1,2

a(2) is prime. a(4) is prime, as pointed out by Poo Sung on Prime Curios for 331997. a(3), a(5) and a(11) are semiprime. a(6), a(7), a(8) and a(10) have 3 prime factors. a(9) has 7 prime factors. a(12) has at least 4 prime factors. From a(4) onwards, all terms are congruent to 31997 mod 10^5. From a(5) onwards, all terms are congruent to 531997 mod 10^6. From a(5) onwards, all terms are congruent to 7531997 mod 10^7. The number of digits of a(n) for n = 1, 2, 3, ..., 12 are 1, 1, 3, 6, 11, 18, 26, 37, 51, 84, 105.

a(n) = SUM[from k = 1 to n] (k!)^k. a(n) = SUM[from k = 1 to n] (A000142(k))^k. a(n) = SUM[from k = 1 to n] A036740(k). a(n) = SUM[from k = 1 to n] A002109(k) * A000178(k-1)

a(1) = (1!)^1 = 1^1 = 1.

a(2) = (1!)^1 + (2!)^2 = 1^1 + 2^2 = 1 + 4 = 5.

a(3) = (1!)^1 + (2!)^2 + (3!)^3 = 1^1 + 2^2 + 6^3 = 1 + 4 + 216 = 221.

a(4) = (1!)^1 + (2!)^2 + (3!)^3 + (4!)^4 = 1^1 + 2^2 + 6^3 + 24^4 = 1 + 4 + 216 + 331776 = 331997.

a(10) = (1!)^1 + (2!)^2 + (3!)^3 + (4!)^4 + (5!)^5 + (6!)^6 + (7!)^7 + (8!)^8 + (9!)^9 + (10!)^10.

Cf. A000142, A000178, A002109, A036740.

Sequence in context: A050617 A066462 A046193 this_sequence A002142 A103732 A065757

Adjacent sequences: A112996 A112997 A112998 this_sequence A113000 A113001 A113002

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 03 2006