A294025 -id:A294025 - cp's OEIS frontend

A231086 Initial members of abundant twins, i.e., values of k such that (k, k+2) are both abundant numbers.

18, 40, 54, 70, 78, 88, 100, 102, 112, 138, 160, 174, 196, 198, 208, 220, 222, 258, 270, 280, 304, 306, 318, 340, 348, 350, 352, 364, 366, 378, 390, 400, 414, 438, 448, 460, 462, 474, 490, 498, 520, 532, 544, 550, 558, 570, 580, 606, 616, 618, 640, 642, 648

Offset: 1

Shyam Sunder Gupta, Nov 03 2013

nonn

The first odd term is <= 76728582876430878992529528245373 (see A294025). Note that there are infinitely many odd terms, since if k is an odd term then 2*t*k*(k+2) + k are odd terms for all t >= 0. - Jianing Song, Nov 13 2022
From Amiram Eldar, May 30 2023: (Start)
The least odd term is larger than 10^11.
The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 7, 81, 820, 8074, 80410, 804623, 8040362, 80414534, 804257458, 8042148484, ... . Apparently, the asymptotic density of this sequence exists and equals 0.08042... . (End)

			18, 20 are abundant, thus the smaller number is listed.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Shyam Sunder Gupta)

Cf. A005101, A096399, A108926, A231088, A231089, A231090, A231092, A231093, A303741.

A:=Filtered([1..700],n->Sigma(n)>2*n);;  a:=List(Filtered([1..Length(A)-1],i->A[i+1]=A[i]+2),j->A[j]); # Muniru A Asiru, Jun 24 2018

withnumtheory: select(n->sigma(n)>2*n and sigma(n+1)<2*(n+1) and sigma(n+2)>2*(n+2),[$1..700]); # Muniru A Asiru, Jun 24 2018

AbundantQ[n_] := DivisorSigma[1, n] > 2n; m = 0; a2 = {}; Do[If[AbundantQ[n], m = m + 1; If[m > 1, AppendTo[a2, n - 2]], m = 0], {n, 2, 100000, 2}];a2
Module[{nn=650,sa},sa=Table[If[DivisorSigma[1,n]>2n,1,0],{n,nn}];Transpose[ SequencePosition[sa,{1,0,1}]]][[1]] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, May 20 2016 *)

is(n)=sigma(n,-1)>2 && sigma(n+2,-1)>2 \\ Charles R Greathouse IV, Feb 21 2017

a(n) = A005101(A303741(n)). - Amiram Eldar, Mar 01 2025

A333503 Differences between consecutive odd abundant numbers.

Original entry on oeis.org

630, 630, 630, 630, 630, 630, 630, 420, 210, 450, 180, 210, 420, 180, 450, 210, 330, 90, 420, 210, 420, 210, 630, 630, 630, 630, 420, 210, 630, 630, 630, 210, 420, 630, 630, 630, 630, 630, 630, 90, 330, 210, 630, 630, 630, 210, 330, 90, 630, 210, 420, 630, 630

Offset: 1

Jeppe Stig Nielsen, Mar 24 2020

nonn

a(n) = A005231(n+1) - A005231(n).
Since 945+1890*k is odd and abundant for all k, 1890 is the maximal term that occurs. The first time it occurs is for a(274)=1890, and the involved odd abundant numbers are 135135 and 137025.
Clearly 2 is the minimal term that can occur. For an example, see comment to A294025.

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..1000

Cf. A005231, A294025.

Differences@ Select[Range[947, 3*10^4 + 1, 2], DivisorSigma[1, #] > 2 # &] (* Michael De Vlieger, Mar 25 2020 *)

my(oldn=945); forstep(n=947, 5*10^4, 2, if(sigma(n)>2*n, print1(n-oldn, ", "); oldn=n))

A294026 Odd unitary abundant numbers with a record small gap to the next odd unitary abundant number.

Original entry on oeis.org

15015, 19635, 21945, 25935, 33495, 1752135, 1915095, 1915305, 119104635, 134877405

Offset: 1

Amiram Eldar, Oct 22 2017

The corresponding gaps are 4620, 2310, 1260, 630, 420, 330, 210, 180, 120, 30.
The upper ends are 19635, 21945, 23205, 26565, 33915, 1752465, 1915305, 1915485, 119104755, 134877435, ...
The unitary version of A294025.
No more terms below 10^9.
10^13 < a(11) <= 42229304608764255 (gap 18), while t = 220730839027951785 and t+6 are a pair with gap 6. - Giovanni Resta, May 07 2020

			Odd unitary abundant numbers are 15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, ...
Their differences are 4620, 2310, 1260, 2730, 630, 4830, 2100, 420, ...
The records of small differences are 4620, 2310, 1260, 630, 420, ...
And the corresponding terms are 15015, 19635, 21945, 25935, 33495, ...

Cf. A129485, A294025.

usigma[n_] := If[n == 1, 1, Times @@ (1 + Power @@@ FactorInteger[n])]; ouaQ[n_] := OddQ[n] && usigma[n] > 2 n; s = Select[Range[100000], ouaQ]; a={}; dmin = 5000; Do[d=s[[j+1]]-s[[j]]; If[d

usig(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d));
isok(n) = (n%2) && (usig(n) > 2*n);
lista(nn) = {last = 0; gap = oo; forstep(n=1, nn, 2, if (isok(n), if (last, if (n - last < gap, print1(last, ", "); gap = n - last)); last = n;););} \\ Michel Marcus, Dec 15 2017

A294027 Odd bi-unitary abundant numbers with a record small gap to the next term odd bi-unitary abundant number.

Original entry on oeis.org

945, 8505, 15015, 19305, 21735, 33345, 34125, 21961263753, 39891817245

Offset: 1

Amiram Eldar, Oct 22 2017

The corresponding gaps are 7560, 1890, 1050, 330, 210, 150, 30, 12, 6.
The upper ends are 8505, 10395, 16065, 19635, 21945, 33495, 34155, 21961263765, 39891817251.
The bi-unitary version of A294025.

			Odd bi-unitary abundant numbers are 945, 8505, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, ...
Their differences are 7560, 1890, 1890, 2730, 1050, 1890, 1350, 330, 2100, 210, ...
The records of small differences are 7560, 1890, 1050, 330, 210, ...
And the corresponding terms are 945, 8505, 15015, 19305, 21735, ...

Cf. A293186, A294025.

f[n_] := Select[Divisors[n], Function[d, CoprimeQ[d, n/d]]]; bsigma[m_] := DivisorSum[m, # &, Last@Intersection[f@#, f[m/#]] == 1 &]; bOddAbundantQ[n_] := OddQ[n] && bsigma[n] > 2 n; s = Select[Range[1000000], bOddAbundantQ]; a = {}; dmin = 50000; Do[d = s[[j + 1]] - s[[j]]; If[d < dmin, AppendTo[a, s[[j]]]; dmin = d], {j, 1, Length[s] - 1}]; a

udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
biusig(n) = vecsum(biudivs(n));
isok(n) = (n % 2) && (biusig(n) > 2*n);
lista(nn) = {last = 0; gap = oo; forstep(n=1, nn, 2, if (isok(n), if (last, if (n - last < gap, print1(last, ", "); gap = n - last)); last = n;););} \\ Michel Marcus, Dec 15 2017

a(1) added by Amiram Eldar, Aug 22 2018.

a(8)-a(9) from Giovanni Resta, Aug 22 2018

cp's OEIS Frontend

A231086 Initial members of abundant twins, i.e., values of k such that (k, k+2) are both abundant numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

A333503 Differences between consecutive odd abundant numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A294026 Odd unitary abundant numbers with a record small gap to the next odd unitary abundant number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A294027 Odd bi-unitary abundant numbers with a record small gap to the next term odd bi-unitary abundant number.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions