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

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, 3939, 14126, 28, 1, 9, 51, 194, 6981, 35367, 113015, 36, 1, 10, 66, 310, 4855, 111702, 318310, 1808248

Offset

1,3

Comments

The numbers in the row b of the array are constructed in base b, but are converted to base 10 for display here.

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.

References

Richard K. Guy, Unsolved Problems In Number Theory, 2nd Edn., Springer Verlag, 1994.

Links

Table of n, a(n) for n=1..44.

Formula

A[b,n] = n-th integer concatenated from consecutive integers in base b.

Example

The array begins:
====================================================================
....|n=1.|.n=2.|.n=3.|.n=4.|..n=5.|..n=6.|...n=7.|.....n=8.|.in OEIS
b=1.|.1..|...3.|...6.|..10.|...15.|...21.|....28.|......36.|.A000217
b=2.|.1..|...6.|..27.|.220.|.1765.|.14126|.113015|.1808248.|.A047778
b=3.|.1..|...5.|..48.|.436.|.3929.|.35367|.318310|.2864798.|.A048435
b=4.|.1..|...6.|..27.|.436.|.6981.|111702|1787239|28595832.|.A048436
b=5.|.1..|...7.|..38.|.194.|.4855.|121381|3034532|75863308.|.A048437
b=6.|.1..|...8.|..51.|.310.|.1865.|.67146|2417263|87021476.|.A048438
b=7.|.1..|...9.|..66.|.466.|.3267.|.22875|1120882|54923226.|.A048439
b=8.|.1..|..10.|..83.|.668.|.5349.|.42798|.342391|21913032.|.A048440
...
b=10|.1..|..12.|.123.|1234.|12345.|123456|1234567|12345678.|.A007908
=====================================================================

Crossrefs

Cf. A000217, A007908, A033307, A047778, A048435, A048436, A048437, A048438, A048439, A048440, A048441, A048442, A048443, A048444, A048445, A048446, A048447.

Sequence in context: A210801 A153091 A210593 * A235706 A124847 A249251

Adjacent sequences: A179066 A179067 A179068 * A179070 A179071 A179072

Keyword

easy,nonn,tabl,base

Author

Jonathan Vos Post, Jun 27 2010

Extensions

Should be revised to start with base 2, rather than the ill-defined "base 1". - N. J. A. Sloane, Jul 05 2010

Status

approved