A164347 -id:A164347 - cp's OEIS frontend

A091440 Smallest number m such that m#/phi(m#) >= n, where m# indicates the primorial (A034386) of m and phi is Euler's totient function.

1, 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, 342243479, 600036989

Offset: 1

T. D. Noe, Jan 09 2004

nonn

Does the ratio of adjacent terms converge?
It appears that lim_{n->infinity} a(n+1)/a(n) = 1.7532... - Jon E. Schoenfield, Feb 21 2019
For n > 1, a(n) is smallest prime p = prime(k) such that no fewer than (n-1)/n of any p# consecutive integers are divisible by a prime not greater than p. Cf. A053144(k)/A002110(k). - Peter Munn, Apr 29 2017
Also, the smallest prime p such that the sum of the reciprocals of the p-smooth numbers converges to at least n. - Keith F. Lynch, Apr 29 2023
Also, if m is a random integer much larger than the square of a(n), and m is not divisible by any prime less than or equal to a(n), the probability that m is prime is n/log(m). - Keith F. Lynch, Dec 17 2023

			7#/phi(7#) = (2*3*5*7)/(1*2*4*6) = 4.375 >= 4, 5#/phi(5#) = 3.75. Hence a(4) = 7.

Jon E. Schoenfield, Table of n, a(n) for n = 1..44
Eric Weisstein's World of Mathematics, Primorial.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A091439, A000010, A002110, A005579, A034386, A053144, A038110, A060753, A164347.

Mathematica
```
prod=1; i=0; Table[While[prod
				
```

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. */ \\ Fred Schneider, Aug 13 2009, modified by Franklin T. Adams-Watters, Aug 29 2009

a(n) = prime(A005579(n)) for n >= 4. - Amiram Eldar, Apr 18 2025

More terms from David W. Wilson, Sep 28 2005

Sequence reference in name corrected by Peter Munn, Apr 29 2017

cp's OEIS Frontend

A091440 Smallest number m such that m#/phi(m#) >= n, where m# indicates the primorial (A034386) of m and phi is Euler's totient function.

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