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A111057 Minimal set in the sense of A071062 of prime-strings in base 12 for primes of the form 4n+1. 0
5, 13, 37, 73, 97, 109, 313, 337, 373, 409, 421, 577, 601, 661, 709, 1009, 1033, 1093, 1129, 1489, 1609, 1669, 3457, 7537, 12721, 13729, 17401, 17569, 19009, 19141, 20593, 20641, 165877, 208501, 221173, 225781, 226201, 226357, 228793, 246817, 246937, 248821, 1097113, 2695813, 2735269, 2736997, 2737129, 32555521, 388177921 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Maple worksheet available upon request. Here is the minimal set of primes of the form 4n+1 in base 12, where X is ten and E is eleven. 5, 11, 31, 61, 81, 91, 221, 241, 271, 2X1, 2E1, 401, 421, 471, 4E1, 701, 721, 771, 7X1, X41, E21, E71, 2001, 4441, 7441, 7E41, X0X1, X201, E001, E0E1, EE01, EE41, 7EEE1, X07E1, X7EE1, XX7E1, XXXX1, XXEE1, E04X1, EXX01, EXXX1, EEEE1, 44XXX1, XX00E1, XEXXE1, XEEXE1, XEEEX1, XXX0001, XX000001. Note that the last prime in the set is the same as the last prime in the minimal set of all primes. See A110600. I am checking certain ranges past this last prime but flow-charting the possibilities leads me to believe I have found the full sequence. The minimal set of prime strings in base 12 for primes of the form 4n+3 is [3, 7, E] since every 4n+3 prime greater than 3 ends in either 7 or E.

LINKS

Table of n, a(n) for n=1..49.

J. Shallit, Minimal primes, J. Recreational Mathematics, vol. 30.2, pp. 113-117, 1999-2000.

EXAMPLE

a(11)=421="2E1" since the pattern "*2*E*1*" does not occur in any previously found prime of the form 4n+1. Assuming all previous members of the list have been similarly recursively constructed, then "401" (577 in base 10) is the next prime in the list. The basic rule is: if no substring of p matches any previously found prime, add p to the list. The basic theorem of minimal sets says that this process will terminate, that is, the minimal set is always finite.

CROSSREFS

Cf. A071062, A071070, A110600, A110615.

Sequence in context: A137815 A089523 A058507 * A019268 A083413 A232879

Adjacent sequences:  A111054 A111055 A111056 * A111058 A111059 A111060

KEYWORD

base,fini,full,nonn,uned

AUTHOR

Walter Kehowski, Oct 06 2005

STATUS

approved

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Last modified February 19 04:09 EST 2020. Contains 332037 sequences. (Running on oeis4.)