A091440 -id:A091440 - 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

A073087 Least k such that sigma(k^k)>=n*k^k.

Original entry on oeis.org

1, 6, 30, 210, 30030, 223092870, 13082761331670030, 3217644767340672907899084554130, 1492182350939279320058875736615841068547583863326864530410

Offset: 1

Benoit Cloitre, Aug 18 2002

nonn

Does a(n) = the product of primes less than or equal to prime(n+1) = A002110(n+1)? Answer from Lambert Klasen (Lambert.Klasen(AT)gmx.net), Sep 14 2005: No, this is not true.
Note that sigma(k^k) = prod (p^(k r + 1) - 1)/(p - 1). - Mitch Harris, Sep 14 2005
I have proved to my own satisfaction that for n >= 4, A073087(n) = p#, where p is the smallest prime satisfying p#/phi(p#) >= n. See link. - David W. Wilson, Sep 14 2005

David W. Wilson, Comments on this sequence

Cf. A023199.

a(n)=if(n<0,0,s=1; while(sigma(s^s)

a(n) = A091440(n)# = A002110(A112873(n)) for n >= 4.

More terms from David W. Wilson, Sep 15 2005

A061556 a(n) is the least k > 0 such that sigma(k!) >= n*k!.

Original entry on oeis.org

1, 1, 3, 5, 9, 14, 23, 43, 79, 149, 263, 461, 823, 1451, 2549, 4483, 7879, 13859, 24247, 42683, 75037, 131707, 230773, 405401, 710569, 1246379, 2185021, 3831913, 6720059, 11781551, 20657677

Offset: 0

Labos Elemer, May 17 2001

It seems that, for n > 1, a(n+1) < 2*a(n). Does lim_{n -> infinity} a(n+1)/a(n) = 2? - Benoit Cloitre, Aug 18 2002
Smallest number m such that the abundancy-index of m! is at least n.
Floor(sigma(m!)/m!) = n; note that abundancy-index [= sigma(u)/u] here is not necessarily an integer.
It appears that a(n) = A091440(n) for n >= 13. - Daniel Suteu, Sep 03 2019

			floor(sigma(842!)/842!) = 11 while floor(sigma(843!)/843!) = 12.

Achim Flammenkamp, The Multiply Perfect Numbers Page.
Fred Helenius, Multiperfect numbers.

Cf. A000142, A000203, A023199, A062569.

PARI
```
a(n)=if(n<0,0,s=1; while(sigma(s!)
				
```

a(n) = Min{w | floor(sigma(w!)/w!) = n}.

More terms from David Wasserman, Jun 18 2002

a(1) inserted and a(21)-a(30) added by Daniel Suteu, Sep 03 2019

A167348 Let a(n) be the n-th term of the sequence. Let m = primorial(a(n)); m is the minimum positive integer such that m/phi(m) >= n.

Original entry on oeis.org

2, 2, 3, 7, 13, 23, 43, 79, 149, 257, 461, 821, 1451, 2549, 4483, 7879, 13859, 24247, 42683, 75037, 131707, 230773, 405401, 710569, 1246379, 2185021, 3831913, 6720059, 11781551, 20657677, 36221753, 63503639, 111333529, 195199289

Offset: 1

Fred Schneider, Aug 13 2009

nonn

A variant of A091440, which is the main entry for this sequence.

			primorial(7) = 210; 210/phi(210) = 210/48 >= 4;

Cf. A091440.

al(lim) = local(mm,n,m); mm=3; n=2; m=1; forprime(x=3,lim, n*=x; m*= (x-1); if (n\m >= mm, print1(x","); mm++)); /* This will generate all terms of this sequence from the 3rd onward, up to lim. The computation slows down for large values because of the size of the internal values. */

Edited and extended by Franklin T. Adams-Watters and N. J. A. Sloane, Aug 29 2009

A360895 Decimal expansion of exp(exp(-gamma)) where gamma is the Euler-Mascheroni constant A001620.

Original entry on oeis.org

1, 7, 5, 3, 2, 2, 9, 4, 4, 3, 4, 9, 5, 6, 9, 4, 5, 8, 2, 2, 9, 7, 3, 6, 5, 4, 2, 9, 6, 4, 4, 0, 6, 1, 2, 8, 7, 6, 0, 5, 7, 4, 5, 8, 0, 2, 0, 2, 0, 7, 5, 4, 4, 5, 6, 1, 9, 0, 2, 9, 5, 1, 5, 6, 3, 1, 5, 3, 9, 8, 8, 9, 4, 0, 8, 7, 8, 0, 7, 2, 0, 6, 0, 7, 2, 4, 5, 3, 1, 0, 5, 5, 5, 8, 8, 8, 6, 7, 4, 0, 5, 2, 0, 2, 4, 3, 4, 3, 7, 6, 8, 4, 6, 4, 2

Offset: 1

Artur Jasinski, Feb 25 2023

Theorem: Let p(n) be the smallest prime such that Product_{prime p<=p(n)} 1/(1-1/p) >= n. Then lim_{n->oo} p(n+1)/p(n) = exp(exp(-gamma)).
Proof. Follow Mertens's Third Theorem Product_{p<=x} 1/(1-1/p) ~ log(x)/exp(-gamma).
For any particular integer n, it follows from the equations n = log(x_n)/exp(-gamma) -> x_n = exp(n*exp(-gamma)) and n+1 = log(x_n+1)/exp(-gamma) -> x_n+1 = exp((n+1)*exp(-gamma)) that lim_{n->oo} exp((n+1)*exp(-gamma))/exp((n)*exp(-gamma)) = exp(exp(-gamma)).
Convergence table:
n   p(n)        truncated Euler product up to p(n)   ratio p(n)/p(n-1)
42  17427088769 42.0000000010939727723681242652955   1.7532416978341651
43  30553756811 43.0000000012946363551468233325186   1.7532335558736718
44  53567706007 44.0000000002803554088007272169139   1.7532281329055578
45  93916601047 45.0000000002681963271546340553884   1.7532317145469581
46 164657625967 46.0000000002470257389410099668348   1.7532323799133028
47 288682860119 47.0000000001305074313442036255929   1.7532310357544971
48 506127311983 48.0000000000705764045487316221655   1.7532295189758258
oo  oo           oo                                  1.7532294434956945

			1.753229443495694582297365429644...

Wikipedia, Mertens' theorems

Cf. A001620, A080130, A061556, A091440, A167348.

RealDigits[N[Exp[Exp[-EulerGamma]], 115]][[1]]

exp(exp(-Euler)) \\ Michel Marcus, Feb 25 2023

Equals exp(A080130).
Limit_{n->oo} A091440(n+1)/A091440(n).
Limit_{n->oo} A061556(n+1)/A061556(n).
Limit_{n->oo} A167348(n+1)/A167348(n).

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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A073087 Least k such that sigma(k^k)>=n*k^k.

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A061556 a(n) is the least k > 0 such that sigma(k!) >= n*k!.

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A167348 Let a(n) be the n-th term of the sequence. Let m = primorial(a(n)); m is the minimum positive integer such that m/phi(m) >= n.

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Examples

Crossrefs

Programs

PARI

Extensions

A360895 Decimal expansion of exp(exp(-gamma)) where gamma is the Euler-Mascheroni constant A001620.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula