A176316 -id:A176316 - cp's OEIS frontend

A176781 Smallest prime prime(i) such that concatenation 2//0_(n)//prime(i) is prime.

3, 11, 3, 17, 3, 3, 3, 11, 89, 41, 257, 3, 29, 131, 353, 3, 3, 11, 89, 521, 257, 3, 17, 3, 467, 89, 149, 17, 71, 47, 293, 17, 191, 47, 3, 41, 23, 11, 401, 41, 443, 41, 293, 479, 311, 23, 587, 41, 1289, 1013, 29, 41, 59, 293, 1031, 17, 23, 17, 347, 401, 599, 11, 227, 827, 401

Offset: 0

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 26 2010

We search for the prime such that the first prime (=2) concatenated with n zeros and concatenated with that prime is again a prime number.
If p = prime(i) is a d(i)-digit prime: q = 2 * 10^(n+d(i)) + p has to be prime.
Necessarily prime(i) is congruent to 2 (mod 3).
It is conjectured that prime(i) = 3 occurs infinitely often: at n= 0, 2, 4, 5, 6, 11, 15, 16, 21, 23, 34, 114, 119,...

			n = 0: 2//3 = 23 = prime(9), 3 = prime(2) is first term
n = 1: 2//0//11 = 2011 = prime(305), 11 = prime(5) is 2nd term
n = 2: 2//00//3 = 2003 = prime(304), 3 = prime(2) is 3rd term

E. I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig/Jena/ Berlin 1982

Cf. A164968, A173291, A176316

Offset corrected and sequence extended by R. J. Mathar, Apr 28 2010

A176833 Smallest prime p = prime(i) such that concatenation q(i) = 13//0_(k)//prime(i) (k = 0, 1, 2, ...) is prime.

Original entry on oeis.org

7, 3, 3, 3, 151, 61, 7, 3, 19, 3, 109, 109, 19, 19, 37, 409, 109, 97, 61, 19, 73, 109, 139, 139, 619, 31, 127, 31, 193, 3, 43, 19, 337, 7, 73, 367, 109, 373, 139, 139

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Apr 27 2010

See comments in A176781
Necessarily p = 3 or p of form 3 * n + 1
In recreational mathematics some authors call a prime that is composed of mostly naughts, i.e. zeros, a naughty prime

			q(0) = 13//7 = 137 = prime(33), 7 = prime(4) is 1st term
q(1) = 13//0//3 = 1303 = prime(213), 3 = prime(2) is 2nd term
q(26) = 13000000000000000000000000031 is a palindromic prime

A164968, A173291, A176316, A176781

A176955 Primes p such that q=3//p, r=p//3, R(q) and R(r) are primes.

Original entry on oeis.org

7, 11, 37, 73, 191, 373, 719, 929, 1033, 1193, 3301, 3461, 3911, 3931, 10223, 10771, 12071, 12451, 13669, 13931

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 29 2010

q and r are emirps (see A006567) as R(q) and R(r) are requested to be primes too
3 = prime(2) is first prime for such a "construction" with prefixed/appended number and reversals
Necessarily FSD (First Significant Digit) of p is 1, 3, 7 or 9
If such a p is a palindromic prime, i.e. p = R(p), then q = R(r) and r = R(q):
Proof: q = 3//p = 3//R(p) = R(p//3) = R(r), r = p//3 = R(p)//3 = R(3//p) = R(q).
If such a p is an emirp, i.e. R(p) also prime, then R(p) is also term of sequence:
Proof: 3//R(p) = R(p//3) = R(r), R(p)//3 = R(3//p) = R(q), R(3//R(p)) = p//3 = r, R(R(p)//3) = 3//p = q.
List of (p: q, r, R(q), R(r))
(7=palprime(4): 37, 73, 73, 37), (11=palprime(5): 311, 113, 113, 311),
(37=emirp(4): 337, 373, 733, 373), (73=emirp(6): 373, 733, 373, 337),
(191=palprime(10): 3191, 1913, 1913, 3191), (373=palprime(13): 3373, 3733, 3733, 3373),
(719=prime(128): 3719, 7193, 9173, 3917), (929=palprime(20): 3929, 9293, 9293, 3929),
(1033=emirp(40): 31033, 10333, 33013, 33301), (1193=emirp(50): 31193, 11933, 39113, 33911),
(3301=emirp(115): 33301, 33013, 10333, 31033), (3461=prime(484): 33461, 34613, 16433, 31643),
(3911=emirp(145): 33911, 39113, 11933, 31193), (3931=prime(546): 33931, 39313, 13933, 31393),
(10223=prime(1254): 310223, 102233, 322013, 332201), (10771=prime(1312): 310771, 107713, 177013, 317701),
(12071=emirp(326): 312071, 120713, 170213, 317021), (12451=prime(1486): 312451, 124513, 154213, 315421),
(13669=prime(1614): 313669, 136693, 966313, 396631), (13931=palprime(31): 313931, 139313, 139313, 313931)

			q=3//7=37=prime(12), r=7//3=73=prime(21), R(q)=r, R(r)=q, p=7=prime(4) is 1st term
q=3//719=3719=prime(519), r=719//3=7193=prime(919), R(q)=9173=prime(1137), R(r)=3917=prime(452),
p=719=prime(2^7) is 7th term and the first where q, r, R(q), R(r) are four different primes

Peter Bundschuh: Einfuehrung in die Zahlentheorie, 6. Auflage, Springer, Berlin, 2008
Martin Gardner: Die magischen Zahlen des Dr. Matrix , Wolfgang Krueger Verlag FrankfurtMain, 1987

Cf. A000040, A006567, A176316, A176601, A176838

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A176781 Smallest prime prime(i) such that concatenation 2//0_(n)//prime(i) is prime.

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A176833 Smallest prime p = prime(i) such that concatenation q(i) = 13//0_(k)//prime(i) (k = 0, 1, 2, ...) is prime.

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A176955 Primes p such that q=3//p, r=p//3, R(q) and R(r) are primes.

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