A220402 - OEIS

A220402

Bases with a record -- new or tie -- smallest 'stem value' for which a prime by concatenation of decrements exists.

2, 3, 4, 10, 16, 22, 25, 28, 31, 34, 40, 42, 46, 58, 60, 88, 100, 148, 166, 250, 252, 400, 1666, 5104, 8011, 19246, 23791, 36664, 55612, 67006, 79420, 84226, 87802, 89482, 102886, 116386, 144766, 201520, 354106, 505606, 652891, 684712, 759946, 1184626, 1432426, 1744366, 3043756, 4800436, 9602776, 24374026, 25069276, 26586886, 28041136, 287217406

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OFFSET

1,1

COMMENTS

'Stem value' here refers to numbers like 4 in 43 or 82 in the larger not-as-well-known prime 8281807978...321, using decimal examples of the types of primes under consideration. To the author's knowledge, this term is not in standard accepted usage but also has no preferred substitute.

The bases listed are such that the smallest positive integer in them from which the sequence of numbers generated by concatenation of incrementally decreasing numbers down to 1 contains at least one prime is at least as great as the greatest such value for smaller bases.

The sequence here is marked by including ties, in contrast to the ordinary submission, in order to highlight the great regularity of 6 being a final digit, including for all of the terms from a(43)=759946 through a(54)=287217406. 25 of the first 54 entries would not be included were the sequence restricted to just record-setters. At least a partial explanation for the coincidence is that 6 as a final digit constrains more tested small numbers to be composite and that the effect becomes more profound the larger the base considered. The degree to which this still leaves an accidental component to the coincidence is not known by the author at time of submission. This includes the possibility that heuristically there may be a small chance that final digits ever depart again from 6.

A217681 deals with record stem values rather than bases. The sequence here first appeared by the author at primenumbers@yahoogroups.com some months before submission here (and with no additional discussion or information), and the referred sequence was computed after and in consideration of this one.

LINKS

Table of n, a(n) for n=1..54.

EXAMPLE

In base 2 and base 3, respectively, both 101 (5 in decimal) and 21 (7 in decimal) are primes, so 4 is guaranteed membership in the list. Reverse concatenations 21, 32 and 321 in base 4, however, are all composite while 103 in base 4 (67 in decimal) is prime; so following bases that produce primes with stem values of 2 or 3 are excluded. Not until base a(12)=42 are all of 21, 32, 321, 43, 432 and 4321 composite, raising the bar to a stem value of 5, so the list indicates that bases a(k) for k from 4 through 11 produce at least one prime with and not before a stem value of 4.

PROG

(PARI) \\ gives record-stem-values:base

{

b=2; rec=2;

while(1, n=2; g=1; e=b;

while(g, N=n; k=n-1; if(k==e, e*=b); E=e;

while(k, N*=E; N+=k; if(ispseudoprime(N), g=0; break());

if(k==E/b, E/=b); k--;

next());

if(g, n++); next());

if(n>=rec, rec=n; print1(n":"b"\n")); b++;

next())