A077359 -id:A077359 - cp's OEIS frontend

A069686 Primes whose internal digits form a prime.

127, 131, 137, 139, 151, 157, 173, 179, 223, 227, 229, 233, 239, 251, 257, 271, 277, 331, 337, 353, 359, 373, 379, 421, 431, 433, 439, 457, 479, 521, 523, 557, 571, 577, 631, 653, 659, 673, 677, 727, 733, 739, 751, 757, 773, 821, 823, 827, 829, 839, 853

Offset: 1

Amarnath Murthy, Nov 05 2002

Primes that remain prime upon deleting the first and last digits.

Jason Yuen, Table of n, a(n) for n = 1..10000 (correcting Christian N. K. Anderson previous bfile).

Cf. A077359, A077360.

Select[Range[100, 853], PrimeQ[#] && PrimeQ[FromDigits[Rest[Most[IntegerDigits[#]]]]] &] (* T. D. Noe, Apr 05 2013 *)

{indigs(n)=local(j,a,d); n=n\10; j=1; a=0; while(n>10,d=divrem(n,10); n=d[1]; a=a+j*d[2]; j=10*j); a}
forprime(p=1,855,if(isprime(indigs(p)),print1(p,","))) \\ Klaus Brockhaus, Nov 06 2002

from sympy import isprime
for p in filter(isprime, range(100, 855)):
    if isprime(int(str(p)[1:-1])): print(p) # Jason Yuen, Mar 28 2024

Edited and extended by Klaus Brockhaus, Nov 06 2002

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 21 2007

A077360 Primes whose external digits as well as internal digits form a prime.

Original entry on oeis.org

127, 131, 137, 139, 151, 157, 173, 179, 223, 229, 233, 239, 331, 337, 421, 431, 433, 457, 523, 631, 677, 733, 739, 751, 773, 823, 829, 839, 853, 859, 937, 977, 1021, 1031, 1033, 1039, 1051, 1117, 1171, 1193, 1231, 1237, 1291, 1297, 1319, 1373, 1433, 1439

Offset: 1

Amarnath Murthy, Nov 05 2002

Intersection of A069686 and A077359.

			139 is a term as 19 and 3 are both primes.

Jason Yuen, Table of n, a(n) for n = 1..10000

Cf. A069686, A077359.

forprime(p=1,1440,if(isprime(indigs(p))&&isprime(exdigs(p)),print1(p,",")))

from sympy import isprime, primerange
for p in primerange(100, 1440):
    if isprime(int(str(p)[1:-1])) and isprime(int(str(p)[0]+str(p)[-1])):
        print(p)  # Jason Yuen, Apr 21 2024

Edited and extended by Klaus Brockhaus, Nov 06 2002

A225082 Centrally deletable primes.

Original entry on oeis.org

101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 229, 233, 239, 263, 269, 283, 293, 307, 311, 317, 331, 337, 347, 367, 397, 401, 421, 431, 433, 443, 457, 461, 463, 467, 487, 491, 503, 509, 523, 563

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, Apr 26 2013

Prime numbers that remain primes when their central digit is (or two central digits are) deleted.
At the 1886th prime number (16229), there are exactly 943 centrally deletable primes, and 943 that become composites. It appears that there are always more non-deletable primes thereafter.
Subset of A080603 and of A077359.

			a(5) = 1(1)3, and 13 is a prime.

Christian N. K. Anderson, Table of n, a(n) for n = 1..10000

Cf. A080603, A077359

dcd[n_] := Block[{d = IntegerDigits@n, z}, z = Length@d; FromDigits@ Delete[d, Floor[(z + {{1}, {2}})/2]]]; Select[Prime@ Range@ 103, PrimeQ@ dcd@ # &] (* Giovanni Resta, Apr 29 2013 *)

library(gmp)
sumsubstrpow<-function(n) {
no0<-function(s){ while(substr(s,1,1)=="0" && nchar(s)>1) s=substr(s,2,nchar(s)); s}
tot=as.bigz(0); s=as.character(n); len=nchar(s)
for(i in 1:len) for(j in i:len) tot=tot+as.bigz(no0(substr(s,i,j)))^(j-i+1)
tot
}
#recursive
n=as.bigz(10); for(y in 1:4) n[y+1]=sumsubstrpow(n[y])

A077361 Smallest n-digit prime whose external digits as well as internal digits form a prime, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 127, 1021, 10037, 100057, 1000033, 10000079, 100000037, 1000000021, 10000000033, 100000000057, 1000000000039, 10000000000037, 100000000000031, 1000000000000037, 10000000000000079, 100000000000000021

Offset: 0

Amarnath Murthy, Nov 05 2002

Conjecture: no entry is zero for n>2.

Harvey P. Dale, Table of n, a(n) for n = 0..500

Cf. A069686, A077359, A077360, A077362.

eifpQ[n_]:=Module[{idn=IntegerDigits[n]},PrimeQ[FromDigits[{First[ idn], Last[ idn]}]]&&PrimeQ[FromDigits[Rest[Most[idn]]]]]; ndp[pwr_]: = Module[ {p=NextPrime[10^pwr]},While[!eifpQ[p],p=NextPrime[p]];p]; Join[ {0,0}, Table[ndp[i],{i,2,20}]] (* Harvey P. Dale, Jan 03 2015 *)

More terms from Sascha Kurz, Jan 11 2003

A077362 Largest n-digit prime whose external digits as well as internal digits form a prime, or 0 if no such number exists.

Original entry on oeis.org

0, 0, 977, 9677, 99377, 998717, 9998777, 99999617, 999999017, 9999996437, 99999997397, 999999997277, 9999999986477, 99999999993317, 999999999997337, 9999999999990797, 99999999999998837, 999999999999995717, 9999999999999997397, 99999999999999994877

Offset: 0

Amarnath Murthy, Nov 05 2002

Conjecture: no entry is zero for n>2.

Conjecture: each term after the first two terms ends with 7. - Harvey P. Dale, May 26 2018

Harvey P. Dale, Table of n, a(n) for n = 0..100

Cf. A069686, A077359, A077360, A077361.

LastDigit[n_] := n - 10*Floor[n/10]; FirstDigit[n_] := Floor[n/(10^(Ceiling[Log[10, n]] - 1))]; MiddleDigits[n_] := Floor[(n - Floor[n/(10^(Ceiling[Log[10, n]] - 1))]*10^(Ceiling[Log[10, n]] - 1))/10]; IntExtPrimeTest2[n_] := TrueQ[(Boole[PrimeQ[FirstDigit[n]*10 + LastDigit[ n]]] + Boole[PrimeQ[MiddleDigits[n]]] + Boole[PrimeQ[n]]) == 3]; finder[digits_] := (maxj = 10^digits; For[j = maxj, IntExtPrimeTest2[j] == False, j-- ]; j); Table[finder[n], {n, 3, 20}] (* Joshua Albert (jba138(AT)psu.edu), Feb 22 2006 *)
eidQ[n_]:=Module[{idn=IntegerDigits[n]},AllTrue[{FromDigits[Join[ {idn[[1]]}, {idn[[-1]]}]],FromDigits[Most[Rest[idn]]]},PrimeQ]]; Join[ {0,0},Table[Module[{np=NextPrime[10^n-1,-1]},While[ !eidQ[np],np = NextPrime[ np,-1]];np],{n,3,18}]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 26 2018 *)

Corrected and extended by Joshua Albert (jba138(AT)psu.edu), Feb 22 2006

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A069686 Primes whose internal digits form a prime.

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A077360 Primes whose external digits as well as internal digits form a prime.

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A225082 Centrally deletable primes.

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A077361 Smallest n-digit prime whose external digits as well as internal digits form a prime, or 0 if no such number exists.

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A077362 Largest n-digit prime whose external digits as well as internal digits form a prime, or 0 if no such number exists.

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Mathematica

Extensions