A066272 -id:A066272 - cp's OEIS frontend

A209271 Primes p such that A066272(p)*p+1 is also prime, where A066272 is the number of anti-divisors.

5, 13, 181, 613, 761, 1201, 8581, 9661, 21013, 26681, 34061, 59513, 68821, 101701, 156241, 584281, 637321, 718801, 782501, 787513, 1078981, 1193513, 1336613, 1470613, 1529501, 1639861, 1757813, 2103301, 2257813, 2287661, 2601481, 3540461, 4307113

Offset: 1

M. F. Hasler, Jan 15 2013

nonn

Could be called "Sophie Germain anti-primes" or "anti-Sophie Germain primes". Inspired by the Gerasimov link.
Sophie Germain primes are such that 2p+1 is also prime, where 2 is the number of divisors of p. Here this is replaced with the number of anti-divisors.
There are only 47 such primes below 10^7.

Donovan Johnson, Table of n, a(n) for n = 1..1000
J. S. Gerasimov, Sophie Germain nonprimes [title corrected], SeqFan mailing list, Jan 15 2013

Cf. A066272.

{forprime(n=1,default(primelimit),isprime(A066272(n)*n+1) & print1(n","))}

A073638 Number of anti-divisors of n (A066272) sets a record.

Original entry on oeis.org

1, 3, 5, 7, 13, 17, 32, 38, 67, 137, 203, 247, 472, 578, 682, 787, 1463, 2047, 2363, 3465, 5197, 5198, 8662, 13513, 15593, 22522, 22523, 29452, 60638, 67567, 67568, 98753, 112612, 157658, 202702, 337837, 337838, 427927, 713212, 788287, 788288, 1013512

Offset: 1

Jason Earls, Sep 01 2002

nonn

antid(n) > antid(k) for all k < n.
Note that several of these come in pairs, i.e., 5197 & 5198, 22522 & 22523, 67567 & 67568, 337837 & 337838, 788287 & 788288, 1013512 & 1013513 and 1914412 & 1914413 to name a few. See A093071 for more. - Robert G. Wilson v, Mar 17 2004
See A066272 for definition of anti-divisor.

Donovan Johnson, Table of n, a(n) for n = 1..117 (terms < 5*10^11)
Jon Perry, Anti-divisors.
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 &], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n &]; a = 0; Do[b = Length[ antid[ n]]; If[b > a, Print[n]; a = b], {n, 1, 1013513}] (* Robert G. Wilson v, Mar 17 2004 *)

More terms from Robert G. Wilson v, Mar 17 2004

A066980 Numbers k such that antid(k) > antid(k+1) and santid(k) > santid(k+1), where antid(k) = A066272 and santid(k) = A066417.

Original entry on oeis.org

5, 7, 13, 15, 18, 23, 25, 28, 32, 33, 35, 38, 43, 45, 47, 50, 53, 55, 60, 63, 67, 68, 73, 77, 78, 83, 85, 88, 95, 98, 105, 108, 110, 113, 115, 117, 123, 127, 128, 130, 133, 137, 138, 140, 143, 145, 150, 153, 155, 158, 162, 163, 165, 168, 172, 173, 175, 176, 178, 182

Offset: 1

Jason Earls, Sep 09 2002

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

Cf. A066272, A066417.

A073547 Numbers k such that antid(k) = antid(k+1), where antid(k) = A066272(k).

Original entry on oeis.org

1, 3, 8, 10, 14, 19, 20, 22, 27, 29, 40, 42, 46, 49, 52, 58, 65, 70, 74, 75, 82, 87, 90, 91, 94, 102, 103, 112, 116, 118, 122, 124, 131, 135, 148, 149, 151, 154, 157, 159, 171, 180, 183, 187, 188, 198, 204, 205, 208, 212, 213, 214, 217, 220, 222, 227, 231, 232

Offset: 1

Jason Earls, Aug 31 2002

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A066272.

N:= 1000: # to get all terms <= N-1
V:= Vector(N):
for k from 1 to floor(N/3) do
  R1:= [seq(i, i=3*k .. N, 2*k)];
  V[R1]:= map(`+`,V[R1],1);
  R2:= [seq(i, i=3*k+1 .. N, 2*k+1)];
  V[R2]:= map(`+`,V[R2],1);
  R3:= [seq(i,i=3*k+2 .. N, 2*k+1)];
  V[R3]:= map(`+`,V[R3],1);
od:
select(t -> V[t]=V[t+1], [$1..N-1]); # Robert Israel, Sep 26 2016

at[n_] := Count[Flatten[Quotient[#, Rest[Select[Divisors[#], OddQ]]] & /@ (2 n + Range[-1, 1])], Except[1]]; Select[Range[232], at[#] == at[# + 1] &] (* Jayanta Basu, Jul 01 2013 *)

A093071 Lesser of a pair of records in A066272.

Original entry on oeis.org

5197, 22522, 67567, 337837, 788287, 1013512, 1914412, 17229712, 50923372, 65472907, 132094462, 254616862, 327364537, 2291551762, 5856187837, 12548973937, 72784048837, 158117071612, 218352146512, 363920244187

Offset: 1

Robert G. Wilson v and Jon Perry, Mar 17 2004

nonn

Or the lesser of a pair in A073638.

See A066272 for definition of anti-divisor.

Jon Perry, Anti-divisors.
Jon Perry, The Anti-divisor [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors [Cached copy]

A073569 Numbers k such that A002034(k) = A066272(k).

Original entry on oeis.org

63, 72, 126, 140, 180, 189, 210, 225, 252, 275, 325, 378, 405, 500, 504, 560, 585, 715, 756, 810, 819, 875, 891, 900, 1001, 1080, 1134, 1152, 1232, 1292, 1530, 1600, 1650, 1680, 1785, 1872, 2093, 2128, 2250, 2295, 2340, 2376, 2464, 2565, 2574

Offset: 1

Jason Earls, Aug 31 2002

nonn

			a(1) = 63 since A002034(63) = 7 and 63 has A066272(63) = 7 anti-divisors: {2, 5, 6, 14, 18, 25, 42}.

Cf. A002034, A066272.

More terms from Max Alekseyev, Jan 31 2012

1 removed by Sean A. Irvine, Dec 09 2024

A032766 Numbers that are congruent to 0 or 1 (mod 3).

Original entry on oeis.org

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103

Offset: 0

Patrick De Geest, May 15 1998

Omitting the initial 0, a(n) is the number of 1's in the n-th row of the triangle in A118111. - Hans Havermann, May 26 2002
Binomial transform is A053220. - Michael Somos, Jul 10 2003
Smallest number of different people in a set of n-1 photographs that satisfies the following conditions: In each photograph there are 3 women, the woman in the middle is the mother of the person on her left and is a sister of the person on her right and the women in the middle of the photographs are all different. - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Partial sums of A000034. - Richard Choulet, Jan 28 2010
Starting with 1 = row sums of triangle A171370. - Gary W. Adamson, Feb 15 2010
a(n) is the set of values for m in which 6k + m can be a perfect square (quadratic residues of 6 including trivial case of 0). - Gary Detlefs, Mar 19 2010
For n >= 2, a(n) is the smallest number with n as an anti-divisor. - Franklin T. Adams-Watters, Oct 28 2011
Sequence is also the maximum number of floors with 3 elevators and n stops in a "Convenient Building". See A196592 and Erich Friedman link below. - Robert Price, May 30 2013
a(n) is also the total number of coins left after packing 4-curves patterns (4c2) into a fountain of coins base n. The total number of 4c2 is A002620 and voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 26 2013
Number of partitions of 6n into two even parts. - Wesley Ivan Hurt, Nov 15 2014
Number of partitions of 3n into exactly 2 parts. - Colin Barker, Mar 23 2015
Nonnegative m such that floor(2*m/3) = 2*floor(m/3). - Bruno Berselli, Dec 09 2015
For n >= 3, also the independence number of the n-web graph. - Eric W. Weisstein, Dec 31 2015
Equivalently, nonnegative numbers m for which m*(m+2)/3 and m*(m+5)/6 are integers. - Bruno Berselli, Jul 18 2016
Also the clique covering number of the n-Andrásfai graph for n > 0. - Eric W. Weisstein, Mar 26 2018
Maximum sum of degeneracies over all decompositions of the complete graph of order n+1 into three factors. The extremal decompositions are characterized in the Bickle link below. - Allan Bickle, Dec 21 2021
Also the Hadwiger number of the n-cocktail party graph. - Eric W. Weisstein, Apr 30 2022
The number of integer rectangles with a side of length n+1 and the property: the bisectors of the angles form a square within its limits. - Alexander M. Domashenko, Oct 17 2024
The maximum possible number of 5-cycles in an outerplanar graph on n+4 vertices. - Stephen Bartell, Jul 10 2025

Indranil Ghosh, Table of n, a(n) for n = 0..10000
Allan Bickle, Nordhaus-Gaddum Theorems for k-Decompositions, Congr. Num. 211 (2012) 171-183.
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
Erich Friedman, Problem of the month November 2009
Z. Füredi, A. Kostochka, M. Stiebitz, R. Skrekovski, and D. West, Nordhaus-Gaddum-type theorems for decompositions into many parts, J. Graph Theory 50 (2005), 273-292.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 282. [Book's website]
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
International Mathematical Olympiad 2001, Hong Kong Preliminary Selection Contest, Problem #20. [Broken link; Cached copy]
Matroids Matheplanet, Number of d-generator groups of order 2^(d+1) and exponent-p class 2 (in German).
Emanuele Munarini, Topological indices for the antiregular graphs, Le Mathematiche, Vol. 76, No. 1 (2021), pp. 277-310, see p. 302.
Kival Ngaokrajang, Illustration of initial terms (U).
Eric Weisstein's World of Mathematics, Andrásfai Graph
Eric Weisstein's World of Mathematics, Clique Covering Number
Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Hadwiger Number
Eric Weisstein's World of Mathematics, Independence Number
Eric Weisstein's World of Mathematics, Web Graph
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A006578 (partial sums), A000034 (first differences), A016789 (complement).
Essentially the same: A049624.
Column 1 (the second leftmost) of triangular table A026374.
Column 1 (the leftmost) of square array A191450.
Row 1 of A254051.
Row sums of A171370.
Cf. A066272 for anti-divisors.
Cf. A253888 and A254049 (permutations of this sequence without the initial zero).
Cf. A254103 and A254104 (pair of permutations based on this sequence and its complement).
Cf. A000035, A000217, A000982, A001651, A002620, A002717, A003945.
Cf. A004396, A004526, A007310, A007494, A008619, A030308, A035360.
Cf. A047270, A049615, A052928, A053220, A070893, A084056, A092942.
Cf. A118111, A132463, A133083, A196592, A249745.

a032766 n = div n 2 + n  -- Reinhard Zumkeller, Dec 13 2014
(MIT/GNU Scheme) (define (A032766 n) (+ n (floor->exact (/ n 2)))) ;; Antti Karttunen, Jan 24 2015

&cat[ [n, n+1]: n in [0..100 by 3] ]; // Vincenzo Librandi, Nov 16 2014

a[0]:=0:a[1]:=1:for n from 2 to 100 do a[n]:=a[n-2]+3 od: seq(a[n], n=0..69); # Zerinvary Lajos, Mar 16 2008
seq(floor(n/2)+n, n=0..69); # Gary Detlefs, Mar 19 2010
select(n->member(n mod 3,{0,1}), [$0..103]); # Peter Luschny, Apr 06 2014

a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = 0; a[1] = 1; a[2] = 3; a[3] = 4; Array[a, 60, 0] (* Robert G. Wilson v, Mar 28 2011 *)
Select[Range[0, 200], MemberQ[{0, 1}, Mod[#, 3]] &] (* Vladimir Joseph Stephan Orlovsky, Feb 11 2012 *)
Flatten[{#,#+1}&/@(3Range[0,40])] (* or *) LinearRecurrence[{1,1,-1}, {0,1,3}, 100] (* or *) With[{nn=110}, Complement[Range[0,nn], Range[2,nn,3]]] (* Harvey P. Dale, Mar 10 2013 *)
CoefficientList[Series[x (1 + 2 x) / ((1 - x) (1 - x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Nov 16 2014 *)
Floor[3 Range[0, 69]/2] (* L. Edson Jeffery, Jan 14 2017 *)
Drop[Range[0,110],{3,-1,3}] (* Harvey P. Dale, Sep 02 2023 *)

PARI
```
{a(n) = n + n\2}
```

concat(0, Vec(x*(1+2*x)/((1-x)*(1-x^2)) + O(x^100))) \\ Altug Alkan, Dec 09 2015

[int(3*n//2) for n in range(101)] # G. C. Greubel, Jun 23 2024

G.f.: x*(1+2*x)/((1-x)*(1-x^2)).
a(-n) = -A007494(n).
a(n) = A049615(n, 2), for n > 2.
From Paul Barry, Sep 04 2003: (Start)
a(n) = (6n - 1 + (-1)^n)/4.
a(n) = floor((3n + 2)/2) - 1 = A001651(n) - 1.
a(n) = sqrt(2) * sqrt( (6n-1) (-1)^n + 18n^2 - 6n + 1 )/4.
a(n) = Sum_{k=0..n} 3/2 - 2*0^k + (-1)^k/2. (End)
a(n) = 3*floor(n/2) + (n mod 2) = A007494(n) - A000035(n). - Reinhard Zumkeller, Apr 04 2005
a(n) = 2 * A004526(n) + A004526(n+1). - Philippe Deléham, Aug 07 2006
a(n) = 1 + ceiling(3*(n-1)/2). - Fung Cheok Yin (cheokyin_restart(AT)yahoo.com.hk), Sep 22 2006
Row sums of triangle A133083. - Gary W. Adamson, Sep 08 2007
a(n) = (cos(Pi*n) - 1)/4 + 3*n/2. - Bart Snapp (snapp(AT)coastal.edu), Sep 18 2008
A004396(a(n)) = n. - Reinhard Zumkeller, Oct 30 2009
a(n) = floor(n/2) + n. - Gary Detlefs, Mar 19 2010
a(n) = 3n - a(n-1) - 2, for n>0, a(0)=0. - Vincenzo Librandi, Nov 19 2010
a(n) = n + (n-1) - (n-2) + (n-3) - ... 1 = A052928(n) + A008619(n-1). - Jaroslav Krizek, Mar 22 2011
a(n) = a(n-1) + a(n-2) - a(n-3). - Robert G. Wilson v, Mar 28 2011
a(n) = Sum_{k>=0} A030308(n,k) * A003945(k). - Philippe Deléham, Oct 17 2011
a(n) = 2n - ceiling(n/2). - Wesley Ivan Hurt, Oct 25 2013
a(n) = A000217(n) - 2 * A002620(n-1). - Kival Ngaokrajang, Oct 26 2013
a(n) = Sum_{i=1..n} gcd(i, 2). - Wesley Ivan Hurt, Jan 23 2014
a(n) = 2n + floor((-n - (n mod 2))/2). - Wesley Ivan Hurt, Mar 31 2014
A092942(a(n)) = n for n > 0. - Reinhard Zumkeller, Dec 13 2014
a(n) = floor(3*n/2). - L. Edson Jeffery, Jan 18 2015
a(n) = A254049(A249745(n)) = (1+A007310(n)) / 2 for n >= 1. - Antti Karttunen, Jan 24 2015
E.g.f.: (3*x*exp(x) - sinh(x))/2. - Ilya Gutkovskiy, Jul 18 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)) + log(3)/2. - Amiram Eldar, Dec 04 2021

Better description from N. J. A. Sloane, Aug 01 1998

A066417 Sum of anti-divisors of n.

Original entry on oeis.org

0, 0, 2, 3, 5, 4, 10, 8, 8, 14, 12, 13, 19, 16, 18, 14, 28, 28, 18, 24, 22, 36, 34, 23, 39, 24, 42, 46, 24, 36, 42, 58, 48, 30, 52, 32, 50, 70, 52, 55, 41, 66, 56, 40, 86, 58, 60, 56, 72, 80, 42, 94, 88, 52, 74, 56, 74, 96, 90, 107, 57, 78, 112, 46, 84, 86, 132, 112, 54, 102

Offset: 1

Jon Perry, Dec 28 2001

nonn

See A066272 for definition of anti-divisor.

			For n = 18: 2n-1, 2n, 2n+1 are 35, 36, 37 with odd divisors > 1 {5,7,35}, {3,9}, {37} and quotients 7, 5, 1, 12, 4, 1, so the anti-divisors of 12 are 4, 5, 7, 12. Therefore a(18) = 28.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Jon Perry, The Anti-divisor. [Cached copy]
Jon Perry, The Anti-divisor: Even More Anti-Divisors. [Cached copy]

Cf. A000203, A007814, A066272, A066416, A066418, A058838, A064277.

# Uses antidivisors() implemented in A066272.
A066417 := proc(n) add(d,d=antidivisors(n)) ; end proc: # R. J. Mathar, Jul 04 2011
# faster alternative with Alekseyev formula
A066417 := proc(n)
    k := A007814(n) ;
    numtheory[sigma](2*n-1)+numtheory[sigma](2*n+1) +numtheory[sigma(n/2^k)*2^(k+1) -6*n-2 ;
end proc: # R. J. Mathar, Nov 11 2014

antid[n_] := Select[ Union[ Join[ Select[ Divisors[2n - 1], OddQ[ # ] && # != 1 & ], Select[ Divisors[2n + 1], OddQ[ # ] && # != 1 & ], 2n/Select[ Divisors[ 2n], OddQ[ # ] && # != 1 &]]], # < n &]; Table[ Plus @@ antid[n], {n, 70}] (* Robert G. Wilson v, Mar 15 2004 *)
a066417[n_Integer] := Total[Cases[Range[2, n - 1], ?(Abs[Mod[n, #] - #/2] < 1 &)]]; Array[a066417, 120] (* _Michael De Vlieger, Aug 08 2014, after Harvey P. Dale at A066272 *)

al(n)=Vec(sum(k=1,n,2*k*(x^(3*k)+x*O(x^n))/(1-x^(2*k))+(2*k+1)*(x^(3*k+1)+x^(3*k+2)+x*O(x^n))/(1-x^(2*k+1)))) \\ Franklin T. Adams-Watters, Sep 11 2009

{ a(n) = my(k); if(n>1, k=valuation(n,2); sigma(2*n+1)+sigma(2*n-1)+sigma(n/2^k)*2^(k+1)-6*n-2, 0); } \\ Max Alekseyev, Apr 27 2010

from sympy import divisors
A066417 = [sum([2*d for d in divisors(n) if n > 2*d and n%(2*d)] + [d for d in divisors(2*n-1) if n > d >=2 and n%d] + [d for d in divisors(2*n+1) if n > d >=2 and n%d]) for n in range(1,10**6)] # Chai Wah Wu, Aug 12 2014

G.f.: Sum_{k>0} (2k x^(3k) / (1 - x^(2k)) + (2k+1)(x^(3k+1) + x^(3k+2)) / (1 - x^(2k+1))). - Franklin T. Adams-Watters, Sep 11 2009
For n>1, a(n) = A000203(2*n-1) + A000203(2*n+1) + A000203(n/2^k)*2^(k+1) - 6*n - 2, where k=A007814(n). - Max Alekseyev, Apr 27 2010
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/8 - 3 = 0.70110165... . - Amiram Eldar, Jan 19 2024

A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.

Original entry on oeis.org

2, 3, 2, 3, 4, 2, 3, 5, 3, 5, 2, 6, 3, 4, 7, 2, 3, 7, 5, 8, 2, 3, 5, 9, 3, 4, 9, 2, 6, 10, 3, 11, 2, 3, 5, 7, 11, 4, 5, 7, 12, 2, 3, 13, 3, 8, 13, 2, 6, 14, 3, 4, 5, 9, 15, 2, 3, 5, 9, 15, 7, 16, 2, 3, 7, 10, 17, 3, 4, 17, 2, 5, 6, 11, 18, 3, 5, 8, 11, 19, 2, 3, 19, 4, 12, 20, 2, 3, 7

Offset: 3

Diana L. Mecum, Jul 17 2007

A066272 gives the number of terms in each row.
See A066272 for definition of anti-divisor.
2n-1 and 2n+1 are twin primes (that is, n is in A040040) iff n has no odd anti-divisors. For example, because n=15 has no odd anti-divisors, 29 and 31 are twin primes. - Jon Perry, Sep 12 2012
Row n is all the numbers which are: (a) 2n divided by its odd divisors (except 1), and (b) the divisors of 2n-1 and 2n+1 (except 1, 2n+1 and 2n-1). For example, n=18: odd divisors of 36 are {3,9} and 36/{3,9} = {4,12}; divisors of 35 are {5,7} and divisors of 37 are null (37 is prime).  Therefore row 18 is 4,5,7 and 12.  See A066542 for further explanation. - Bob Selcoe, Feb 24 2014

			Anti-divisors of 3 through 20:
3: 2
4: 3
5: 2, 3
6: 4
7: 2, 3, 5
8: 3, 5
9: 2, 6
10: 3, 4, 7
11: 2, 3, 7
12: 5, 8
13: 2, 3, 5, 9
14: 3, 4, 9
15: 2, 6, 10
16: 3, 11
17: 2, 3, 5, 7, 11
18: 4, 5, 7, 12
19: 2, 3, 13
20: 3, 8, 13

T. D. Noe, Rows n=3..1000 of triangle, flattened
Diana Mecum, Rows 3 through 500

f[n_] := Complement[ Sort@ Join[ Select[ Union@ Flatten@ Divisors[{2 n - 1, 2 n + 1}], OddQ@ # && # < n &], Select[ Divisors[2 n], EvenQ@ # && # < n &]], Divisors@ n]; Flatten@ Table[ f@n, {n, 3, 32}] (* Robert G. Wilson v, Jul 17 2007 *)
Table[Select[Range[2, n - 1], Abs[Mod[n, #] - #/2] < 1 &], {n, 3, 31}] // Flatten (* Michael De Vlieger, Jun 14 2016, after Harvey P. Dale at A066272 *)

A073930 Numbers that are equal to the sum of their anti-divisors.

Original entry on oeis.org

5, 8, 41, 56, 946, 5186, 6874, 8104, 17386, 27024, 84026, 167786, 2667584, 4921776, 27914146, 505235234, 3238952914, 73600829714, 455879783074, 528080296234, 673223621664, 4054397778846, 4437083907194, 4869434608274, 6904301600914, 7738291969456

Offset: 1

Jason Earls, Sep 03 2002

nonn

See A066272 for definition of anti-divisor.

			For k=5186, the anti-divisor sum: 3+4+11+23+41+253+451+943+3457 = 5186.

Jon Perry, Anti-perfects, anti-amicables and other records.

Cf. A066417, A192272.

sad(n) = vecsum(select(t->n%t && tA066417
isok(n) = sad(n) == n; \\ Michel Marcus, Oct 12 2019

from sympy import divisors
A073930 = [n for n in range(1,10**5) if sum([2*d for d in divisors(n) if n > 2*d and n % (2*d)] + [d for d in divisors(2*n-1) if n > d >=2 and n % d] + [d for d in divisors(2*n+1) if n > d >=2 and n % d]) == n] # Chai Wah Wu, Aug 14 2014

a(16)-a(17) from Lior Manor, Mar 03 2004
a(18) from Donovan Johnson, Jun 19 2010
a(19)-a(21) by Jud McCranie, Aug 31 2019
a(22)-a(26) by Jud McCranie, Oct 10 2019

cp's OEIS Frontend

A209271 Primes p such that A066272(p)*p+1 is also prime, where A066272 is the number of anti-divisors.

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A073638 Number of anti-divisors of n (A066272) sets a record.

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Extensions

A066980 Numbers k such that antid(k) > antid(k+1) and santid(k) > santid(k+1), where antid(k) = A066272 and santid(k) = A066417.

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A073547 Numbers k such that antid(k) = antid(k+1), where antid(k) = A066272(k).

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Maple

Mathematica

A093071 Lesser of a pair of records in A066272.

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A073569 Numbers k such that A002034(k) = A066272(k).

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Extensions

A032766 Numbers that are congruent to 0 or 1 (mod 3).

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Haskell

Magma

Maple

Mathematica

PARI

PARI

SageMath

Formula

Extensions

A066417 Sum of anti-divisors of n.

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Maple

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PARI

PARI

Python

Formula

A130799 Triangle read by rows in which row n (n>=3) list the anti-divisors of n.