Displaying 1-10 of 73 results found.
Odd abundant numbers ( A005231) which are not in A136446, i.e., not sum of some of their proper divisors > 1.
+20
6
COMMENTS
It is conjectured that there are no odd weird numbers ( A006037), i.e., that all odd abundant numbers ( A005231) are pseudoperfect ( A005835); this sequence lists those which are not equal to the sum of a subset of proper divisors > 1. No second term in the range <= 53850001. - R. J. Mathar, Mar 21 2011 No other terms congruent to 21 (mod 30) below 10^9. - M. F. Hasler, Jul 16 2016
EXAMPLE
a(1) = 351351 = 3^3 * 7 * 11 * 13^2 is the sum of all its 47 proper divisors (including 1) except 7 and 11, but it is not possible to get the same sum without using the trivial divisor 1: The sum of all proper divisors *larger than 1* yields 351351 + 7 + 11 - 1 = 351351 + 17, and it is not possible to get 17 as sum of a subset of {3, 7, 9, 11, 13, 21, ...}. Thus, 351351 is not in A136446, and therefore in this sequence. - M. F. Hasler, Jul 16 2016, edited Mar 15 2021
PROG
(PARI) is_ A122036(n)={n>351350 && !is_ A005835(n, n=divisors(n)[2..-2]) && n && vecsum(n)>=n[1]*n[#n] && n[1]>2} \\ (Checking for abundant & odd after is_ A005835() rather than before, to make it faster when operating on candidates known to satisfy these conditions.) Updated for current PARI syntax by M. F. Hasler, Jul 16 2016, further edits Jan 31 2020 forstep(n=1, 10^7, 2, is_ A122036(n) && print1(n", "))
Largest odd abundant number ( A005231) equal to the product of n consecutive primes.
+20
5
15015, 255255, 4849845, 111546435, 33426748355, 1236789689135, 50708377254535, 2180460221945005, 102481630431415235, 5431526412865007455, 320460058359035439845, 19548063559901161830545, 1309720258513377842646515, 1357656019974967471687377449, 107254825578022430263302818471
COMMENTS
The smallest term is a(5) = 3*5*7*11*13, there is no odd abundant number ( A005231) equal to the product of less than 5 consecutive primes. The smallest odd abundant number ( A005231) equal to the product of n consecutive primes is equal (when it exists, i.e., for n >= 5) to the least odd number with n (distinct) prime divisors, equal to the product of the first n odd primes = A070826(n+1) = A002110(n+1)/2. See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number ( A006038) with n distinct prime factors, and A275449 for the least odd primitive abundant number with n prime factors counted with multiplicity. The terms are in general not primitive abundant numbers ( A091191), in particular this cannot be the case when a(n) is a multiple of a(n-1), as is the case for most of the terms, for which a(n) = a(n-1)* A117366(a(n-1)). In the other event, spf(a(n)) = nextprime(spf(a(n-1))), and a(n) is in A007741(2,3,4...). These are exactly the primitive terms in this sequence.
FORMULA
a(n) >= a(n-1)*p where p = A117366(a(n-1)) = A151800( A006530(a(n-1))) = nextprime(gpf(a(n-1))), an odd abundant number equal to the product of n consecutive primes. We have strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = a(n-1)*p*p'/q, where p' = nextprime(p), q = least prime factor of a(n-1). This is the case if a(n) is in A007741.
EXAMPLE
For n < 5, there is no odd abundant number equal to the product of n distinct primes.
For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*prime(n+1).
For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*prime(n+2).
For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*prime(n+3).
For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*prime(n+4).
For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*prime(n+5).
PROG
(PARI) a(r, f=vector(r, i, prime(i+1)), o)={ while(sigma(factorback(f), -1)>2, o=f; f=concat(f[^1], nextprime(f[r]+1))); factorback(o)} \\ Intentionally throws an error when n < 5.
CROSSREFS
A subsequence of A112643 (odd squarefree abundant numbers); see also A108227 (~ A107705) which give indices of primitive terms = those with smallest prime factor larger than that of earlier terms.
Largest prime factor of A285993(n), the largest odd abundant number ( A005231) equal to the product of n consecutive primes.
+20
1
13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353
COMMENTS
The smallest term is a(5), there is no odd abundant number ( A005231) equal to the product of less than 5 consecutive primes. The corresponding abundant numbers are A285993(n) = prime(k-n+1)*...*prime(k), with prime(k) = a(n).
FORMULA
a(n) = A006530( A285993(n)) >= A151800(a(n-1)) = nextprime(a(n-1)), with strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = nextprime(nextprime(a(n-1))). This is the case if A285993(n) is in A007741.
EXAMPLE
For n < 5, there is no odd abundant number equal to the product of n distinct primes.
For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*a(n) with a(n) = prime(n+1).
For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*a(n) with a(n) = prime(n+2).
For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*a(n) with a(n) = prime(n+3).
For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*a(n) with a(n) = prime(n+4).
For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*a(n) with a(n) = prime(n+5).
PROG
(PARI) a(r, f=vector(r, i, prime(i+1)), o)={ while(sigma(factorback(f), -1)>2, o=f; f=concat(f[^1], nextprime(f[r]+1))); o[#o]} \\ Intentionally throws an error when n < 5.
Abundant numbers (sum of divisors of m exceeds 2m).
(Formerly M4825)
+10
346
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264, 270
COMMENTS
A number m is abundant if sigma(m) > 2m (this sequence), perfect if sigma(m) = 2m (cf. A000396), or deficient if sigma(m) < 2m (cf. A005100), where sigma(m) is the sum of the divisors of m ( A000203). While the first even abundant number is 12 = 2^2*3, the first odd abundant is 945 = 3^3*5*7, the 232nd abundant number!
If m is a term so is every positive multiple of m. "Primitive" terms are in A091191. If m=6k (k>=2), then sigma(m) >= 1 + k + 2*k + 3*k + 6*k > 12*k = 2*m. Thus all such m are in the sequence.
According to Deléglise (1998), the abundant numbers have natural density 0.2474 < A(2) < 0.2480. Thus the n-th abundant number is asymptotic to 4.0322*n < n/A(2) < 4.0421*n. - Daniel Forgues, Oct 11 2015 From Bob Selcoe, Mar 28 2017 (prompted by correspondence with Peter Seymour): (Start) Applying similar logic as the proof that all multiples of 6 >= 12 appear in the sequence, for all odd primes p:
i) all numbers of the form j*p*2^k (j >= 1) appear in the sequence when p < 2^(k+1) - 1;
ii) no numbers appear when p > 2^(k+1) - 1 (i.e., are deficient and are in A005100); iii) when p = 2^(k+1) - 1 (i.e., perfect numbers, A000396), j*p*2^k (j >= 2) appear. Note that redundancies are eliminated when evaluating p only in the interval [2^k, 2^(k+1)].
The first few even terms not of the forms i or iii are {70, 350, 490, 550, 572, 650, 770, ...}. (End)
REFERENCES
L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Pure Appl. Math., Vol. 44 (1913), pp. 264-296.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B2, pp. 74-84.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 59.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Eric Weisstein's World of Mathematics, Abundance.
FORMULA
a(n) is asymptotic to C*n with C=4.038... (Deléglise, 1998). - Benoit Cloitre, Sep 04 2002
MAPLE
with(numtheory): for n from 1 to 270 do if sigma(n)>2*n then printf(`%d, `, n) fi: od:
isA005101 := proc(n)
simplify(numtheory[sigma](n) > 2*n) ;
option remember ;
local a ;
if n =1 then
12 ;
else
a := procname(n-1)+1 ;
while numtheory[sigma](a) <= 2*a do
a := a+1 ;
end do ;
a ;
end if ;
PROG
(Haskell)
a005101 n = a005101_list !! (n-1)
a005101_list = filter (\x -> a001065 x > x) [1..]
(Python)
from sympy import divisors
def ok(n): return sum(divisors(n)) > 2*n
(Python)
from sympy import divisor_sigma
from itertools import count, islice
def A005101_gen(startvalue=1): return filter(lambda n:divisor_sigma(n) > 2*n, count(max(startvalue, 1))) # generator of terms >= startvalue
CROSSREFS
Cf. A094268 (n consecutive abundant numbers). Cf. A173490 (even abundant numbers).
Pseudoperfect (or semiperfect) numbers n: some subset of the proper divisors of n sums to n.
(Formerly M4094)
+10
71
6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, 126, 132, 138, 140, 144, 150, 156, 160, 162, 168, 174, 176, 180, 186, 192, 196, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 234, 240, 246, 252, 258, 260, 264
COMMENTS
In other words, some subset of the numbers { 1 <= d < n : d divides n } adds up to n. - N. J. A. Sloane, Apr 06 2008 Deficient numbers cannot be pseudoperfect. This sequence includes the perfect numbers ( A000396). By definition, it does not include the weird, i.e., abundant but not pseudoperfect, numbers ( A006037). The first odd pseudoperfect number is a(233) = 945.
An empirical observation (from the graph) is that it seems that the n-th pseudoperfect number would be asymptotic to 4n, or equivalently that the asymptotic density of pseudoperfect numbers would be 1/4. Any proof of this? (End)
Deléglise (1998) shows that abundant numbers have asymptotic density < 0.2480, resolving the question which he attributes to Henri Cohen of whether the abundant numbers have density greater or less than 1/4. The density of pseudoperfect numbers is the difference between the densities of abundant numbers ( A005101) and weird numbers ( A006037), since the remaining integers are perfect numbers ( A000396), which have density 0. Using the first 22 primitive pseudoperfect numbers ( A006036) and the fact that every multiple of a pseudoperfect number is pseudoperfect it can be shown that the density of pseudoperfect numbers is > 0.23790. - Jaycob Coleman, Oct 26 2013 The odd terms of this sequence are given by the odd abundant numbers A005231, up to hypothetical (so far unknown) odd weird numbers ( A006037). - M. F. Hasler, Nov 23 2017 The term "pseudoperfect numbers" was coined by Sierpiński (1965). The alternative term "semiperfect numbers" was coined by Zachariou and Zachariou (1972). - Amiram Eldar, Dec 04 2020
REFERENCES
Richard K. Guy, Unsolved Problems in Number Theory, 3rd edition, Springer, 2004, Section B2, pp. 74-75.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Stan Benkoski, Problem E2308, Amer. Math. Monthly, Vol. 79, No. 7 (1972), p. 774.
EXAMPLE
6 = 1+2+3, 12 = 1+2+3+6, 18 = 3+6+9, etc.
70 is not a member since the proper divisors of 70 are {1, 2, 5, 7, 10, 14, 35} and no subset adds to 70.
MAPLE
with(combinat):
isA005835 := proc(n)
local b, S;
b:=false;
S:=subsets(numtheory[divisors](n) minus {n});
while not S[finished] do
if convert(S[nextvalue](), `+`)=n then
b:=true;
break
end if ;
end do;
b
end proc:
for n from 1 do
if isA005835(n) then
print(n);
end if;
MATHEMATICA
pseudoPerfectQ[n_] := Module[{divs = Most[Divisors[n]]}, MemberQ[Total/@Subsets[ divs, Length[ divs]], n]]; A005835 = Select[Range[300], pseudoPerfectQ] (* Harvey P. Dale, Sep 19 2011 *) A005835 = {}; n = 0; While[Length[ A005835] < 100, n++; d = Most[Divisors[n]]; c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; If[c > 0, AppendTo[ A005835, n]]]; A005835 (* T. D. Noe, Dec 29 2011 *)
PROG
(PARI) is_ A005835(n, d=divisors(n)[^-1], s=vecsum(d), m=#d)={ m||return; while(d[m]>n, s-=d[m]; m--||return); d[m]==n || if(n<s, is_ A005835(n-d[m], d, s-d[m], m-1) || is_ A005835(n, d, s-d[m], m-1), n==s)} \\ Returns nonzero iff n is the sum of a subset of d, which defaults to the set of proper divisors of n. Improved using more recent PARI syntax by M. F. Hasler, Jul 15 2016, Jul 27 2016. NOTE: This function is also used (with 2nd optional arg) in A136446, A122036 and possibly in A006037. - M. F. Hasler, Jul 28 2016 (Haskell)
a005835 n = a005835_list !! (n-1)
a005835_list = filter ((== 1) . a210455) [1..]
EXTENSIONS
Better description and more terms from Jud McCranie, Oct 15 1997
Odd primitive abundant numbers.
(Formerly M5486)
+10
33
945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 28215
COMMENTS
Dickson proves that there are only a finite number of odd primitive abundant numbers having n distinct prime factors. Sequence A188342 lists the smallest such numbers. - T. D. Noe, Mar 28 2011 Sequence A188439 sorts the numbers in this sequence by the number of distinct prime factors. Eight numbers have exactly three prime factors; 576 have exactly four prime factors. - T. D. Noe, Apr 04 2011 Any multiple of an abundant number ( A005101) is again an abundant number. Primitive abundant numbers ( A091191) are those not of this form, i.e., without an abundant proper divisor. We don't know any odd perfect number ( A000396), so the (odd) terms here have only deficient proper divisors ( A071395), and their prime factors p are less than sigma(n/p)/deficiency(n/p). See A005231 (odd abundant numbers) for an explanation why all terms have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity ( A001222), whence a(1) = 3^3*5*7. All known terms are semiperfect ( A005835, and thus in A006036): no odd weird number ( A006037) is known, but if it exists, the smallest one is in this sequence. - M. F. Hasler, Jul 28 2016 So far, a(173) = 351351 is the only known term of A122036, i.e., which can't be written as sum of its proper divisors > 1. - M. F. Hasler, Jan 26 2020
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
MAPLE
isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
isA005100 := proc(n) is(numtheory[sigma](n) < 2*n ); end proc:
isA006038 := proc(n) local d; if type(n, 'odd') and isA005101(n) then for d in numtheory[divisors](n) minus {1, n} do if not isA005100(d) then return false; end if; end do: return true; else false; end if; end proc:
n := 1 ; for a from 1 by 2 do if isA006038(a) then printf("%d %d\n", n, a) ; n := n+1 ; end if; end do: # R. J. Mathar, Mar 28 2011
MATHEMATICA
t = {}; n = 1; While[Length[t] < 50, n = n + 2; If[DivisorSigma[1, n] > 2 n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)
PROG
(PARI) is_ A006038(n)=bittest(n, 0) && sigma(n)>2*n && !for(i=1, #f=factor(n)[, 1], sigma(n\f[i], -1)>2&&return) \\ More than 5 times faster. - M. F. Hasler, Jul 28 2016 (Haskell)
a006038 n = a006038_list !! (n-1)
a006038_list = filter f [1, 3 ..] where
f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
where pdivs = a027751_row x
Smallest abundant number (sigma(x) > 2x) which is not divisible by any of the first n primes.
+10
22
12, 945, 5391411025, 20169691981106018776756331, 49061132957714428902152118459264865645885092682687973, 7970466327524571538225709545434506255970026969710012787303278390616918473506860039424701
COMMENTS
a(n) exists for every n, since the sum of the inverses of the primes is infinite.
Heuristic: Add the squares of several successive primes and then add successive primes until the number is abundant.
a(2) = 5^2 * 7 * 11 * 13 * 17 * 19 * 23 * 29;
a(3) = 7^2 * 11^2 * 13 * 17 * ... * 61 * 67;
a(4) = 11^2 * 13^2 * 17 * 19 * ... * 131 * 137;
a(5) = 13^2 * 17^2 * 19 * 23 * ... * 223 * 227. (End)
a(6) = 17^2 * 19^2 * 23^2 * 29 * 31 * ... * 347 * 349;
a(7) = 19^2 * 23^2 * 29^2 * 31 * 37 * ... * 491 * 499 (both coming from the D. Iannucci paper). - Michel Marcus, May 01 2013 The known terms of this sequence provide Egyptian decompositions of unity in which all the denominators lack the first n primes, as follows: Every term listed in this sequence is a semiperfect number, which means that a subset of its divisors add up to the number itself. The decomposition 1 = 1/a + 1/b + ... + 1/m, where the denominators are a(n) divided by those divisors, is the desired decomposition. - Javier Múgica, Nov 15 2017 a(n) is the product of consecutive primes starting from prime(n+1) raised to nonincreasing powers. - Jianing Song, Apr 10 2021 By definition, Omega(a(n)) >= A108227(n+1) for all n, where Omega = A001222. For 0 <= n <= 12 we have Omega(a(n)) = A108227(n+1), but this is not true for n = 13, where Omega(a(13)) = 335 > A108227(14) = 334. We also have omega(a(n)) >= A001276(n+1) for all n, where omega = A001221. The differences for known terms are 0, 0, 1, 1, 2, 3, 2, 3, 4, 4, 5, 6, 6, 6 respectively. Conjecture: other than a(1) = 945, all terms are cubefree. (End)
REFERENCES
M. T. Whalen and C. L. Miller, Odd abundant numbers: some interesting observations, Journal of Recreational Mathematics 22 (1990), pp. 257-261.
EXAMPLE
a(0) = 12, the first abundant number; a(1) = 945, the first odd abundant number; a(5) is the first abundant number not divisible by 2,3,5,7 or 11.
AUTHOR
Ulrich Schimke (ulrschimke(AT)aol.com)
Odd infinitary abundant numbers.
+10
17
945, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, 23205, 23625, 25245, 25935, 26565, 27405, 28215, 28875, 29295, 29835, 31395, 33345, 33495, 33915, 34125, 34155, 34965, 35805, 37125, 38745, 39585, 40635, 41055, 42315
COMMENTS
This is also the sequence of odd integers whose infinitary aliquot sequences initially increase. Based on empirical evidence (up to 10 million), this applies to only about 0.1% of odd integers.
The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 1, 77, 473, 5703, 53569, 561610, 5525461, 54979537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0005... . - Amiram Eldar, Sep 09 2022
FORMULA
Odd values of n for which A126168(n)>n.
EXAMPLE
a(5)=16065 because 16065 is the fifth odd number that is exceeded by the sum of its proper infinitary divisors.
MATHEMATICA
ExponentList[n_Integer, factors_List]:={#, IntegerExponent[n, # ]}&/@factors; InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g]==g][ #, Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #, factors]&/@d]], _?(And@@#&), {1}]] ]] ] Null; properinfinitarydivisorsum[k_]:=Plus@@InfinitaryDivisors[k]-k; Select[Range[1, 50000, 2], properinfinitarydivisorsum[ # ]># &] (* end of program *)
fun[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; isigma[1] = 1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[1, 50000, 2], isigma[#] > 2 # &] (* Amiram Eldar, Jun 09 2019 *)
PROG
(PARI) A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))}
120, 180, 240, 300, 360, 420, 480, 504, 540, 600, 630, 660, 720, 780, 840, 900, 924, 960, 990, 1008, 1020, 1050, 1080, 1092, 1140, 1170, 1200, 1260, 1320, 1380, 1440, 1470, 1500, 1512, 1560, 1620, 1650, 1680, 1740, 1800, 1848, 1860, 1890, 1920, 1980, 2016, 2040, 2100, 2160, 2184, 2220, 2280, 2310, 2340, 2400, 2460
COMMENTS
Provided that there are no odd perfect numbers, then these are equal to numbers k for which A003961(k) is in A005231, i.e., numbers that become odd abundant numbers when prime-shifted once. Not all terms are even. The first odd term is a(8313165) = 334639305 = A064989( A115414(1)). (See A337385). For any odd term x present, A064989(x) is also present, for example, A064989(334639305) = 19399380 = a(482324).
MATHEMATICA
Select[Range[2500], If[# == 1, 1, DivisorSigma[1, # ]] >= 2# &@ Apply[Times, FactorInteger[#] /. {p_, e_} /; e > 0 :> Prime[PrimePi@ p + 1]^e] &] (* Michael De Vlieger, Aug 27 2020 *)
PROG
(PARI)
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
Odd bi-unitary abundant numbers: odd numbers k such that bsigma(k) > 2*k, where bsigma is the sum of the bi-unitary divisors function ( A188999).
+10
13
945, 8505, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, 23205, 23625, 25245, 25515, 25935, 26565, 27405, 28215, 28875, 29295, 29835, 31185, 31395, 33345, 33495, 33915, 34125, 34155, 34965, 35805, 36855, 37125, 38745, 39585, 40635, 41055
COMMENTS
The numbers of terms not exceeding 10^k, for k = 3, 4, ..., are 1, 2, 82, 559, 6493, 61831, 642468, 6339347, 63112602, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00063... . - Amiram Eldar, Sep 02 2022
EXAMPLE
945 is in the sequence since bsigma(945) = 1920 > 2*945.
MATHEMATICA
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; Select[Range[1000], bOddAbundantQ] (* after Michael De Vlieger at A188999 *)
PROG
(PARI) 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); \\ Michel Marcus, Dec 15 2017
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