A332173 a(n) = 7*(10^(2n+1)-1)/9 - 4*10^n.

3, 737, 77377, 7773777, 777737777, 77777377777, 7777773777777, 777777737777777, 77777777377777777, 7777777773777777777, 777777777737777777777, 77777777777377777777777, 7777777777773777777777777, 777777777777737777777777777, 77777777777777377777777777777, 7777777777777773777777777777777

Offset: 0

Author

M. F. Hasler, Feb 06 2020

Keywords

• nonn
• base
• easy

Comments

According to M. Kamada, n = 0 and n = 2 are the only indices of a prime up to n = 2*10^4.

Links

• Makoto Kamada, Factorization of 77...77377...77, updated Dec 11 2018.
• Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Crossrefs

Cf. A138148 (cyclops numbers with binary digits only).

Cf. A002275 (repunits R_n = (10^n-1)/9), A002281 (7*R_n), A011557 (10^n).

Cf. A332171 .. A332179 (variants with different middle digit 1, ..., 9).

Programs

Maple

A332173 := n -> 7*(10^(n*2+1)-1)/9 - 4*10^n;

Mathematica

Array[7 (10^(2 # + 1) - 1)/9 - 4*10^# &, 15, 0]

PARI

apply( {A332173(n)=10^(n*2+1)\9*7-4*10^n}, [0..15])

Python

def A332173(n): return 10**(n*2+1)//9*7-4*10^n

Formula

a(n) = 7*A138148(n) + 3*10^n.

G.f.: (1 + 404*x - 1100*x^2)/((1 - x)*(1 - 10*x)*(1 - 100*x)).

a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n>2.

E.g.f.: exp(x)*(70*exp(99*x) - 36*exp(9*x) - 7)/9. - Stefano Spezia, Feb 19 2025