A069686 -id:A069686 - cp's OEIS frontend

A077358 Duplicate of A069686.

127, 131, 137, 139, 151, 157, 173, 179, 223, 227, 229, 233, 239, 251, 257, 271, 277

Offset: 1

dead

A077359 Primes whose external digits form a prime. Or primes from which deleting the internal digits leaves a prime.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 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

Offset: 1

Amarnath Murthy, Nov 05 2002

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A069686, A077360.

edpQ[n_]:=Module[{idn=IntegerDigits[n]},PrimeQ[10idn[[1]]+idn[[-1]]]]; Join[ {2,3,5,7},Select[Prime[Range[200]],edpQ]] (* Harvey P. Dale, Nov 20 2020 *)

{exdigs(n)=local(a,j,d); d=divrem(n,10); a=d[2]; n=d[1]; j=1; while(n>10,d=divrem(n,10); n=d[1]; j=10*j); 10*n+a} forprime(p=1,335,if(isprime(exdigs(p)),print1(p,",")))

Edited and extended by Klaus Brockhaus, Nov 06 2002

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

A225235 Emirps whose internal digits are also an emirp.

Original entry on oeis.org

1979, 3319, 3371, 3373, 3719, 3733, 7177, 7717, 9133, 9173, 9791, 10177, 10711, 10739, 11071, 11497, 11579, 11677, 13477, 13591, 13597, 17011, 17393, 17519, 19531, 19913, 30139, 30319, 30971, 31139, 31799, 31991, 37619, 39371, 39419, 39839, 70313, 70373, 70717

Offset: 1

Kevin L. Schwartz and Christian N. K. Anderson, May 03 2013

This sequence (like the emirps) experiences large gaps when the most-significant-digit is {2,4,5,6,8}.

			7177 and 10177 are in the sequence because both are emirps, and both become the emirp 17 upon deletion of their first and last digits.

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

Cf. A210498, A006567, A069686, A210498.

library(gmp); isemirp<-function(x) isprime(x) & (j=paste(rev(unlist(strsplit(as.character(x), split=""))), collapse=""))!=x & isprime(j);
no0<-function(s){ while(substr(s,1,1)=="0" & nchar(s)>1) s=substr(s,2,nchar(s)); s};
i=as.bigz(0); y=as.bigz(rep(0, 100)); len=0;
while(len<100) if(isemirp((i=nextprime(i)))) if(isemirp(as.bigz(no0(substr(i,2,nchar(as.character(i))-1))))) y[(len=len+1)]=i;
as.vector(y)

A235687 Semiprimes which remain semiprimes when the rightmost digit is removed.

Original entry on oeis.org

46, 49, 62, 65, 69, 91, 93, 94, 95, 106, 141, 142, 143, 145, 146, 155, 158, 159, 213, 214, 215, 217, 218, 219, 221, 226, 253, 254, 259, 262, 265, 267, 334, 335, 339, 341, 346, 355, 358, 381, 382, 386, 391, 393, 394, 395, 398, 466, 469, 493, 497, 511, 514

Offset: 1

Colin Barker, Jan 14 2014

			514 is in the sequence because 51 = 3*17.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A235688, A235689.

Cf. A227919, A227916, A069686.

Select[Range[600],PrimeOmega[#]==2==PrimeOmega[FromDigits[ Most[ IntegerDigits[ #]]]]&] (* Harvey P. Dale, Oct 02 2014 *)
Select[Range[600],PrimeOmega[#]==PrimeOmega[Quotient[#,10]]==2&] (* Harvey P. Dale, Mar 18 2023 *)

list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
issemiprime(n) = n>0 && bigomega(n)==2
t=list(1000); for(n=1, #t, if(issemiprime(t[n]\10), print1(t[n],", ")))

A235688 Semiprimes which remain semiprimes when the leftmost digit is removed.

Original entry on oeis.org

14, 26, 34, 39, 46, 49, 69, 74, 86, 94, 106, 115, 121, 122, 133, 134, 146, 155, 158, 169, 177, 185, 187, 194, 206, 209, 214, 215, 221, 226, 235, 249, 262, 265, 274, 287, 291, 295, 309, 314, 321, 326, 334, 335, 339, 346, 355, 358, 362, 365, 377, 382, 386, 391

Offset: 1

Colin Barker, Jan 14 2014

			249 is in the sequence because 49 = 7*7.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A235687, A235689.

Cf. A227919, A227916, A069686.

Select[Range[400],PrimeOmega[#]==PrimeOmega[Mod[#,10^(IntegerLength[ #]-1)]] == 2&] (* Harvey P. Dale, Jun 07 2017 *)

list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
delleft(a) = my(b, c); b=#Str(a); c=a\(10^(b-1)); a-c*(10^(b-1))
issemiprime(n) = n>0 && bigomega(n)==2
t=list(500); for(n=1, #t, if(issemiprime(delleft(t[n])), print1(t[n],", ")))

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

A224953 Number of ways a digit can be appended or prepended to n and form a prime.

Original entry on oeis.org

4, 9, 3, 9, 3, 3, 2, 9, 2, 6, 4, 6, 1, 7, 1, 2, 2, 5, 1, 9, 0, 4, 3, 6, 1, 2, 2, 6, 2, 5, 1, 8, 0, 5, 2, 2, 1, 6, 2, 6, 2, 6, 1, 7, 2, 1, 3, 6, 1, 5, 2, 3, 2, 5, 2, 1, 2, 8, 1, 6, 2, 7, 0, 6, 3, 2, 1, 7, 1, 4, 2, 5, 1, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 7, 0, 3, 1

Offset: 0

Christian N. K. Anderson and Kevin L. Schwartz, Apr 20 2013

The prime number may be formed by adding a digit either before or after n, though only odd numbers can become prime by having digits added before n.
Appending a zero before n produces a prime if and only if n is prime. Conversely, for all prime numbers p, a(p) > 0.
In theory, a maximum of 7 digits could be added before any n, and 3 of the odd digits after n in cases where [10*n, 10*n+9] contains a number that is a factor of 3, 5 and 7 (the three single-digit odd primes). In practice, it appears that all 10 possibilities are never realized. There are 9 possibilities for n = {1, 3, 7, 19}.
The only example of a prime being formed two different ways is for n = 1, which can become 11 if a 1 is appended to either the front or the back. These are naively counted as two distinct alternatives. [This would also be true for n = A002275(A004023(k) - 1) for k > 1 as appending a 1 to either the front or the back forms the k-th repunit prime. - Michael S. Branicky, May 22 2024]
The term a(29587) is the first occurrence of 10. The primes are 29587, 129587, 329587, 429587, 729587, 929587, 295871, 295873, 295877, and 295879. This is the only occurrence of 10 for n < 10^8. - T. D. Noe, Apr 21 2013

			a(0) = 4 because there are 4 ways to concatenate a digit to 0 to produce a prime number: 02, 03, 05, and 07.
a(3) = 9 because a digit can be concatenated to 3 in 9 ways to produce a prime number: 03, 13, 23, 43, 53, 73, 83, 31, and 37.

Christian N. K. Anderson, Table of n, a(n) for n = 0..9999
Christian N. K. Anderson, Primes formed by appending digits to n for n = 0..9999

Cf. A087388, A087389, A087390.
Cf. A069686.
Cf. A075595.
Cr. A002275, A004023.
Index of zeros in this sequence: A124665.

Table[num = IntegerDigits[n]; cnt = 0; Do[If[PrimeQ[FromDigits[Prepend[num, k]]], cnt++], {k, 0, 9}]; Do[If[PrimeQ[FromDigits[Append[num, k]]], cnt++], {k, 0, 9}]; cnt, {n, 0, 86}] (* T. D. Noe, Apr 20 2013 *)

sapply(1:100, function(x) sum(sapply(as.numeric(c(paste0(0:9,x), paste0(x,c(1,3,7,9)))), is_prime  ))) # Christian N. K. Anderson, Apr 30 2024

A235689 Semiprimes which remain semiprimes when the leftmost and rightmost digits are removed.

Original entry on oeis.org

141, 142, 143, 145, 146, 161, 166, 169, 194, 247, 249, 262, 265, 267, 291, 295, 298, 299, 341, 346, 361, 362, 365, 391, 393, 394, 395, 398, 445, 446, 447, 466, 469, 493, 497, 542, 543, 545, 562, 565, 566, 591, 597, 649, 662, 667, 669, 694, 695, 697, 698, 699

Offset: 1

Colin Barker, Jan 14 2014

			169 = 13^2 is in the sequence because 6 = 2*3.

Cf. A235687, A235688.

Cf. A227919, A227916, A069686.

Select[Range[100,700],PrimeOmega[#]==PrimeOmega[FromDigits[ Rest[ Most[ IntegerDigits[ #]]]]] ==2&] (* Harvey P. Dale, Nov 22 2018 *)

list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, listput(v, t*q))); vecsort(Vec(v)) \\ From A001358
delleft(a) = my(b, c); b=#Str(a); c=a\(10^(b-1)); a-c*(10^(b-1))
issemiprime(n) = n>0 && bigomega(n)==2
t=list(700); for(n=1, #t, if(issemiprime(delleft(t[n]\10)), print1(t[n],", ")))

cp's OEIS Frontend

A077358 Duplicate of A069686.

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A077359 Primes whose external digits form a prime. Or primes from which deleting the internal digits leaves a prime.

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Mathematica

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

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PARI

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A225235 Emirps whose internal digits are also an emirp.

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A235687 Semiprimes which remain semiprimes when the rightmost digit is removed.

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A235688 Semiprimes which remain semiprimes when the leftmost digit is removed.

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Mathematica

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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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Mathematica

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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

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A224953 Number of ways a digit can be appended or prepended to n and form a prime.

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