A060646 Bonse sequence: a(n) = minimal j such that n-j+1 < prime(j).

1, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18

Offset

1,2

Comments

For 3<n and any a(n-1)<a(n) use a(n)=a(n+1)=a(n+2) to show prime(j+1)^3 < prime(1)*...*prime(j) for j>5.

References

R. Remak, Archiv d. Math. u. Physik (3) vol. 15 (1908) 186-193

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

H. Bonse, Über eine bekannte Eigenshaft der Zahl 30 und ihre Verallgemeinerung, Archiv d. Math. u. Physik (3) vol. 12 (1907) 292-295.

H. Rademacher and O. Toeplitz, Eine Eigenschaft der Zahl 30, (A property of the number 30), Von Zahlen und Figuren (1930, reprint Springer 1968), ch. 22.

Formula

From Ridouane Oudra, Jan 13 2026: (Start)

a(n) = n + 2 - A112231(n+1).

a(n) = A036234(A112231(n)).

a(A014688(n)) = n + 1.

a(A095116(n)) = n + 1.

a(n) = a(n-1) + 1 if n is in A095116, otherwise a(n) = a(n-1); i.e., a(n) - a(n-1) is the characteristic function of A095116. (End)

Example

For n=5, j=3 gives 5-3+1 = 3 < prime(3) = 5, true; but if j=2 we get 5-2+1 = 4 which is not < prime(2) = 3; hence a(5) = 3.
a(75)=18 because 75-18+1=58 < 61=prime(18), but 75-17+1=59=prime(17).

Mathematica

Table[j=0; While[j++; n-j+1 >= Prime[j]]; j, {n, 1, 76}] (* Jean-François Alcover, Aug 30 2011 *)

Prog

(Haskell)
import Data.List (findIndex)
import Data.Maybe (fromJust)
a060646 n = (fromJust $ findIndex ((n+1) <) a014688_list) + 1
-- Reinhard Zumkeller, Sep 16 2011
(Python)
from sympy import nextprime
from itertools import count, islice
def agen(): # generator of terms
    n, pj = 1, 2
    for j in count(1):
        while n - j + 1 < pj: yield j; n += 1
        pj = nextprime(pj)
print(list(islice(agen(), 76))) # Michael S. Branicky, Aug 09 2022

Crossrefs

Cf. A014688, A112231, A036234, A095116.

Sequence in context: A006670 A132914 A370461 * A103298 A082429 A047744

Adjacent sequences: A060643 A060644 A060645 * A060647 A060648 A060649

Keyword

easy,nonn,nice

Author

Frank Ellermann, Apr 17 2001

Status

approved