A296008 Number of days for which concat(n,mm,dd) is prime, for mm = 01..12, dd = 01..number of days in month mm of year n, according to the Gregorian calendar.

18, 22, 23, 19, 17, 19, 26, 22, 21, 15, 24, 17, 21, 25, 26, 25, 21, 30, 25, 20, 21, 19, 19, 22, 16, 20, 17, 13, 20, 21, 22, 23, 17, 18, 22, 31, 25, 24, 24, 22, 27, 26, 26, 17, 18, 17, 19, 23, 22, 23, 25, 20, 14, 24, 20, 16, 21, 27, 23, 21, 23, 21, 26, 22, 27, 21, 21, 26, 19, 20, 23, 25

Offset: 1582

Author

M. F. Hasler, Dec 02 2017

Keywords

• nonn
• base

Comments

The Gregorian calendar entered in vigor on October 15, 1582, the day following October 4, 1582 according to the Julian calendar. Therefore a(1582) is tentatively taken to be the count of prime days for that year, subtracting the two inexisting prime days October 9 and 11 (15821009 and 15821011).

It appears that 12 <= a(n) <= 35. a(n) = 12 for 1939, 2244, ... and a(n) = 35 for n = 2384 ; a(n) = 34 for n = 1980. The year 1980 is also the only year between 1582 and 2112 for which yyyymmdd is prime more than 5 times, when dd is taken to be the last day of the respective month.

Examples

			Starting with year 2000, the sequence reads (27, 16, 20, 18, 27, 26, 16, 20, 24, 24, 25, 19, 22, 19, 26, 13, 19, 19, 18, 19, 25, 20, 23, 17, 21, 21, 24, 18, 29, 18, 26,  17, 19, 22, 25, 20, 19, 20, 20, 21, 22, 25, 22, 22, 26, 22, 13, ...)
a(2015) = 13 since the year 2015 had only 13 "prime days", namely: Jan. 11 and 31, Feb. 27, March 3 and 27, April 11, May 13, Aug. 21, Oct. 11 and 31, Nov. 27 and Dec. 21 and 27. These days correspond to the 13 numbers { 2015.01.11, ..., 2015.12.27 }, with dots deleted, which are prime.
		

Crossrefs

Cf. A008685.

Programs

PARI

dom(m,y)=if(m<8==m%2,31,30-2^!(y%4==0 && y%100!=0 || y%400==0));
a(y)=sum(m=1,12,sum(d=1,dom(m,y),isprime(y*10^4+m*100+d)))