login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A211681 Numbers such that all the substrings of length <= 2 are primes. 56
2, 3, 5, 7, 23, 37, 53, 73, 237, 373, 537, 737, 2373, 3737, 5373, 7373, 23737, 37373, 53737, 73737, 237373, 373737, 537373, 737373, 2373737, 3737373, 5373737, 7373737, 23737373, 37373737, 53737373, 73737373, 237373737, 373737373, 537373737, 737373737 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The terms are primes for n = 1, 2, 3, 4, 5, 6, 7, 8, 10, 21, 23, 27, 31, 43, 45, 60, 67, 82, 91, .... The further terms until index 102 are composite. For the subsequence with prime terms see A211682. - [updated by Hieronymus Fischer, Oct 02 2018]

From Hieronymus Fischer, Oct 02 2018: (Start)

For indices n > 8, prime terms satisfy n mod 24 = 1, 3, 5, 7, 10, 12, 19, 21, 23. However, this condition is not sufficient. Indeed, for n <= 200 most of those terms are proven composite unless the terms with n = 103, 106, 123, 156, 165, 175, 178, 191, 193 and 195 which are potentially prime.

The terms are composite for n > 10 and n mod 24 = 0, 2, 4, 6, 8, 9, 11, 13, 14, 15, 16, 17, 18, 20, 22 (see formula section for the details).

(End)

Cf. A213299 for the partial sums.

LINKS

Hieronymus Fischer, Table of n, a(n) for n = 1..250

FORMULA

a(1+8*k) =  2*10^(2k) + 37*(10^(2k)-1)/99,

a(2+8*k) =  3*10^(2k) + 73*(10^(2k)-1)/99,

a(3+8*k) =  5*10^(2k) + 37*(10^(2k)-1)/99,

a(4+8*k) =  7*10^(2k) + 37*(10^(2k)-1)/99,

a(5+8*k) = 23*10^(2k) + 73*(10^(2k)-1)/99,

a(6+8*k) = 37*10^(2k) + 37*(10^(2k)-1)/99,

a(7+8*k) = 53*10^(2k) + 73*(10^(2k)-1)/99,

a(8+8*k) = 73*10^(2k) + 73*(10^(2k)-1)/99, for k >= 0.

a(n) = ((2*n+7) mod 8 + dn3 - dn2)*10^dn_1 + floor((37+36*(dn2-dn1))*10^dn_1/99), where dn1 = floor((n+1)/4), dn2 = floor((n+2)/4), dn3 = floor((n+3)/4), dn_1 = floor((n-1)/4). [updated by Hieronymus Fischer, Oct 02 2018]

From Hieronymus Fischer, Oct 02 2018: (Start)

a(24k + 0)  = 73*(10^(6k-2) + (10^(6k-2)-1)/99), for k > 0.

a(24k + 2)  =  3*(1245790*(10^(6k)-1)/999999 + 1),

a(24k + 4)  =  7*(1053390*(10^(6k)-1)/999999 + 1),

a(24k + 6)  = 37*(10^(6k) + (10^(6k)-1)/99),

a(24k + 8)  = 73*(10^(6k) + (10^(6k)-1)/99),

a(24k + 9)  =  3*(79124500*(10^(6k)-1)/999999 + 79),

a(24k + 11) =  3*(79124500*(10^(6k)-1)/999999 + 79 + 10^(6k+2)),

a(24k + 13) =  3*(791245000*(10^(6k)-1)/999999 + 791),

a(24k + 14) = 37*(10^(6k+2) + (10^(6k+2)-1)/99),

a(24k + 15) =  3*(791245000*(10^(6k)-1)/999999 + 791 + 10^(6k+3)),

a(24k + 16) = 73*(10^(6k+2) + (10^(6k+2)-1)/99),

a(24k + 17) =  7*(3391050000*(10^(6k)-1)/999999 + 3391),

a(24k + 18) =  7*(5339100000*(10^(6k)-1)/999999 + 5339),

a(24k + 20) =  3*(24579100000*(10^(6k)-1)/999999 + 24579),

a(24k + 22) = 37*(10^(6k+4) + (10^(6k+4)-1)/99), for k >= 0.

(End)

Recursion for n>8:

a(n) = 10*(1+a(n-4)) - a(n-4) mod 10.

G.f.: (2*x*(1+x^10) + 3*x^2*(1 + x^3 + x^5 + x^6) + 5*x^3*(1+x^6) + 7*x^4*(1+x^2))/((1-10*x^4)*(1-x^8)). [corrected by Hieronymus Fischer, Sep 03 2012]

EXAMPLE

a(11)=537, since all substrings of length <= 2 are primes (5, 3, 7, 53 and 37).

a(21)=237373, the substrings of length <= 2 are 2, 3, 7, 23, 37, 73.

MATHEMATICA

Table[FromDigits/@Select[Tuples[{2, 3, 5, 7}, n], AllTrue[FromDigits/@ Partition[ #, 2, 1], PrimeQ]&], {n, 9}]//Flatten (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 13 2020 *)

CROSSREFS

Cf. A019546, A035232, A039996, A046034, A085823, A211682, A213299.

Sequence in context: A019546 A104179 A096148 * A124674 A177061 A020994

Adjacent sequences:  A211678 A211679 A211680 * A211682 A211683 A211684

KEYWORD

nonn,base,easy

AUTHOR

Hieronymus Fischer, Apr 30 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 28 06:42 EST 2021. Contains 349401 sequences. (Running on oeis4.)