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A319852
Difference between 3^n and the product of primes less than or equal to n.
2
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
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
KEYWORD
nonn
AUTHOR
Alonso del Arte, Sep 29 2018
EXTENSIONS
Many thanks to Amiram Eldar for several bibliographic citations on this topic.
STATUS
approved