A068679 -id:A068679 - cp's OEIS frontend

A069246 Primes which yield a prime whenever a 1 is inserted anywhere in them (including at the beginning or end).

3, 7, 13, 31, 103, 109, 151, 181, 193, 367, 571, 601, 613, 811, 1117, 1831, 4519, 6871, 11119, 11317, 11467, 13171, 16141, 17167, 18211, 18457, 27241, 38917, 55381, 71317, 81199, 81931, 86743, 114031, 139861, 141667, 151687, 179203, 200191

Offset: 1

N. J. A. Sloane, Apr 14 2002

Giovanni Resta, Table of n, a(n) for n = 1..270 (terms 1..77 from Vincenzo Librandi, terms 78..145 from Zak Seidov)

Prime subsequence of A068679.

Cf. A215417, A215419 - A215421.

[p: p in PrimesUpTo(210000) | forall{m: t in [0..#Intseq(p)] | IsPrime(m) where m is (Floor(p/10^t)*10+1)*10^t+p mod 10^t}]; // Bruno Berselli, Sep 03 2012

d[n_]:=IntegerDigits[n]; ins[n_]:=FromDigits/@Table[Insert[d[n],1,k],{k,Length[d[n]]+1}]; Select[Prime[Range[20000]],And@@PrimeQ/@ins[#] &] (* Jayanta Basu, May 20 2013 *)

More terms from Vladeta Jovovic, Apr 16 2002

A164329 Numbers which yield a prime whenever a zero is inserted between any two digits.

Original entry on oeis.org

11, 13, 17, 19, 37, 41, 49, 53, 59, 61, 67, 71, 79, 89, 97, 109, 113, 119, 121, 131, 133, 149, 161, 169, 191, 197, 203, 227, 239, 253, 269, 281, 283, 299, 301, 319, 323, 337, 367, 379, 383, 401, 403, 407, 421, 449, 457, 473, 493, 499, 503, 509, 511, 539, 551

Offset: 1

Farideh Firoozbakht, Sep 22 2009

Single-digit numbers 0, ..., 9 seem to be excluded but would satisfy the condition voidly. - M. F. Hasler, May 10 2018

			998471 is in the sequence because all the five numbers 9098471, 9908471, 9980471, 9984071 and 9984701 are primes.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A216169 (subset of composite terms), A215417 (subset of primes), A159236 (0 is inserted between all digits).
Cf. A068679 (1 is prefixed, appended or inserted anywhere), A069246 (primes among these), A068673 (1 is prefixed, or appended).
Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these).
Cf. A069832 (7 is prefixed, appended or inserted anywhere), A215420 (primes among these), A068677 (7 is prefixed or appended).
Cf. A069833 (9 is prefixed, appended or inserted anywhere), A215421 (primes among these).
Cf. A158232 (13 is prefixed or appended).

f[n_]:=(r=IntegerDigits[n];l=Length[r];For[k=2,PrimeQ[FromDigits[Insert
[r,0,k]]],k++ ];If[k==l+1,n,0]);Select[Range[11,560],f[ # ]>0&]

is(n, L=logint(n+!n, 10)+1, P)={!for(k=1, L-1, isprime([10*P=10^(L-k),1]*divrem(n, P))||return) && n>9} \\ M. F. Hasler, May 10 2018

Erroneous comment and cross-references deleted by M. F. Hasler, May 10 2018

A216169 Composite numbers > 9 which yield a prime whenever a 0 is inserted between any two digits.

Original entry on oeis.org

49, 119, 121, 133, 161, 169, 203, 253, 299, 301, 319, 323, 403, 407, 473, 493, 511, 539, 551, 581, 611, 667, 679, 713, 869, 901, 913, 943, 1007, 1067, 1079, 1099, 1211, 1273, 1691, 1729, 1799, 1909, 2021, 2047, 2101, 2117, 2359, 2407, 2533, 2717, 2759, 2899

Offset: 1

Paolo P. Lava, Sep 03 2012

			2359 is not prime but 23509, 23059 and 20359 are all primes.

Paolo P. Lava and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1500 terms from Paolo P. Lava)

Subset of composite numbers in A164329. - M. F. Hasler, May 10 2018

Cf. A068673, A068674, A068677, A068679, A069246, A215417, A215419-A215421, A216165-A216168.

A216169:=proc(q,x)
local a,b,c,i,n,ok;
for n from 10 to q do
if not isprime(n) then
  a:=n; b:=0; while a>0 do b:=b+1; a:=trunc(a/10); od; a:=n; ok:=1;
  for i from 1 to b-1 do c:=a+9*10^i*trunc(a/10^i)+10^i*x;
    if not isprime(c) then ok:=0; break; fi; od;
  if ok=1 then print(n); fi;
fi; od; end: A216169(1000,0);

Select[Range[10,3000],CompositeQ[#]&&AllTrue[Table[FromDigits[ Insert[ IntegerDigits[ #],0,n]],{n,2,IntegerLength[#]}],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 13 2018 *)

is(n, L=logint(n+!n, 10)+1, P)={!isprime(n) && !for(k=1, L-1, isprime([10*P=10^(L-k),1]*divrem(n, P))||return) && n>9} \\ M. F. Hasler, May 10 2018

Name edited by M. F. Hasler, May 10 2018

A158594 Numbers which yield a prime whenever a 3 is prefixed, appended or inserted.

Original entry on oeis.org

1, 7, 11, 17, 31, 37, 73, 121, 271, 331, 343, 359, 361, 373, 533, 637, 673, 733, 793, 889, 943, 1033, 1183, 2297, 3013, 3119, 3223, 3353, 3403, 3461, 3757, 3827, 3893, 3923, 4313, 4543, 4963, 5323, 5381, 5419, 6073, 6353, 8653, 9103, 9887, 10423, 14257

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 22 2009

1) It is conjectured that sequences of this type are infinite; also that an infinite number of primes is included.
2) Necessarily a(n) has end digit 1,3,7 or 9.
3) Sum of digits of a(n) has form 3k-1 or 3k+1.
4) Sequence is part of A068674 a(n) n=1,...,30: first 14 primes: 7, 11, 17, 31, 37, 73, 271, 331, 359, 373, 673, 733, 2297, 3461.
5) Note the "world record" 2297: smallest prime which yields five other primes 32297, 23297, 22397, 22937, 22973.

			109 is not a term: 3109, 1039, 1093 are primes, but 1309 = 7 * 11 * 17.
121 is a term: 3121 (3 prefixed), 1213 (3 appended), 1321 and 1231 (3 inserted) are primes.

Marcus Du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins. 2004
Bryan Bunch, Kingdom of Infinite Number: A Field Guide, W.H. Freeman & Company, 2001

Jinyuan Wang, Table of n, a(n) for n = 1..500

Cf. A068674, Numbers which yield primes when a 3 is prefixed or appended.
Cf. A068679, Numbers which yield a prime whenever a 1 is inserted anywhere in them (including at the beginning or end).
Cf. A158232, Numbers which yield primes when "13" is prefixed or appended.

Lton := proc(L) local i ; add(op(i,L)*10^(i-1),i=1..nops(L) ) ; end: isA158594 := proc(n) local dgs,i,p; dgs := convert(n,base,10) ; p := [3,op(dgs)] ; if not isprime(Lton(p)) then RETURN(false) ; fi; p := [op(dgs),3] ; if not isprime(Lton(p)) then RETURN(false) ; fi; for i from 1 to nops(dgs)-1 do p := [op(1..i,dgs),3,op(i+1..nops(dgs),dgs)] ; if not isprime(Lton(p)) then RETURN(false) ; fi; od: RETURN(true) ; end: for n from 1 to 25000 do if isA158594(n) then printf("%d,",n) ; fi; od: # R. J. Mathar, Mar 26 2009

isok(n)={i=#digits(n);m=1;k=0;while(kJinyuan Wang, Feb 02 2019

Corrected and extended by Chris K. Caldwell and R. J. Mathar, Mar 26 2009

A069832 Prefixing, suffixing or inserting a 7 in the number anywhere gives a prime.

Original entry on oeis.org

1, 3, 9, 19, 27, 33, 39, 57, 87, 97, 159, 177, 187, 433, 487, 541, 603, 691, 717, 753, 757, 853, 949, 1257, 1471, 1707, 2277, 2367, 2617, 2953, 4317, 4507, 4623, 4779, 4797, 5773, 6481, 6757, 6777, 7017, 7351, 7417, 7471, 7479, 7747, 7797, 7813, 7977

Offset: 1

Amarnath Murthy, Apr 14 2002

Giovanni Resta, Table of n, a(n) for n = 1..2399 (terms < 10^13, first 201 terms from Harvey P. Dale)

Cf. A068674, A068679, A069833.

psi7Q[n_]:=Module[{idn=IntegerDigits[n]},And@@PrimeQ[FromDigits/@ Table[ Insert[ idn,7,i],{i,Length[idn]+1}]]]; Select[Range[8000],psi7Q] (* Harvey P. Dale, Sep 18 2012 *)

More terms from Vladeta Jovovic, Apr 16 2002

Offset changed to 1 by Giovanni Resta, Oct 14 2019

A069833 Prefixing, suffixing or inserting a 9 in the number anywhere gives a prime.

Original entry on oeis.org

7, 19, 37, 41, 91, 199, 209, 239, 311, 539, 587, 661, 749, 923, 931, 941, 967, 1009, 1079, 1139, 1997, 2717, 2959, 3971, 3979, 4559, 4993, 4999, 5393, 5629, 5651, 6401, 6739, 6911, 8213, 8491, 8939, 9109, 9397, 9607, 9679, 9829, 11089, 11227, 13943

Offset: 1

Amarnath Murthy, Apr 14 2002

Giovanni Resta, Table of n, a(n) for n = 1..2128 (terms < 10^13, first 1175 terms from Chai Wah Wu)

Cf. A215421 (subsequence of primes).
Cf. A068679 (1 is prefixed, appended or inserted anywhere), A069246 (primes among these), A068673 (1 is prefixed, or appended).
Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these).
Cf. A069832 (7 is prefixed, appended or inserted anywhere), A215420 (primes among these), A068677 (7 is prefixed or appended).
Cf. A158232 (13 is prefixed or appended).
Cf. A164329 (0 is inserted), A216169 (subset of composite terms), A215417 (subset of primes), A159236 (0 is inserted between all digits).

is(n,L=logint(n+!n,10)+1,d,P)={!for(k=0,L,isprime((d=divrem(n,P=10^(L-k)))[2]+(10*d[1]+9)*P)||return)} \\ M. F. Hasler, May 10 2018

More terms from Vladeta Jovovic, Apr 16 2002

Corrected offset by Chai Wah Wu, Oct 10 2019

A304244 Numbers that yield a prime when prime(k) is inserted after the k-th digit, for any k >= 1, k < number of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 23, 27, 29, 41, 51, 53, 77, 81, 83, 87, 89, 99, 101, 149, 191, 239, 251, 287, 317, 353, 359, 419, 473, 497, 509, 527, 533, 611, 677, 743, 797, 809, 821, 887, 893, 941, 983, 1037, 1043, 1277, 1421, 1841, 1853, 1973, 1979, 2543

Offset: 1

M. F. Hasler, May 21 2018

The primes to insert are: 2 (after the first digit), 3 (after the second digit, if there are at least three), etc.
Inspired by A304243 and analog sequences given in cross-references.
The sequence is finite: if insertion of 3 after the second digit yields a prime, then the sum of digits must be congruent to 1 or 2 (mod 3). However, insertion of 2 after the first digit also must yield a prime, so only the second case is possible. But then, insertion of a digit 7 cannot yield a prime, so no term can have 5 digits or more. (Sequence A304243 circumvents this restriction by excluding 3 from the primes to insert, but it is still finite for a similar reason occurring later.)

			The 1-digit numbers 0..9 are included since the condition is voidly satisfied: Nothing can be inserted, therefore each of the resulting numbers is prime.
17 is in the sequence because 127 is prime.
101 is in the sequence because 1201 and 1031 are prime.

M. F. Hasler, Table of n, a(n) for n = 1..100 (complete sequence).

Cf. A304243 (2 is prefixed or prime(k+2) is inserted after the k-th digit).
Cf. A068679 (1 is prefixed, appended or inserted anywhere), A069246 (primes among these), A068673 (1 is prefixed, or appended).
Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these).
Cf. A069832 (7 is prefixed, appended or inserted anywhere), A215420 (primes among these), A068677 (7 is prefixed or appended).
Cf. A069833 (9 is prefixed, appended or inserted anywhere), A215421 (primes among these).
Cf. A158232 (13 is prefixed or appended).
Cf. A164329 (0 is inserted), A216169 (subset of composite terms), A215417 (subset of primes), A159236 (0 is inserted between all digits).

is(n,L=logint(n+!n,10)+1,d,p,P)={!for(k=1,L-1, isprime((d=divrem(n,P=10^(L-k)))[2]+(10^logint(10*p=prime(k),10)*d[1]+p)*P)|| return)}

A216165 Composite numbers and 1 which yield a prime whenever a 1 is inserted anywhere in them, including at the beginning or end.

Original entry on oeis.org

1, 49, 63, 81, 91, 99, 117, 123, 213, 231, 279, 319, 427, 459, 621, 697, 721, 801, 951, 987, 1113, 1131, 1261, 1821, 1939, 2101, 2149, 2211, 2517, 2611, 3151, 3219, 4011, 4411, 4887, 5031, 5361, 6231, 6487, 7011, 7209, 8671, 9141, 9801, 10051, 10161, 10281

Offset: 1

Paolo P. Lava, Sep 03 2012

			7209 is not prime but 72091, 72019, 72109, 71209 and 17209 are all primes.

Paolo P. Lava, Table of n, a(n) for n = 1..250

Cf. A068673, A068674, A068677, A068679, A069246, A215417, A215419-A215421, A216166-A216169.

[n: n in [1..11000] | not IsPrime(n) and forall{m: t in [0..#Intseq(n)] | IsPrime(m) where m is (Floor(n/10^t)*10+1)*10^t+n mod 10^t}]; // Bruno Berselli, Sep 03 2012

with(numtheory);
A216165:=proc(q,x)
local a,b,c,i,n,ok;
for n from 1 to q do
if not isprime(n) then
  a:=n; b:=0; while a>0 do b:=b+1; a:=trunc(a/10); od; a:=n; ok:=1;
  for i from 0 to b do c:=a+9*10^i*trunc(a/10^i)+10^i*x;
    if not isprime(c) then ok:=0; break; fi;
  od;
  if ok=1 then print(n); fi;
fi;
od; end:
A216165(1000,1);

Join[{1},Select[Range[11000],CompositeQ[#]&&AllTrue[FromDigits/@ Table[ Insert[ IntegerDigits[#],1,i],{i,IntegerLength[#]+1}],PrimeQ]&]] (* Harvey P. Dale, Mar 24 2017 *) (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 24 2017 *)

A304243 Numbers that yield a prime when prime(k+2) is inserted after the k-th digit (or prime(1) = 2 before the 1st digit for k=0), for 0 <= k <= number of digits.

Original entry on oeis.org

27, 33, 39, 57, 93, 333, 3747, 5073, 5997, 7239, 10053, 22419, 349731, 425991, 714807, 1719279, 81453303, 406253439, 481683189, 886662423, 2653294371

Offset: 1

M. F. Hasler, May 10 2018

The primes to insert are 2 (in front) or 5, 7, 11, 13, ... after the number's first, second, third, ... digit. So there cannot be any 1 digit solution because if 5 is appended this cannot yield a prime. One can show that the terms cannot have more than 21 digits.

The prime 3 is excluded from the strings to insert, because else no term could have more than 2 digits: to be prime with 2 prefixed or with 3 inserted, the number must be congruent to 2 (mod 3), so it cannot be prime with 7 appended or inserted. See also the Rivera link and A304244.

			a(1) = 27 because 2|27 = 227, 2|5|7 = 257 and 27|7 = 277 are all prime.
Similarly for a(6) = 333, because 2333, 3533, 3373 and 33311 are all prime.

Carlos Rivera, Puzzle 791. Interesting consecutive primes, The Prime Puzzles & Problems Connection.

Cf. A304244 (prime(k) is inserted after the k-th digit), A304245 (2 is inserted after the first digit, or prime(k+1) is inserted after the k-th digit for k > 1).
Cf. A068679 (1 is prefixed, appended or inserted anywhere), A069246 (primes among these), A068673 (1 is prefixed, or appended).
Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these).
Cf. A069832 (7 is prefixed, appended or inserted anywhere), A215420 (primes among these), A068677 (7 is prefixed or appended).
Cf. A069833 (9 is prefixed, appended or inserted anywhere), A215421 (primes among these).
Cf. A158232 (13 is prefixed or appended).
Cf. A164329 (0 is inserted), A216169 (subset of composite terms), A215417 (subset of primes), A159236 (0 is inserted between all digits).

is(n,L=logint(n+!n,10)+1,d,p,P)={isprime(n+2*10^L) && !for(k=1,L, isprime((d=divrem(n,P=10^(L-k)))[2]+(10^logint(10*p=prime(2+k),10)*d[1]+p)*P)|| return)}

A304245 Numbers that yield a prime when '2' is inserted between the first and second digit, or prime(k+1) is inserted after the k-th digit for any k > 1, k < number of digits.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 17, 23, 27, 29, 41, 51, 53, 77, 81, 83, 87, 89, 99, 101, 113, 129, 149, 159, 179, 191, 203, 213, 221, 237, 251, 267, 269, 273, 281, 287, 293, 297, 321, 329, 357, 359, 401, 417, 419, 429, 441, 461, 471, 497, 509, 531, 561, 581, 603, 611, 663, 669, 687, 699, 707, 711

Offset: 1

M. F. Hasler, May 21 2018

The primes to be inserted are: 2 between 1st and 2nd digit, or 5 between 2nd and 3rd digit, or 7 between 3rd and 4th digit, etc.
The prime 3 is excluded because it would restrict the terms to have no more than 4 digits; see A304244 and the Rivera link in A304243.
The two terms 27 and 87 are the only numbers (with more than one digit) for which 2, 5 or 7 can be inserted between any two digits to yield a prime: all of 227, 257, 277, 827, 857 an 877 are prime. There is no other such number with more than 2 digits.

			The 1-digit numbers 0..9 are included since the condition is voidly satisfied: nothing can be inserted, therefore each of the resulting numbers is prime.
17 is in the sequence because 127 is prime.
101 is in the sequence because 1201 and 1051 are prime.

M. F. Hasler, Table of n, a(n) for n = 1..348 (all terms < 10^7).

Cf. A304243 (2 is prefixed or prime(k+2) is inserted after the k-th digit), A304244 (prime(k) is inserted after the k-th digit) .
Cf. A068679 (1 is prefixed, appended or inserted anywhere), A069246 (primes among these), A068673 (1 is prefixed, or appended).
Cf. A158594 (3 is prefixed, appended or inserted anywhere), A215419 (primes among these).
Cf. A069832 (7 is prefixed, appended or inserted anywhere), A215420 (primes among these), A068677 (7 is prefixed or appended).
Cf. A069833 (9 is prefixed, appended or inserted anywhere), A215421 (primes among these).
Cf. A158232 (13 is prefixed or appended).
Cf. A164329 (0 is inserted), A216169 (subset of composite terms), A215417 (subset of primes), A159236 (0 is inserted between all digits).

is(n,L=logint(n+!n,10)+1,d,p,P)={!for(k=1,L-1, isprime((d=divrem(n,P=10^(L-k)))[2]+(10^logint(10*p=prime(k+(k>1)),10)*d[1]+p)*P)|| return)}

cp's OEIS Frontend

A069246 Primes which yield a prime whenever a 1 is inserted anywhere in them (including at the beginning or end).

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A164329 Numbers which yield a prime whenever a zero is inserted between any two digits.

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Mathematica

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A216169 Composite numbers > 9 which yield a prime whenever a 0 is inserted between any two digits.

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Maple

Mathematica

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A158594 Numbers which yield a prime whenever a 3 is prefixed, appended or inserted.

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A069832 Prefixing, suffixing or inserting a 7 in the number anywhere gives a prime.

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A069833 Prefixing, suffixing or inserting a 9 in the number anywhere gives a prime.

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A304244 Numbers that yield a prime when prime(k) is inserted after the k-th digit, for any k >= 1, k < number of digits.

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A216165 Composite numbers and 1 which yield a prime whenever a 1 is inserted anywhere in them, including at the beginning or end.

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