A090584 Smallest number m >= 0 such that n with m threes appended yields a prime or -1 if no such m exists.

1, 0, 0, 1, 0, -1, 0, 1, -1, 1, 0, -1, 0, 2, -1, 1, 0, -1, 0, 3, -1, 1, 0, -1, 8, 1, -1, 1, 0, -1, 0, 4, -1, 2, 1, -1, 0, 1, -1, 483, 0, -1, 0, 1, -1, 1, 0, -1, 2, 1, -1, 1, 0, -1, 3, 1, -1, 6, 0, -1, 0, 5, -1, 1, 1, -1, 0, 1, -1, 5, 0, -1, 0, 1, -1, 3, 1, -1, 0, 4, -1, 1, 0, -1, 1, 1, -1, 1, 0, -1, 2, 3, -1, 2, 1, -1, 0, 1, -1, 3, 0, -1, 0, 2, -1, 1, 0, -1, 0, 1

Offset: 1

Author

Chuck Seggelin, Dec 02 2003

Keywords

• base
• sign

Comments

a(n) = 0 if n is already prime. a(n) = -1 for n = any multiple of 3 other than 3 itself. The first 5 record holders in this sequence are 1, 14, 20, 25, 40 with the values 1, 2, 3, 8, 483 respectively. 410 may be the next record holder as no solution has been found for it yet. 410 was tested out to 1250 threes with no prime formed.

From Toshitaka Suzuki, May 19 2024: (Start)

The first 6 record holders in this sequence are 1, 14, 20, 25, 40, 410 with the values 1, 2, 3, 8, 483, 37398 respectively. 817 may be the next record holder as no solution has been found for it yet. 817 was tested out to 300000 threes with no prime formed.

a(n) = -1 when n = 37037*k + 2808, 3666, 4070, 9287, 18799, 21574, 28083, 30558, 33300, 33740, 36663 or 36707, because n followed by any positive number, m say, of 3's is divisible by at least one of the primes {7,11,13,37}. (End)

Examples

			a(25) = 8 because eight 3's must be appended to 25 before a prime is formed (2533333333).
a(6) = -1 because no matter how many 3's are appended to 6, the resulting number is always divisible by 3 and can therefore not be prime. [Similarly for any larger multiple of 3. - _M. F. Hasler_, Jun 06 2024]
		

Links

• Toshitaka Suzuki, Table of n, a(n) for n = 1..816

Crossrefs

Cf. A372262 (m > 0).

Cf. A083747 (The Wilde Primes, i.e. same operation using ones), A090464 (using sevens), A090465 (using nines).

Programs

PARI

apply( {A090584(n, LIM=500)=n%3 && for(m=0, LIM, ispseudoprime(n) && return(m); n=n*10+3); -(n>3)}, [1..55]) \\ Retun value -1 means that a(n) = -1 or, for non-multiples of 3, a(n) > LIM, the search limit given as 2nd (optional) parameter. - M. F. Hasler, Jun 05 2024

Extensions

Name edited by M. F. Hasler, Jun 06 2024