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

Author

Alonso del Arte, Sep 29 2018

Keywords

• nonn

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A000244, A034386, A319857.

Programs

Mathematica

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

PARI

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

Formula

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

Extensions

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