A319852 Difference between 3^n and the product of primes less than or equal to n.

0, 2, 7, 21, 75, 213, 699, 1977, 6351, 19473, 58839, 174837, 529131, 1564293, 4752939, 14318877, 43016691, 128629653, 386909979, 1152561777, 3477084711, 10450653513, 31371359919, 93920085957, 282206443611, 847065516573, 2541642735459, 7625374392117, 22876569362091

Offset

0,2

Comments

From Rosser (1941), it seems that the tightest possible upper bound is somewhere between e^n and 2.83^n. Therefore 3^n is the best possible upper bound with an integer base and integer exponent. - Alonso del Arte, Oct 22 2018

Links

Table of n, a(n) for n=0..28.

Barkley Rosser, "Explicit Bounds for Some Functions of Prime Numbers", Amer. J. Math., 1941, 63 (1) p. 228, Lemma 21.

J. Barkley Rosser, Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 1962, 64-94.

Formula

a(n) = 3^n - n#, where n# = A034386(n) is the product of the primes less than or equal to n.

Example

3^5 = 243. The primes less than or equal to 5 are: 2, 3, 5. Then 2 * 3 * 5 = 30 and hence a(5) = 243 - 30 = 213.

Mathematica

Table[3^n - Times@@Select[Range[n], PrimeQ], {n, 0, 26}]

Prog

(PARI) a(n) = 3^n-factorback(primes(primepi(n))) \\ David A. Corneth, Oct 22 2018

Crossrefs

Cf. A000244, A034386, A319857.

Sequence in context: A347302 A127540 A339041 * A060900 A305850 A151289

Adjacent sequences: A319849 A319850 A319851 * A319853 A319854 A319855

Keyword

nonn

Author

Alonso del Arte, Sep 29 2018

Extensions

Many thanks to Amiram Eldar for several bibliographic citations on this topic.

Status

approved