cp's OEIS Frontend

For the first 240000 terms, also the conversion of a digital clock display H:MM:SS to seconds, i.e., a(HMMSS) yields the number of seconds after midnight corresponding to that time, with leading zeros allowed: 0 <= H <= 24, 0 <= MM, SS <= 59.

Numbers where SS or MM etc. are > 59 are strictly spoken invalid in base 60. However, in order to "fill in the gaps" in a meaningful way, we simply do allow SS, MM, etc. to take values up to 99. (The ISO specification for writing time and date does actually allow SS to take values including 60 in order to account for leap seconds, as it allows H to be equal to 24 to denote midnight at the end of a day.) Therefore a(60+k) = 60+k for k < 40, but a(100+k) = 60+k, a(200+k) = 120+k etc, for 0 <= k <= 99.

Could be called "super prime time primes": all of H, MM, SS, HMMSS (6-digit clock display) and H*3600 + MM*60 + SS (seconds after midnight) are prime.

This sequence lists the prime time primes A295013 for which the number of seconds A295003 is prime. These are also the times HMMSS (obtained through A055643) which correspond to the subset A295002 of primes in A295003.

Sequence A295002 lists the number of seconds after midnight corresponding to the "prime times" = 6-digit clock displays listed here.

Sequences A295003 and A295004 list the number of seconds after midnight corresponding to the prime time primes A295013 and prime time numbers A295014, i.e., primes and all numbers of the form HMMSS where H < 24 and MM, SS < 60 are prime.

See A295000(n) = A055643(a(n)) for the corresponding "prime time primes", i.e., 6-digit clock displays. Reciprocally, the number of seconds given here is obtained through A292579 from the "super prime time primes" A295000.

The sequence is periodic of period 1440 since the term a(1440) = a(0) = 24 corresponds to the time 00:00.

The sequence is periodic of period 1440 since the term a(1440) = a(0) = 24 corresponds to the time 00:00.

The sequence is periodic of period 1440 since the term a(1440) = a(0) = 24 corresponds to the time 00:00.

The sequence is periodic of period 1440 since the term a(1440) = a(0) = 24 corresponds to the time 00:00.

The sequence is periodic of period 1440 since the term a(1440) = a(0) = 24 corresponds to the time 00:00.

Exercise 14 on page 30 of the Long textbook is "Let m_1, m_2, m_3 ... be an infinite sequence of integers such that m_i >= 2 for all i. Let M_0 = 1 and M_i = Product_{j=1..i} m_j for all i >= 1. Show that every nonnegative integer r can be written uniquely in the form r = c_n M_n + c_(n-1) M_(n-1) + ... + c_1 M_1 + c_0 where c_n <> 0 for r <> 0 and 0 <= c_i < m_(i+1) for all i." The current sequence of terms a(r) = (c_n c_(n-1) ... c_1 c_0 concatenated) is one example of an infinite family of hybrid representations (just using only 2 and 3). For the m_i, this sequence uses A010693. Then the corresponding M_i are A026549. Thus the places reading from right have values (1,2,2*3,2*3*2,2*3*2*3,...) = A026549. The (ternary) digit 2 may only appear in the even positions counting from the rightmost as position 1. Appending "00" to any term multiplies the number by 6.

However, appending a single "0" to a term multiplies the number by 2 or by 3 or produces an invalid string of digits -- or even none of the above (110 => 1100, 8 becomes 18) -- depending upon the original number and its length.

Each sexagesimal digit appears as a pair of decimal digits as on a digital clock. Any leading zeros are truncated. Thus decimal 81 appears as "121" and not "0121".