A075600 -id:A075600

A075595

Smallest prime that is obtained by placing digits on both sides of the n-th prime. Or smallest prime that encompasses the n-th prime.

+10
11

127, 131, 151, 173, 1117, 2131, 1171, 1193, 1231, 1291, 1319, 1373, 2411, 1433, 1471, 1531, 1597, 1613, 2671, 2711, 1733, 2791, 1831, 2897, 1973, 21011, 21031, 11071, 11093, 11131, 11273, 11311, 21377, 11393, 11491, 11519, 11579, 11633

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

For small primes, the prefix required is usually 1 or 2.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

MAPLE

f:= proc(n) local m, d, d1, v, x, y, y0, z, found;

m:= ilog10(n);

v:= infinity;

for d from 2 do

for d1 from 1 to d-1 do

found:= false;

for x from 10^(d1-1) to 10^d1-1 while not found do

if d-d1=1 then y0:= 1 else y0:= 10^(d-d1-1)+1 fi;

for y from y0 to 10^(d-d1)-1 by 2 do

z:= y+10^(d-d1)*n + 10^(d-d1+m+1)*x;

if isprime(z) then v:= min(v, z); found:= true; break fi

od od;

od;

if v < infinity then return v fi

od

end proc:

map(f@ithprime, [$1..100]); # Robert Israel, Aug 28 2018

CROSSREFS

Cf. A075596, A075597, A075598, A075599, A075600, A075601, A075602.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 28 2002

EXTENSIONS

Corrected and extended by Sascha Kurz, Jan 20 2003

STATUS

approved

A075597

a(1) = 3 and then the smallest prime that is obtained by placing digits on both sides of the previous term. Or smallest prime that encompasses a(n-1).

+10
8

3, 131, 11311, 3113111, 231131111, 22311311119, 3223113111199, 132231131111993, 11322311311119931, 7113223113111199313, 171132231131111993137, 31711322311311119931373, 3317113223113111199313733

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..13.

MATHEMATICA

NestList[Select[Flatten[Table[FromDigits[Join[{a}, IntegerDigits[#], {b}]], {a, 9}, {b, 1, 9, 2}]], PrimeQ][[1]]&, 3, 12] (* Harvey P. Dale, Nov 11 2022 *)

CROSSREFS

Cf. A075595, A075596, A075598, A075599, A075600, A075601, A075602.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 28 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 20 2003

One more term from David Wasserman, Jan 20 2005

STATUS

approved

A075598

a(1) = 5 and then the smallest prime that is obtained by placing digits on both sides of the previous term. Or smallest prime that encompasses a(n-1).

+10
8

5, 151, 11519, 2115193, 121151939, 21211519397, 4212115193971, 342121151939719, 43421211519397199, 2434212115193971993, 224342121151939719937, 122434212115193971993787, 51224342121151939719937871, 2512243421211519397199378719, 325122434212115193971993787197, 93251224342121151939719937871973

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Robert Israel, Table of n, a(n) for n = 1..330

MAPLE

f:= proc(n) local m, d, d1, v, x, y, y0, z, found;

m:= ilog10(n);

v:= infinity;

for d from 2 do

for d1 from 1 to d-1 do

found:= false;

for x from 10^(d1-1) to 10^d1-1 while not found do

if d-d1=1 then y0:= 1 else y0:= 10^(d-d1-1)+1 fi;

for y from y0 to 10^(d-d1)-1 by 2 do

z:= y+10^(d-d1)*n + 10^(d-d1+m+1)*x;

if isprime(z) then v:= min(v, z); found:= true; break fi

od od;

od;

if v < infinity then return v fi

od

end proc:

A[1]:= 5:

for n from 2 to 20 do

A[n]:= f(A[n-1])

od:

seq(A[n], n=1..20); # Robert Israel, Aug 28 2018

CROSSREFS

Cf. A075595, A075596, A075597, A075599, A075600, A075601, A075602.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 28 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 20 2003

Terms a(13) and beyond from Robert Israel, Aug 28 2018

STATUS

approved

A075596

a(1) = 2 and then the smallest prime that is obtained by placing digits on both sides of the previous term. Or smallest prime that encompasses a(n-1).

+10
7

2, 127, 11273, 1112731, 411127319, 34111273199, 3341112731993, 233411127319933, 32334111273199339, 3323341112731993391, 633233411127319933919, 66332334111273199339193, 1663323341112731993391937

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..13.

CROSSREFS

Cf. A075595, A075597, A075598, A075599, A075600, A075601, A075602.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 28 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 20 2003

One more term from David Wasserman, Jan 20 2005

STATUS

approved

A075599

a(1) = 7 and then the smallest prime that is obtained by placing digits on both sides of the previous term. Or smallest prime that encompasses a(n-1).

+10
7

7, 173, 11731, 2117317, 421173173, 34211731733, 1342117317337, 813421173173377, 28134211731733771, 4281342117317337713, 142813421173173377131, 21428134211731733771317, 2214281342117317337713171, 2221428134211731733771317113, 922214281342117317337713171139

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..15.

CROSSREFS

Cf. A075595, A075596, A075597, A075598, A075600, A075601, A075602.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 28 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 20 2003

Further terms from Ray G. Opao, Nov 26 2004

STATUS

approved

A075601

a(1) = 13 and then the smallest prime that is obtained by placing digits on both sides of the previous term. Or smallest prime that encompasses a(n-1).

+10
7

13, 2131, 121313, 51213137, 3512131379, 135121313797, 21351213137977, 5213512131379777, 352135121313797771, 63521351213137977713, 16352135121313797771319, 9163521351213137977713193

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..12.

CROSSREFS

Cf. A075595, A075596, A075597, A075598, A075599, A075600, A075602.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 28 2002

EXTENSIONS

Corrected and extended by Sascha Kurz, Jan 23 2003

STATUS

approved

A075602

a(1) = 17 and then the smallest prime that is obtained by placing digits on both sides of the previous term. Or smallest prime that encompasses a(n-1).

+10
7

17, 1171, 211711, 32117117, 6321171179, 263211711799, 42632117117999, 6426321171179993, 564263211711799931, 15642632117117999311, 1156426321171179993119, 311564263211711799931199

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..12.

CROSSREFS

Cf. A075595, A075596, A075597, A075598, A075599, A075600, A075601.

KEYWORD

base,nonn

AUTHOR

Amarnath Murthy, Sep 28 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 23 2003

STATUS

approved