cp's OEIS Frontend

A098103 Consider the succession of single digits of the primes (A000040): 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 ... (A033308). This sequence is the lexicographically earliest derangement of A000040 that produces the same succession of digits.

%I A098103 #30 Jun 09 2016 05:29:38
%S A098103 23,5,7,11,13,17,19,2,3,293,137,41,43,47,53,59,61,67,71,73,79,83,89,
%T A098103 97,101,103,107,109,113,127,131,1371391491511,571,631,67173179181191,
%U A098103 193,197,199,211,223,227,229,233,239,241,251,257,263,269,271,277,281,283
%N A098103 Consider the succession of single digits of the primes (A000040): 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 ... (A033308). This sequence is the lexicographically earliest derangement of A000040 that produces the same succession of digits.
%C A098103 Derangement here means a(n) != A000040(n) for all n.
%C A098103 Original name: "Write each prime number >0 on a single label. Put the labels in numerical order to form an infinite sequence L. Consider the succession of single digits of L: 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 3 7 4 1 4 3 4 7 5 3 5 9 6 1 6 7 7 1 7 3 7 9... (see A033308). The sequence S gives a rearrangement of the labels that reproduces the same succession of digits, subject to the constraints that a label of L cannot represent itself, and the smallest label must be used that does not lead to a contradiction."
%C A098103 This could be roughly rephrased like this: "Rewrite in the most economical way the 'prime numbers pattern' using only prime numbers, but rearranged. Do not use any prime more than once."
%C A098103 a(180) has over 1000 digits. - _Danny Rorabaugh_, Nov 29 2015
%H A098103 Danny Rorabaugh, <a href="/A098103/b098103.txt">Table of n, a(n) for n = 1..179</a>
%H A098103 Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/Sloane01.htm">Sequence re-writing</a>
%H A098103 Eric Angelini, <a href="/A098103/a098103.htm">Sequence re-writing</a> [Cached copy, with permission]
%e A098103 We must begin with "2,3,5,7,11,..." and we cannot have the first term be 2, the first prime, so the smallest available prime is 23.
%t A098103 f[lst_List, k_] := Block[{L = lst, g, a = {}, m = 0}, g[] := {Set[m, First@ FromDigits@ Append[IntegerDigits@ m, First@ #]], Set[L, Last@ #]} &@ TakeDrop[L, 1]; Do[g[]; While[Or[m == Prime[Length@ a + 1], ! PrimeQ@ m, MemberQ[a, m]], g[]]; AppendTo[a, m]; m = 0, {k}]; a]; f[Flatten@ Map[IntegerDigits, Prime@ Range@ 120], 53] (* _Michael De Vlieger_, Nov 29 2015, Version 10.2 *)
%o A098103 (Sage)
%o A098103 def A098103(n):
%o A098103   Pr, p, s, A, i = Primes(), 2, "", [], 1
%o A098103   while len(A)<n:
%o A098103     while len(s)<=i: s, p = s+str(p), next_prime(p)
%o A098103     q = int(s[:i])
%o A098103     if s[i]!="0" and is_prime(q) and Pr.unrank(len(A))!=q and (q not in A):
%o A098103       A.append(q)
%o A098103       s, i = s[i:], 1
%o A098103     else: i += 1
%o A098103   return A
%o A098103 A098103(179) # _Danny Rorabaugh_, Nov 29 2015
%Y A098103 For other sequences of this type, cf. A098067.
%K A098103 base,nice,nonn
%O A098103 1,1
%A A098103 _Eric Angelini_, Sep 22 2004
%E A098103 Name, Comments, and Example edited by _Danny Rorabaugh_, Nov 28 2015
%E A098103 Corrected and extended by _Danny Rorabaugh_, Nov 29 2015