A298643 Array A(n, k) read by antidiagonals downwards: k-th base-n non-repunit prime p such that all numbers resulting from switching any two adjacent digits in the base-n representation of p are prime, where k runs over the positive integers, i.e., the offset of k is 1.

11, 191, 2, 223, 5, 2, 227, 7, 3, 2, 2111, 17, 7, 3, 2, 3847, 31, 13, 7, 3, 2, 229631, 41, 23, 11, 5, 3, 2, 246271, 53, 29, 13, 11, 5, 3, 2, 262111, 157, 47, 17, 31, 11, 5, 3, 2, 786431, 229, 53, 19, 47, 13, 7, 5, 3, 2, 1046527, 239, 101, 23, 71, 17, 13, 7, 5

Offset

2,1

Comments

Conjecture: All rows of the array are infinite.

If the above conjecture is false, then this should have keyword "tabf" rather than "tabl".

Row n is a supersequence of the base-n non-repunit absolute primes. For example, row 10 (A107845) is a supersequence of the decimal non-repunit absolute primes (A129338).

Links

Table of n, a(n) for n=2..65.

Example

The base-3 representation of 251 is 100022. Base-3 numbers that can be obtained by switching any two adjacent base-3 digits are 10022 and 100202. These two numbers are 89 and 263, respectively, when converted to decimal, and both 89 and 263 are prime. Since 251 is the 12th number with this property in base 3, A(3, 12) = 251.
Array starts
11, 191, 223, 227, 2111, 3847, 229631, 246271, 262111, 786431, 1046527, 1047551
2,    5,   7,  17,   31,   41,     53,    157,    229,    239,     241,     251
2,    3,   7,  13,   23,   29,     47,     53,    101,    127,     149,     151
2,    3,   7,  11,   13,   17,     19,     23,     43,    131,     281,     311
2,    3,   5,  11,   31,   47,     71,     83,    103,    107,     151,     191
2,    3,   5,  11,   13,   17,     19,     23,     29,     37,      41,      43
2,    3,   5,   7,   13,   29,     31,     41,     43,     47,      59,      61
2,    3,   5,   7,   11,   13,     17,     19,     23,     37,      43,      47
2,    3,   5,   7,   13,   17,     31,     37,     71,     73,      79,      97
2,    3,   5,   7,   13,   17,     19,     23,     29,     31,      37,      43
2,    3,   5,   7,   11,   17,     61,     67,     71,     89,     137,     163

Prog

(PARI) switchdigits(v, pos) = my(vt=v[pos]); v[pos]=v[pos+1]; v[pos+1]=vt; v
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
is(p, base) = my(db=digits(p, base)); if(vecmin(db)==1 && vecmax(db)==1, return(0)); for(k=1, #db-1, my(x=decimal(switchdigits(db, k), base)); if(!ispseudoprime(x), return(0))); 1
array(n, k) = for(x=2, n+1, my(i=0); forprime(p=1, , if(is(p, x), print1(p, ", "); i++); if(i==k, print(""); break)))
array(6, 10) \\ print initial 6 rows and 10 columns of array

Crossrefs

Cf. A107845 (row 10), A129338.

Sequence in context: A215203 A034787 A001408 * A185123 A036936 A002195

Adjacent sequences: A298640 A298641 A298642 * A298644 A298645 A298646

Keyword

nonn,tabl,base

Author

Felix Fröhlich, Jan 24 2018

Status

approved