 20110, 20259, 20343, 20967, 20997, 22262, 22735, 24342, 25650, 26003, 26471, 27122, 27721, 28914, 28968, 29741, 30203, 30294, 31274, 33322

Offset: 1

N. J. A. Sloane, May 29 2010

nonn

A096399 is the main entry for this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..7000 from Muniru A Asiru)

Cf. A005101, A091194, A096399.

A:=Filtered([1..150000],n->Sigma(n)>2*n);;
  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+1); # Muniru A Asiru, Jun 10 2018

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..150000]):
  a:=select(j->A[j+1]=A[j]+1,[$1..nops(A)-1]); # Muniru A Asiru, Jun 10 2018

fQ[n_] := DivisorSigma[1, n] > 2 n; lst = {}; c = 0; k = 1; While[k < 125000, If[fQ@k, c++; If[fQ[k - 1], AppendTo[lst, c - 1]]]; k++ ]; lst (* Robert G. Wilson v, Jun 11 2010 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 1, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

a(n) = A091194(A096399(n)). - Amiram Eldar, Mar 01 2025

a(10) onwards from Robert G. Wilson v, Jun 11 2010

A228382 Abundant numbers that differ from the next abundant number by 3.

Original entry on oeis.org

942, 945, 1572, 1575, 2202, 2205, 2832, 2835, 3462, 3465, 4092, 4095, 4722, 4725, 5352, 5355, 5772, 5985, 6432, 6435, 6612, 6615, 6822, 6825, 7242, 7245, 7425, 7872, 7875, 8082, 8085, 8412, 8415, 8502, 8505, 8922, 8925, 9132, 9135, 9552, 9555, 9762, 9765

Offset: 1

Michel Marcus, Aug 21 2013

nonn

Apparently these numbers come up mostly by pairs m, m+3 with m even; the odd terms being a subsequence of A005231. But this is not always the case (e.g., note the term 7425).

The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 2, 43, 393, 3635, 37599, 374092, 3731903, 37338208, 373256850, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00373... . - Amiram Eldar, May 30 2023

			942 is abundant, 943 and 944 are deficient, 945 is abundant.
945 is abundant, 946 and 947 are deficient, 948 is abundant.

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

Subsequence of A005101.

Cf. A005231, A096399, A316095.

a:=Filtered([1..130000],n->Sigma(n)>2*n and Sigma(n+1)<2*(n+1) and Sigma(n+2)<2*(n+2) and Sigma(n+3)>2*(n+3)); # Muniru A Asiru, Jun 09 2018

with(numtheory): select(n->sigma(n)>2*n and sigma(n+1)<2*(n+1) and sigma(n+2)<2*(n+2) and sigma(n+3)>2*(n+3),[$1..12000]); # Muniru A Asiru, Jun 09 2018

With[{abs = Select[Range[10000], DivisorSigma[-1, #] > 2 &]}, abs[[Position[Differences[abs], 3] // Flatten]]] (* Amiram Eldar, May 30 2023 *)
SequencePosition[Table[If[DivisorSigma[1,n]>2n,1,0],{n,10000}],{1,0,0,1}][[;;,1]] (* Harvey P. Dale, Aug 31 2024 *)

isok(n) = (sigma(n)> 2*n) && (sigma(n+1)< 2*(n+1)) && (sigma(n+2) < 2*(n+2)) && (sigma(n+3) > 2*(n+3)); \\ Michel Marcus, Aug 21 2013

a(n) = A005101(A316095(n)). - Amiram Eldar, Mar 01 2025

A353538 Numbers k such that k and k+1 both have an abundancy index that is larger than Pi^2/6.

Original entry on oeis.org

44, 63, 75, 104, 135, 164, 188, 189, 195, 224, 230, 231, 255, 284, 285, 315, 344, 374, 375, 404, 405, 434, 435, 440, 441, 464, 494, 495, 524, 567, 584, 675, 692, 735, 764, 819, 824, 825, 854, 855, 944, 945, 975, 1034, 1035, 1070, 1071, 1124, 1155, 1196, 1215, 1274

Offset: 1

Amiram Eldar, Apr 25 2022

nonn

The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 3, 43, 399, 4292, 41306, 413310, 4155925, 414963651, 4149603572, ...

Apparently, the asymptotic density of this sequence is 0.04149...

			44 is a term since sigma(44)/44 = 84/44 = 1.909... and sigma(45)/45 = 78/45 = 1.733... are both larger than Pi^2/6 = 1.644... .

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

Cf. A000203, A013661.
Subsequence of A353537.
Subsequences: A096399, A353539, A353540, A353541.

q[n_] := DivisorSigma[-1, n] > Pi^2/6; Select[Range[1300], q[#] && q[# + 1] &]

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

A303741 Numbers k such that A(k+1) = A(k) + 2, where A() = A005101() are the abundant numbers.

Original entry on oeis.org

2, 7, 10, 14, 16, 19, 22, 23, 26, 31, 36, 39, 44, 45, 48, 51, 52, 59, 62, 65, 70, 71, 74, 79, 81, 82, 83, 86, 87, 90, 93, 96, 99, 104, 107, 110, 111, 114, 118, 120, 125, 128, 131, 133, 135, 138, 141, 146, 149, 150, 155, 156, 158, 164, 169, 170, 175, 178, 179

Offset: 1

Muniru A Asiru, Jun 22 2018

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Muniru A Asiru)

A231086 is the main entry for this sequence.

Cf. A005101, A091194, A096399, A169822.

A:=Filtered([1..1000],n->Sigma(n)>2*n);;  a:=Filtered([1..Length(A)-1],i->A[i+1]=A[i]+2);

with(numtheory): A:=select(n->sigma(n)>2*n,[$1..1000]):  a:=select(j->A[j+1]=A[j]+2,[$1..nops(A)-1]);

Position[Differences[Select[Range[750], DivisorSigma[1, #] > 2*# &]], 2] // Flatten (* Amiram Eldar, Mar 15 2024 *)

list(lim) = {my(k = 1, k2, m = 0); for(k2 = 2, lim, if(sigma(k2, -1) > 2, if(k2 == k1 + 2, print1(m, ", ")); m++; k1 = k2));} \\ Amiram Eldar, Mar 01 2025

Sequence is { k | A005101(k+1) = A005101(k) + 2 }.

a(n) = A091194(A231086(n)). - Amiram Eldar, Mar 01 2025

A094268 Starting term of smallest consecutive n-tuples of abundant numbers.

Original entry on oeis.org

12, 5775, 171078830

Offset: 1

Lekraj Beedassy, Jun 02 2004

The triple 171078830, 171078831, 171078832 was apparently found by Laurent Hodges and Michael Reid in 1995.
The starting term of the smallest consecutive 4-tuple of abundant numbers is at most 141363708067871564084949719820472453374 - Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Nov 01 2007
Paul Erdős showed that there are two absolute constants c1, c2 such that for all large n there are at least c1 log log log n but not more than c2 log log log n consecutive abundant numbers less than n. - Bruno Mishutka (bruno.mishutka(AT)googlemail.com), Nov 01 2007
From Jianing Song, Apr 10 2021: (Start)
a(n) exists for all n. Proof: since the infinite product Product_{p prime} (1 + 1/p) diverges, we can find a strictly increasing sequence {b(m)} such that b(0) = 0, Product_{k=b(m)+1..b(m+1)} (1 + 1/prime(k)) > 2 for all m. Given n, by Chinese Remainder Theorem, we can find N such that N + m divides Product_{k=b(m)+1..b(m+1)} prime(k) for m = 0..n-1, then sigma(N + m)/(N + m) >= Product_{k=b(m)+1..b(m+1)} (1 + 1/prime(k)) > 2.
For example, if N is divisible by 2*3*5, N+1 is divisible by 7*11*...*73, N+2 is divisible by 79*83*...*7499, N+3 is divisible by 7507*7517*...*57081677, N+4 is divisible by 57081679*57081697*...*(some very large prime), then N through N+4 are consecutive abundant numbers.
Of course, the number N found using this method will be extremely large, since Product_{k=1..K} (1 + 1/prime(k)) ~ log(log(K)). (End)

J.-M. De Koninck and A. Mercier, 1001 Problemes en Theorie Classique Des Nombres, Problem 771, pp. 98, 327, Ellipses, Paris, 2004.
S. Kravitz, Three Consecutive Abundant Numbers, Journal of Recreational Mathematics, 26:2 (1994), 149. Solution by L. Hodges and M. Reid, JRM, 27:2 (1995), 156-157.

Paul Erdős, Note on consecutive abundant numbers, J. London Math. Soc. 10, 128-131 (1935).
Carlos Rivera, Puzzle 878. Consecutive abundant integers

Cf. A005101, A005231, A096399.

A103289 Numbers k such that both sigma(k) >= 2k-1 and sigma(k+1) >= 2(k+1)-1.

Original entry on oeis.org

1, 4095, 5775, 5984, 7424, 11024, 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

Offset: 1

Max Alekseyev, Jan 28 2005

nonn

A004125(n) is less than A004125(n-1) and A004125(n+1) iff n belongs to A023196 but not to the current sequence.
Numbers k that both k and k+1 are in A103288.
Union of sequences A096399 and {2^m-1} for m in A103291.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2500 from Muniru A Asiru)

Cf. A103288, A096399, A103291, A004125, A023196.

Filtered([1..120000],n->Sigma(n)>=2*n-1 and Sigma(n+1)>2*(n+1)-1); # Muniru A Asiru, Jun 26 2018

with(numtheory): a:=`if`(sigma(n)>=2*n-1 and sigma(n+1)>=2*(n+1)-1,n,NULL): seq(a(n),n=1..120000); # Muniru A Asiru, Jun 26 2018

Select[Range[120000], And[DivisorSigma[1, #] >= 2 #1 - 1, DivisorSigma[1, # + 1] >= 2 (#1 + 1) - 1] &] (* Michael De Vlieger, Jun 29 2018 *)

for(i=1,1000000,if(sigma(i)>=2*i-1 && sigma(i+1)>=2*i+1, print1(i, ", ")));

A103291 Numbers k such that sigma(2^k-1) >= 2*(2^k-1)-1, i.e., the number 2^k-1 is perfect, abundant, or least deficient.

Original entry on oeis.org

1, 12, 24, 36, 40, 48, 60, 72, 80, 84, 90, 96, 108, 120, 132, 140, 144, 156, 160, 168, 180, 192, 200, 204, 210, 216, 220, 228, 240, 252, 264, 270, 276, 280, 288, 300, 312, 320, 324, 330, 336, 348, 360, 372, 384, 396, 400, 408, 420, 432, 440, 444, 450, 456, 468

Offset: 1

Max Alekseyev, Jan 28 2005

Is there an odd term besides 1? Numbers 2^a(i)-1 form set difference of sequences A103289 and A096399.

Odd terms > 1 exist, but there are none < 10^7. If k > 1 is an odd term, then 2^k-1 must have more than 900000 distinct prime factors and all of them must be members of A014663. - David Wasserman, Apr 15 2008

Amiram Eldar, Table of n, a(n) for n = 1..136

Cf. A000203, A075708, A103288, A103289, A103292, A023196.

for(i=1,1000,n=2^i-1;if(sigma(n)>=2*n-1,print1(i, ", ")));

Numbers k such that 2^k-1 is in A103288.

More terms from David Wasserman, Apr 15 2008

A292704 Unitary abundant numbers k such that k+2 is also unitary abundant.

Original entry on oeis.org

1428, 20020, 49740, 63490, 107338, 137170, 142140, 195130, 218218, 315588, 340338, 380380, 382380, 497418, 514668, 555828, 578578, 580578, 602140, 626890, 672978, 711618, 740740, 786828, 795340, 811578, 860860, 862860, 885610, 897258, 904330, 907060, 940938

Offset: 1

Amiram Eldar, Sep 21 2017

nonn

			1428 is in the sequence since both 1428 and 1430 are unitary abundant: usigma(1428) = 2880 > 2*1428 and usigma(1430) = 3024 > 2*1430.

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

Cf. A034683, A096399.

usigma[n_]:=If[n==1, 1, Times @@ (1+Power @@@ FactorInteger[n])];
uaQ[n_] := usigma[n] > 2 n; Select[Range@100000, uaQ[#] && uaQ[# + 2] &]

cp's OEIS Frontend

A330872 Numbers k such that k and k+1 are both primitive abundant numbers (A071395).

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A125115 Differences between consecutive abundant numbers.

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A169822 Numbers k such that A(k+1) = A(k) + 1, where A() = A005101() are the abundant numbers.

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A228382 Abundant numbers that differ from the next abundant number by 3.

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A353538 Numbers k such that k and k+1 both have an abundancy index that is larger than Pi^2/6.

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A303741 Numbers k such that A(k+1) = A(k) + 2, where A() = A005101() are the abundant numbers.

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A094268 Starting term of smallest consecutive n-tuples of abundant numbers.

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A103289 Numbers k such that both sigma(k) >= 2k-1 and sigma(k+1) >= 2(k+1)-1.