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A179069 Array read by antidiagonals: row b lists the base-b analog of the base-10 sequence 1, 12, 123, ..., 123456789, 12345678910, ... (A007908).

%I A179069 #13 Mar 22 2021 06:28:03
%S A179069 1,1,3,1,6,6,1,5,27,10,1,6,48,220,15,1,7,27,436,1765,21,1,8,38,436,
%T A179069 3939,14126,28,1,9,51,194,6981,35367,113015,36,1,10,66,310,4855,
%U A179069 111702,318310,1808248
%N A179069 Array read by antidiagonals: row b lists the base-b analog of the base-10 sequence 1, 12, 123, ..., 123456789, 12345678910, ... (A007908).
%C A179069 The numbers in the row b of the array are constructed in base b, but are converted to base 10 for display here.
%C A179069 R. K. Guy writes [UPINT, A3, pp. 9-10]: Selfridge asked if the sequence (in decimal notation) 1, 12, 123, 1234, ... [A007908] ... contains infinitely many primes.... The question can be asked for other scales of notation. There are (trivially) an infinite number of primes in the n=2 column, as that converges to k+2. In the n=3 column, the first prime is A[3,8] = 83 (base 10) = 123 (base 8). In the n=7 column, the first prime is A[8,7] = 342391 (base 10) = 1234567 (base 8). This can be continued to bases higher than 10, where A, B, C, ... are conventionally used as numerals. For example, A[12,5] = 12345 (base 12) = 24677 (base 10) is prime, as is A[12,17] = 656998737209054448298001 (base 10). A[13,3] = 227 (base 10) = 123 (base 13) is prime. Similarly, to pick the 9th row but go further than the table shown here, A[9,14] = 1709671414851143033 (base 10) is prime. Existing OEIS sequences stop at A048447, the concatenation of first n numbers in base 16.
%D A179069 Richard K. Guy, Unsolved Problems In Number Theory, 2nd Edn., Springer Verlag, 1994.
%F A179069 A[b,n] = n-th integer concatenated from consecutive integers in base b.
%e A179069 The array begins:
%e A179069 ====================================================================
%e A179069 ....|n=1.|.n=2.|.n=3.|.n=4.|..n=5.|..n=6.|...n=7.|.....n=8.|.in OEIS
%e A179069 b=1.|.1..|...3.|...6.|..10.|...15.|...21.|....28.|......36.|.A000217
%e A179069 b=2.|.1..|...6.|..27.|.220.|.1765.|.14126|.113015|.1808248.|.A047778
%e A179069 b=3.|.1..|...5.|..48.|.436.|.3929.|.35367|.318310|.2864798.|.A048435
%e A179069 b=4.|.1..|...6.|..27.|.436.|.6981.|111702|1787239|28595832.|.A048436
%e A179069 b=5.|.1..|...7.|..38.|.194.|.4855.|121381|3034532|75863308.|.A048437
%e A179069 b=6.|.1..|...8.|..51.|.310.|.1865.|.67146|2417263|87021476.|.A048438
%e A179069 b=7.|.1..|...9.|..66.|.466.|.3267.|.22875|1120882|54923226.|.A048439
%e A179069 b=8.|.1..|..10.|..83.|.668.|.5349.|.42798|.342391|21913032.|.A048440
%e A179069 ...
%e A179069 b=10|.1..|..12.|.123.|1234.|12345.|123456|1234567|12345678.|.A007908
%e A179069 =====================================================================
%Y A179069 Cf. A000217, A007908, A033307, A047778, A048435, A048436, A048437, A048438, A048439, A048440, A048441, A048442, A048443, A048444, A048445, A048446, A048447.
%K A179069 easy,nonn,tabl,base
%O A179069 1,3
%A A179069 _Jonathan Vos Post_, Jun 27 2010
%E A179069 Should be revised to start with base 2, rather than the ill-defined "base 1". - _N. J. A. Sloane_, Jul 05 2010