A006038 -id:A006038 - cp's OEIS frontend

A180332 Primitive Zumkeller numbers.

6, 20, 28, 70, 88, 104, 272, 304, 368, 464, 496, 550, 572, 650, 836, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4030, 4070, 4095, 4216

Offset: 1

T. D. Noe, Sep 07 2010

nonn

A number is called a primitive Zumkeller number if it is a Zumkeller number (A083207) but none of its proper divisors are Zumkeller numbers. These numbers are very similar to primitive non-deficient numbers (A006039), but neither is a subsequence of the other. [See A378538, A378656, A378657].
Because every Zumkeller number has a divisor that is a primitive Zumkeller number, every Zumkeller number z can be factored as z = d*r, where d is the smallest divisor of z that is a primitive Zumkeller number.
Every number of the form p*2^k is a primitive Zumkeller number, where p is an odd prime and k = floor(log_2(p)).
The odd terms are not the same as A006038. For example, 342225 occurs there, but not here, while 4448925 occurs here, but is not in A006038. - Antti Karttunen, Dec 05 2024

Cf. A000396 (subsequence), A006038, A083207, A006039, A378537 (characteristic function), A378538, A378656, A378657.

ZumkellerQ[n_] := ZumkellerQ[n] = Module[{d = Divisors[n], ds, x}, ds = Total[d]; If[OddQ[ds], False, SeriesCoefficient[Product[1 + x^i, {i, d}], {x, 0, ds/2}] > 0]];
Reap[For[n = 1, n <= 5000, n++, If[ZumkellerQ[n] && NoneTrue[Most[Divisors[ n]], ZumkellerQ], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Mar 01 2019 *)

from sympy import divisors
from sympy.utilities.iterables import subsets
def isz(n): # after Peter Luschny in A083207
    divs = divisors(n)
    s = sum(divs)
    if not (s%2 == 0 and 2*n <= s): return False
    S = s//2 - n
    R = [m for m in divs if m <= S]
    return any(sum(c) == S for c in subsets(R))
def ok(n): return isz(n) and not any(isz(d) for d in divisors(n)[:-1])
print(list(filter(ok, range(1, 5000)))) # Michael S. Branicky, Jun 20 2021

# uses[is_Zumkeller from A083207]
def is_primitiveZumkeller(n):
    return (is_Zumkeller(n) and
        not any(is_Zumkeller(d) for d in divisors(n)[:-1]))
print([n for n in (1..4216) if is_primitiveZumkeller(n)]) # Peter Luschny, Jun 21 2021

A007741 a(n) = prime(n)...prime(m), the least product of consecutive primes which is abundant.

Original entry on oeis.org

30, 15015, 33426748355, 1357656019974967471687377449, 7105630242567996762185122555313528897845637444413640621

Offset: 1

Walter Nissen

nonn

Essentially, i.e., except for a(1), identical to A007702. All terms are primitive abundant numbers (A091191) and thus, except for the first term, odd primitive abundant (A006038). The next term is too large to be displayed here, see A007707 (and formula) for many more terms, using a more compact encoding. - M. F. Hasler, Apr 30 2017

Amiram Eldar, Table of n, a(n) for n = 1..14
G. Thimm, Emails to N. J. A. Sloane, Sep. 1994

Cf. A005101, A006038, A007702, A007707, A007708, A091191.

Cf. A108227, A285993.

a[n_] := Module[{p = Prime[n]}, r = 1; prod = 1; While[r <= 2, r *= 1 + 1/p; prod *= p; p = NextPrime[p]]; prod]; Array[a, 5] (* Amiram Eldar, Jun 29 2019 *)

a(n) = {p = prime(n); sig = p+1; prd = p; while (sig <= 2*prd, p = nextprime(p+1); sig *= p+1; prd *= p;); return (prd);} \\ Michel Marcus, Mar 10 2013

a(n) = Product_{k=n..A007707(n)} prime(k) = Product_{0 <= i < A108227(n)} prime(n+i). - M. F. Hasler, Apr 30 2017 and Jun 15 2017

More terms from Don Reble, Nov 10 2005

A249263 Primitive, odd, squarefree abundant numbers.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655, 118965

Offset: 1

Derek Orr, Oct 23 2014

nonn

The subsequence of primitive terms (not multiples of smaller terms) of A112643.
The subsequence of squarefree terms of A006038.
The subsequence of odd terms of A249242.
Not the same as A129485. Does not contain, for example, 195195, 255255, 285285, 333795, 345345, 373065, which are in A129485. - R. J. Mathar, Nov 09 2014
Sequences A287590, A188342 and A287581 list the number, smallest* and largest of all squarefree odd primitive abundant numbers with n prime factors. (*At least whenever A188342(n) is squarefree, which appears to be the case for all n >= 5.) - M. F. Hasler, May 29 2017

Giovanni Resta, Table of n, a(n) for n = 1..10000

Intersection of A112643 and A006038.
Cf. A249242, A129485.
Cf. A188342 (least with n factors), A287581 (largest with n factors), A287590 (number of terms with n factors).

# see A112643 and A006038 for the coding of isA112643 and isA006038
isA249263 := proc(n)
    isA112643(n) and isA006038(n) ;
end proc:
for n from 1 do
    if isA249263(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

PrimAbunQ[n_] := Module[{x, y},
   y = Most[Divisors[n]]; x = DivisorSigma[1, y];
   DivisorSigma[1, n] > 2 n  &&  AllTrue[x/y, # <= 2  &]];
Select[Range[1, 120000, 2], PrimAbunQ[#] &&
AllTrue[FactorInteger[#][[All, 2]], # == 1 &]  &] (* Robert Price, Sep 26 2019 *)

v=[]; for(k=1, 10^5, n=2*k+1; if(issquarefree(n) && sigma(n)>2*n, for(i=1, #v, n%v[i] || next(2)); print1(n, ", "); v=concat(v, n))) \\ Improved (from 20 sec to 0.2 sec) by M. F. Hasler, May 27 2017

A006036 Primitive pseudoperfect numbers.

Original entry on oeis.org

6, 20, 28, 88, 104, 272, 304, 350, 368, 464, 490, 496, 550, 572, 650, 748, 770, 910, 945, 1184, 1190, 1312, 1330, 1376, 1430, 1504, 1575, 1610, 1696, 1870, 1888, 1952, 2002, 2030, 2090, 2170, 2205, 2210, 2470, 2530, 2584, 2590, 2870, 2990, 3010, 3128, 3190, 3230, 3290, 3410, 3465, 3496, 3710, 3770, 3944, 4070, 4095, 4130, 4216, 4270, 4288, 4408, 4510, 4544, 4672, 4690, 4712, 4730, 4970

Offset: 1

N. J. A. Sloane, R. K. Guy

A primitive pseudoperfect number is a pseudoperfect number that is not a multiple of any other pseudoperfect number.
The odd entries so far are identical to the odd primitive abundant A006038. - Walter Kehowski, Aug 12 2005
Zachariou and Zachariou (1972) called these numbers "irreducible semiperfect numbers". - Amiram Eldar, Dec 04 2020

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).

Donovan Johnson, Table of n, a(n) for n = 1..10000
Richard K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Eric Weisstein's World of Mathematics, Primitive Pseudoperfect Number.
Andreas Zachariou and Eleni Zachariou, Perfect, Semi-Perfect and Ore Numbers, Bull. Soc. Math. Grèce (New Ser.), Vol. 13, No. 13A (1972), pp. 12-22; alternative link.

Cf. A005835.

Cf. A210455, A027751.

a006036 n = a006036_list !! (n-1)
a006036_list = filter (all (== 0) . map a210455 . a027751_row) a005835_list
-- Reinhard Zumkeller, Jan 21 2013

with(numtheory): with(combinat): issemiperfect := proc(n) local b, S;
b:=false; S:=subsets(divisors(n) minus {n}); while not S[finished] do if
convert(S[nextvalue](),`+`)=n then b:=true; break fi od; return b end:
L:=remove(proc(z) isprime(z) end,[$1..5000]): PP:=[]: for zz from 1 to 1 do
for n in L do if issemiperfect(n) then PP:=[op(PP),n] fi od od;
sr := proc(l::list) local x, R, S, P, L; S:=sort(l); R:=[]; P:=S;
for x in S do
if not(x in R) then
L:=selectremove(proc(z) z>x and z mod x = 0 end, P);
R:=[op(R),op(L[1])]; P:=L[2];
fi; od; return P; end:
PPP:=sr(PP); # primitive pseudoperfect numbers less than 5000 # Walter Kehowski, Aug 12 2005

(* First run one of the programs for A005835 *) A006036 = A005835; curr = 1; max = A005835[[-1]]; While[curr < Length[A006036], currMult = A006036[[curr]]; A006036 = Complement[A006036, Range[2currMult, Ceiling[max/currMult] currMult, currMult]]; curr++]; A006036 (* Alonso del Arte, Sep 08 2012 *)

More terms from Walter Kehowski, Aug 12 2005

A108227 a(n) is the least number of prime factors for any abundant number with p_n (the n-th prime) as its least factor.

Original entry on oeis.org

3, 5, 9, 18, 31, 46, 67, 91, 122, 158, 194, 238, 284, 334, 392, 456, 522, 591, 668, 749, 835, 929, 1028, 1133, 1242, 1352, 1469, 1594, 1727, 1869, 2019, 2163, 2315, 2471, 2636, 2802, 2977, 3157, 3342, 3534, 3731, 3933, 4145, 4358, 4581, 4811

Offset: 1

Hugo van der Sanden, Jun 17 2005

nonn

If we replace "abundant" in the definition with "non-deficient", we get the same sequence with an initial 2 instead of 3, barring an astronomically unlikely coincidence with some as-yet-undiscovered odd perfect number. [This is sequence A107705. - M. F. Hasler, Jun 14 2017]
It appears that all terms >= 5 correspond to the odd primitive abundant numbers (A006038) which are products of consecutive primes (cf. A285993), i.e., of the form N = Product_{0<=iM. F. Hasler, May 08 2017
From Jianing Song, Apr 21 2021: (Start)
Let x_1 < x_2 < ... < x_k < ... be the numbers of the form p of p^2 + p, where p is a prime >= prime(n). Then a(n) is the smallest N such that Product_{i=1..N} (1 + 1/x_i) > 2. See my link below for a proof.
For example, for n = 3, we have {x_1, x_2, ..., x_k, ...} = {5, 7, 11, 13, 17, 19, 23, 29, 5^2 + 5, ...}, we have Product_{i=1..8} (1 + 1/x_i) < 2 and Product_{i=1..9} (1 + 1/x_i) > 2, so a(3) = 9. (End)

			a(2) = 5 since 945 = 3^3*5*7 is an abundant number with p_2 = 3 as its smallest prime factor, and no such number exists with fewer than 5 prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..500
Jianing Song, Notes for A108227

Cf. A000040, A005101, A006038, A001222.
Cf. A107705.
Cf. A007707 (~ A007684), A007708, A007741.
Cf. A001276 (least number of prime factors for a (p_n)-rough abundant number, counted without multiplicity).

A108227(n, s=1+1/prime(n))=for(a=1, 9e9, if(2M. F. Hasler, Jun 15 2017

isform(k,q) = my(p=prime(k)); if(isprime(q) && (q>=p), 1, if(issquare(4*q+1), my(r=(sqrtint(4*q+1)-1)/2); isprime(r) && (r>=p), 0))
a(n) = my(Prod=1, Sum=0); for(i=prime(n), oo, if(isform(n,i), Prod *= (1+1/i); Sum++); if(Prod>2, return(Sum))) \\ Jianing Song, Apr 21 2021

a(n) = A007684(n)-n+1, for n>1. A007741(n) = Product_{0<=iM. F. Hasler, Jun 15 2017

Data corrected by Amiram Eldar, Aug 08 2019

A188263 Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

Original entry on oeis.org

945, 2205, 7425, 8415, 8925, 31815, 32445, 351351, 442365, 14571585, 20355825, 20487159, 78524145, 159030135, 1756753845, 2586415095, 82014476355, 93128205975, 125208115065, 127595519865, 154063853475, 394247024535, 948907364895

Offset: 1

T. D. Noe, Mar 30 2011

nonn

The abundancy of a number n is defined as sigma(n)/n. Abundant numbers have an abundancy greater than 2. All these numbers must be odd primitive abundant numbers, A006038.
These numbers might be considered the opposite of A119239, which has odd numbers whose abundancy increases. This sequence has terms in common with A171929. A similar sequence for deficient numbers is A188597.
These are odd numbers that are barely abundant. See A071927 for the even version.
a(24) > 10^12. - Donovan Johnson, May 05 2012

Giovanni Resta, Table of n, a(n) for n = 1..31 (terms < 10^13)

Cf. A171929 (odd numbers whose abundancy is closer to 2 than any smaller odd number)

k = 1; minDiff = 1; Table[k = k + 2; While[abun = DivisorSigma[1, k]/k; abun - 2 > minDiff || abun < 2, k = k + 2]; minDiff = abun - 2; k, {10}]

a(15)-a(16) from Donovan Johnson, Mar 31 2011
a(17)-a(22) from Donovan Johnson, Apr 02 2011
a(23) from Donovan Johnson, May 05 2012

A064001 Odd abundant numbers not divisible by 5.

Original entry on oeis.org

81081, 153153, 171171, 189189, 207207, 223839, 243243, 261261, 279279, 297297, 351351, 459459, 513513, 567567, 621621, 671517, 729729, 742203, 783783, 793611, 812889, 837837, 891891, 908523, 960687, 999999, 1024947, 1054053, 1072071

Offset: 1

Harvey P. Dale, Sep 17 2001

nonn

Or, odd abundant numbers that do not end in 5.
All terms below 2000000 are divisible by 21 (so by 3). Moreover, except for a few, most are divisible by 231. - Labos Elemer, Sep 15 2005 [The least term that is not divisible by 21 is a(908) = 28683369. - Amiram Eldar, Jan 27 2025]
An odd abundant number (see A005231) not divisible by 3 nor 5 must have at least 15 distinct prime factors (e.g., 61#/5#*7^2*11*13*17, where # is primorial) and be >= 67#/5#*77 = A047802(3) ~ 2.0*10^25. -- The smallest non-primitive abundant number (cf. A006038) in this sequence is 7*a(1) = 567567 = a(14). - M. F. Hasler, Jul 27 2016
There are 26 terms less than 10^6 and a surprising fact is that 18 of them are doublets (cf. A020338). - Omar E. Pol, Jan 17 2025
The numbers of terms that do not exceed 10^k, for k = 5, 6, ..., are 1, 26, 290, 3071, 31600, 320948, 3174762, 31693948, ... . Apparently, the asymptotic density of this sequence equals 0.000031... . Therefore, the least term not divisible by 3 that was mentioned above is a(~6*10^20) = 20169691981106018776756331. - Amiram Eldar, Jan 27 2025

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Rev. ed. 1997, p. 169.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.

Intersection of A005231 and A047201.
Cf. A006038, A047802, A110585.
Cf. A020338.

Select[ Range[ 1, 10^6, 2 ], DivisorSigma[ 1, # ] - 2# > 0 && Mod[ #, 5 ] != 0 & ]
ta={{0}};Do[g=n;s=DivisorSigma[1, n]-2*n; If[Greater[s, 0]&&!Equal[Mod[n, 2], 0]&& !Equal[Mod[n, 5], 0], Print[n];ta=Append[ta, n]], {n, 1, 2000000}] ta=Delete[ta, 1] (* Labos Elemer, Sep 15 2005 *)

{ n=0; forstep (m=1, 10^9, 2, if (m%5 && sigma(m) > 2*m, write("b064001.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 05 2009

More terms from Robert G. Wilson v, Sep 28 2001
Further terms from Labos Elemer, Sep 15 2005
Entry revised by N. J. A. Sloane, Mar 28 2006

A379949 Primitive abundant numbers (A091191) that are odd squares.

Original entry on oeis.org

342225, 1029447225, 1757705625, 2177622225, 14787776025, 18114198921, 32871503025, 45018230625, 150897287025, 245485566225, 296006724225, 705373218225, 1126920249225, 1329226832241, 1358425215225, 1545732725625, 1555265892609, 1783322538921, 2811755495241, 4627123655625, 5248080775161, 6140855705625, 7683069267225

Offset: 1

Antti Karttunen, Jan 07 2025

nonn

Question: Does A379504(.) obtain generally smaller values for the terms of this subsequence of A156942 than for its non-primitive terms? (See A379951, with A379951(5) = 5969, where A156942(5) = 342225, the first term of this sequence). Is A103977(.) = 1 for all terms, i.e., is this a subsequence of A379503?

Amiram Eldar, Table of n, a(n) for n = 1..3500 (terms 1..725 from Antti Karttunen)

Cf. A103977, A379504, A379950 (square roots).
Intersection of A016754 and A091191.
Intersection of A006038 and A156942.
Subsequences of the following sequences: A306796 (odd terms, but only if there are no odd perfect numbers), A363176, A379503 (conjectured).

is_A379949(n) = if(!(n%2) || !issquare(n) || sigma(n)<=2*n, 0, fordiv(n, d, if(d>1 && sigma(n/d, -1)>2, return(0))); (1));

is1(k) = {my(f = factor(k)); for(i = 1, #f~, f[i, 2] *= 2); if(sigma(f, -1) <= 2, return(0)); for(i = 1, #f~, f[i, 2] -= 1; if(sigma(f, -1) > 2, return(0)); f[i, 2] += 1); 1;}
list(lim) = forstep(k = 1, lim, 2, if(is1(k), print1(k^2, ", "))); \\ Amiram Eldar, Mar 12 2025

a(n) = A379950(n)^2.

A107705 a(n) is the least number of prime factors in any non-deficient number that has the n-th prime as its least prime factor.

Original entry on oeis.org

2, 5, 9, 18, 31, 46, 67, 91, 122, 158, 194, 238, 284, 334, 392, 456, 522, 591, 668, 749, 835, 929, 1028, 1133, 1242, 1352, 1469, 1594, 1727, 1869, 2019, 2163, 2315, 2471, 2636, 2802, 2977, 3157, 3342, 3534, 3731, 3933, 4145, 4358, 4581, 4811, 5053, 5293

Offset: 1

Hugo van der Sanden, Jun 10 2005

nonn

Barring unforeseen odd perfect numbers (which it has been proved must have at least 29 prime factors if they exist at all), if we replace "non-deficient" in the description with "abundant", the value of a(1) becomes 3 and all other values stay the same.

The above mentioned sequence is A108227, see there for a comment on the relation of this sequence to that of primitive abundant numbers (A006038) which are products of consecutive primes, i.e., of the form N = Product_{0<=iA007702. - M. F. Hasler, Jun 15 2017

			a(2) is 5 since 1) there are abundant numbers with a(2)=5 prime factors of which p_2=3 is the least prime factor (such as 945 = 3^3.5.7); 2) there are no non-deficient numbers with fewer than 5 prime factors, of which 3 is the least prime factor.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A000040, A023196, A005101, A001222.

Cf. A007684 (~ A007707), A007708, A007741.

A107705(n,s=1+1/prime(n))=for(a=1,9e9,2>(s*=1+1/prime(n+a))||return(a+1)) \\ M. F. Hasler, Jun 15 2017

a(n) = A007684(n)-n+1. A007702(n) = Product_{0<=iM. F. Hasler, Jun 15 2017

Data corrected by Amiram Eldar, Aug 08 2019

A285993 Largest odd abundant number (A005231) equal to the product of n consecutive primes.

Original entry on oeis.org

15015, 255255, 4849845, 111546435, 33426748355, 1236789689135, 50708377254535, 2180460221945005, 102481630431415235, 5431526412865007455, 320460058359035439845, 19548063559901161830545, 1309720258513377842646515, 1357656019974967471687377449, 107254825578022430263302818471

Offset: 5

M. F. Hasler, Apr 30 2017

nonn

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.

			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).

Amiram Eldar, Table of n, a(n) for n = 5..341

Cf. A005231, A006038, A007707, A007708, A007741.

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.

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.

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.

cp's OEIS Frontend

A180332 Primitive Zumkeller numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

SageMath

A007741 a(n) = prime(n)*...*prime(m), the least product of consecutive primes which is abundant.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A249263 Primitive, odd, squarefree abundant numbers.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A006036 Primitive pseudoperfect numbers.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Extensions

A108227 a(n) is the least number of prime factors for any abundant number with p_n (the n-th prime) as its least factor.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A188263 Odd abundant numbers whose abundancy is closer to 2 than any smaller odd abundant number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A064001 Odd abundant numbers not divisible by 5.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

A007741 a(n) = prime(n)...prime(m), the least product of consecutive primes which is abundant.