A072986 -id:A072986 - cp's OEIS frontend

A005579 a(n) = smallest number k such that Product_{i=1..k} prime(i)/(prime(i)-1) > n.

0, 1, 2, 3, 4, 6, 9, 14, 22, 35, 55, 89, 142, 230, 373, 609, 996, 1637, 2698, 4461, 7398, 12301, 20503, 34253, 57348, 96198, 161659, 272124, 458789, 774616, 1309627, 2216968, 3757384, 6375166, 10828012, 18409028, 31326514, 53354259, 90945529, 155142139

Offset: 0

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

nonn

Laatsch (1986) proved that for n >= 2, a(n) gives the smallest number of distinct prime factors in even numbers having an abundancy index > n.
The abundancy index of a number k is sigma(k)/k. - T. D. Noe, May 08 2006
The first differences of this sequence, A005347, begin the same as the Fibonacci sequence A000045. - T. D. Noe, May 08 2006
Equal to A256968 except for n = 2 and n = 3.  See comment in A256968. - Chai Wah Wu, Apr 17 2015

			The products Product_{i=1..k} prime(i)/(prime(i)-1) for k >= 0 start with 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, 323323/55296, 676039/110592, 2800733/442368, 86822723/13271040, 3212440751/477757440, 131710070791/19110297600, 5663533044013/802632499200, ... = A060753/A038110. So a(3) = 3.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 0..44
Richard K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy).
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.
Popular Computing (Calabasas, CA), Problem 182 (Suggested by Victor Meally), Annotated and scanned copy of page 10 of Vol. 5 (No. 53, Aug 1977).

A001611 is similar but strictly different.

Cf. A000720, A001611, A005580, A060753, A023199, A038110, A072986, A091440, A256968.

(* For speed and accuracy, the second Mathematica program uses 30-digit real numbers and interval arithmetic. *)
prod=1; k=0; Table[While[prod<=n, k++; prod=prod*Prime[k]/(Prime[k]-1)]; k, {n,0,25}] (* T. D. Noe, May 08 2006 *)
prod=Interval[1]; k=0; Table[While[Max[prod]<=n, k++; p=Prime[k]; prod=N[prod*p/(p-1),30]]; If[Min[prod]>n, k, "too few digits"], {n,0,38}]

a(n)=my(s=1,k); forprime(p=2,, s*=p/(p-1); k++; if(s>n, return(k))) \\ Charles R Greathouse IV, Aug 20 2015

from sympy import nextprime
def a_list(upto: int) -> list[int]:
    L: list[int] = [0]
    count = 1; bn = 1; bd = 1; p = 2
    for k in range(1, upto + 1):
        bn *= p
        bd *= p - 1
        while bn > count * bd:
            L.append(k)
            count += 1
        p = nextprime(p)
    return L
print(a_list(1000))  # Chai Wah Wu, Apr 17 2015, adapted by Peter Luschny, Jan 25 2025

a(n) = smallest k such that A002110(k)/A005867(k) > n. - Artur Jasinski, Nov 06 2008

a(n) = PrimePi(A091440(n)) = A000720(A091440(n)) for n >= 4. - Amiram Eldar, Apr 18 2025

Edited by T. D. Noe, May 08 2006
a(26) added by T. D. Noe, Sep 18 2008
Typo corrected by Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
a(27)-a(36) from Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
Comment corrected by T. D. Noe, Apr 04 2010
a(37)-a(39) from T. D. Noe, Nov 16 2010
Edited and terms a(0)-a(1) prepended by Max Alekseyev, Jan 25 2025

A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

Original entry on oeis.org

901800900, 1542132900, 1926332100, 2153888100, 2690496900, 2822796900, 3942584100, 4487660100, 4600908900, 5127992100, 6267888900, 6742052100, 7162236900, 7305120900, 8421732900, 8969984100, 9866448900, 10203020100, 10718460900, 11723411700, 11787444900, 12528324900

Offset: 1

Amiram Eldar, Oct 04 2019

nonn

Aiello et al. found bounds on e-multiperfect numbers, i.e., numbers m such that esigma(m) = k * m for k > 2: 2 * 10^7 for k = 3, and 10^85, 10^320, and 10^1210 for k = 4, 5, and 6. The data of this sequence raise the bound for exponential 3-perfect numbers to 3 * 10^10.
The least odd term is (59#/2)^2 = 924251841031287598942273821762233522616225. The least term which is coprime to 6 is (239#/6)^2 = 3.135... * 10^190.
The least exponential 4-abundant number (esigma(m) >= 4m) is (31#)^2 = 40224510201185827416900. In general, the least exponential k-abundant number (esigma(m) >= k*m), for k > 2, is (A002110(A072986(k)))^2.
The asymptotic density of this sequence is Sum_{n>=1} f(A383699(n)) = 1.325...*10^(-9), where f(n) = (6/(Pi^2*n))*Product_{prime p|n}(p/(p+1)). - Amiram Eldar, May 06 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000
W. Aiello, G. E. Hardy, and M. V. Subbarao, On the existence of e-multiperfect numbers, Fibonacci Quarterly, Vol. 25, No. 1 (1987), pp. 65-71.

Subsequence of A129575.
A383699 is a subsequence.
Cf. A002110, A051377, A072986, A321147.
Cf. A023197, A307112, A285615 (unitary), A293187 (bi-unitary), A300664 (infinitary).

f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[10^10], esigma[#] >= 3 # &]

A072997 Smallest prime p such that Product_{primes q <= p} q+1 >= n*Product_{primes q <= p} q.

Original entry on oeis.org

2, 3, 13, 31, 89, 239, 617, 1571, 4007, 10141, 25673, 64853, 163367, 412007, 1037759, 2614369, 6584857, 16585291, 41764859, 105178831, 264877933, 667038311, 1679809291, 4230219377, 10652786759, 26826453991, 67555877849

Offset: 1

Benoit Cloitre, Aug 14 2002

For k > 2, the primorial number A034386(A072997(k)) = A002110(A072986(k)) is the least unitary k-abundant number, i.e., the least number m such that usigma(m) >= k*m, where usigma(m) = A034448(m) is the sum of unitary divisors of m. The sequence of these primorials is the unitary version of A023199. - Amiram Eldar, Aug 24 2018

Cf. A000040, A002110, A023199, A034386, A034448, A072986, A072997, A362151.

n=x=y=1; Do[x *= (Prime[s] + 1); y *= Prime[s]; If[x >= n*y, Print[Prime[s]]; n++ ], {s, 1, 10^6}] (* Ryan Propper, Jul 22 2005 *)

a(n)=if(n<0,0,s=1; while(prod(i=1,s, prime(i)+1)

It seems that lim_{n -> oo} a(n+1)/a(n) exists and is > 2.
a(n) = A000040(A072986(n)). - Amiram Eldar, Aug 24 2018
Limit_{n -> oo} a(n+1)/a(n) = exp(zeta(2)/exp(gamma)) = 2.518... (A362151). - Amiram Eldar, Aug 26 2025

7 more terms from Ryan Propper, Jul 22 2005
a(18)-a(22) added by Amiram Eldar, Aug 24 2018 from the data at A072986
a(23)-a(27) from Keith F. Lynch, Jan 13 2024

A371702 a(n) is the least positive k such that Product_{i=1..k} (1 + 1/(2*i-1)) >= n.

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 12, 16, 21, 26, 32, 39, 46, 54, 63, 72, 82, 92, 103, 115, 128, 141, 154, 169, 184, 199, 215, 232, 250, 268, 287, 306, 326, 347, 368, 390, 413, 436, 460, 484, 510, 535, 562, 589, 616, 645, 674, 703, 734, 765, 796, 828, 861, 894, 928, 963

Offset: 0

Ilya Gutkovskiy, Apr 03 2024

nonn

			a(4) = 5: (2/1) * (4/3) * (6/5) * (8/7) * (10/9) = 256 / 63 = 4.063492... > 4.

Cf. A000165, A001147, A001803, A046161, A072986, A371701.

a[n_] := For[k = 1, True, k++, If[(2 k)!!/(2 k - 1)!! >= n, Return[k]]]; Table[a[n], {n, 0, 55}]

a(n) = my(k=1); while (prod(i=1, k, 1 + 1/(2*i-1)) < n, k++); k; \\ Michel Marcus, Apr 03 2024

a(n) ~ n^2 / Pi. - Vaclav Kotesovec, Apr 03 2024

cp's OEIS Frontend

A005579 a(n) = smallest number k such that Product_{i=1..k} prime(i)/(prime(i)-1) > n.

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A328135 Exponential 3-abundant numbers: numbers m such that esigma(m) >= 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

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Mathematica

A072997 Smallest prime p such that Product_{primes q <= p} q+1 >= n*Product_{primes q <= p} q.

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Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A371702 a(n) is the least positive k such that Product_{i=1..k} (1 + 1/(2*i-1)) >= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Formula