A109093 Fully-transmutable primes: Transmutable primes such that each transmutation is itself a transmutable prime (A108388).

139119131, 193113191, 319339313, 391331393, 913993919, 931991939, 1319999199391, 1913333133931, 3139999399193, 3931111311913, 9193333933139, 9391111911319, 11333911193113, 11999311139119, 33111933391331

Offset: 0

Author

Rick L. Shepherd, Jun 18 2005

Keywords

• base
• hard
• nonn

Comments

See the definitions of "transmutable" and "transmutation" in A108388. Some primes with two distinct digits, namely all terms of A083983, can be considered trivially fully-transmutable. This subsequence of A108388 considers only transmutable primes with more distinct digits. These are primes such that all permutations of assignments of their distinct digits to their shared digit pattern produces primes. (Contrast this with the absolute primes, A003459, where all permutations of the digits themselves produce primes.). Fully-transmutable primes with three distinct digits occur in sets of 3! = 6. Fully-transmutable primes with four distinct digits, if any, would occur in sets of 4! = 24 and would also be a subsequence of A108389.

Examples

			The first six terms share the digit pattern d1 d2 d3 d1 d1 d3 d1 d2 d1. Each of these terms is a (9-digit) prime corresponding to one of the 3! = 6 bijective mappings of {1,3,9} onto {d1,d2,d3}. There are no other such primes with nine or fewer digits.
		

Crossrefs

Cf. A108388 (transmutable primes), A083983 (transmutable primes with two distinct digits), A108389 (transmutable primes with four distinct digits), A003459 (absolute primes), A108387 (doubly-transmutable primes).