A249336 a(1) = 1; for n>1, a(n) = number of values k in range 1 .. n-1 such that {sum of prime indices in the prime factorization of a(k)} = {sum of prime indices in the prime factorization of a(n-1)}, both counted with multiplicity.

1, 1, 2, 1, 3, 1, 4, 2, 2, 3, 3, 4, 5, 1, 5, 2, 4, 6, 3, 7, 1, 6, 4, 8, 5, 6, 7, 2, 5, 8, 9, 3, 9, 4, 10, 5, 10, 6, 11, 1, 7, 7, 8, 12, 9, 10, 11, 2, 6, 13, 1, 8, 14, 3, 11, 4, 12, 12, 13, 2, 7, 14, 5, 15, 6, 16, 15, 7, 16, 17, 1, 9, 18, 8, 17, 2, 8, 18, 9, 19, 1, 10, 20, 10, 21, 3, 13, 4, 14, 11, 12, 22, 5, 19, 2, 9, 23, 1

Offset: 1

Author

Antti Karttunen, Oct 25 2014

Keywords

• nonn

Comments

The initial occurrences of primes appear in ascending order. After a(1) and a(2), 1's occur only after each such initial occurrence of a prime, followed by that prime's index (in A000040) + 2.

Examples

			a(1) = 1 by definition.
For n = 2, we see that a(n-1) = a(1) = 1, the sum of whose prime indices is 0, and the only integer k for which A056239(k) = 0 is 1, and 1 occurs once among the terms a(1) .. a(1), thus a(2) = 1 also.
For n = 3, we see that a(n-1) = a(2) = 1 occurs two times among the terms a(1) .. a(2), thus a(3) = 2.
For n = 4, we see that a(n-1) = a(3) = 2, and A056239(2) = 1, and so far there are no other terms than a(3) in a(1) .. a(3) which would result the same sum, thus a(4) = 1.
For n = 5, we see that a(n-1) = a(4) = 1 occurs three times in a(1) .. a(4), thus a(5) = 3.
For n = 6, we see that a(n-1) = a(5) = 3, and A056239(3) = 2 (as 3 = p_2), and so far there are no other terms than a(5) in a(1) .. a(5) which would result the same sum, thus a(6) = 1.
For n = 7, we see that a(n-1) = a(6) = 1 occurs four times in a(1) .. a(6), thus a(7) = 4.
For n = 8, we see that a(n-1) = a(7) = 4, and A056239(4) = 2 (as 4 = p_1 * p_1), and so far among the terms a(1) .. a(7) only a(5) results in the same sum, thus a(8) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..12580

Crossrefs

Cf. A056239, A249338 (sum of prime indices of n-th term), A249339 (positions of ones), A249340 (positions of first occurrences of each noncomposite).

Cf. also A249337 (a similar sequence with a slightly different starting condition), A249148.

Programs

PARI

A049084(n) = if(isprime(n), primepi(n), 0); \\ This function from Charles R Greathouse IV
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * A049084(f[i,1]))); }
A249336_write_bfile(up_to_n) = { my(counts, n, a_n); counts = vector(up_to_n); a_n = 1; for(n = 1, up_to_n, write("b249336.txt", n, " ", a_n); counts[1+A056239(a_n)]++; a_n = counts[1+A056239(a_n)]); };
A249336_write_bfile(12580);

Scheme

;; With memoization-macro definec from Antti Karttunen's IntSeq-library:
(definec (A249336 n) (if (<= n 1) n (let ((s (A056239 (A249336 (- n 1))))) (let loop ((i (- n 1)) (k 0)) (cond ((zero? i) k) ((= (A056239 (A249336 i)) s) (loop (- i 1) (+ k 1))) (else (loop (- i 1) k))))))) ;; Slow, quadratic time implementation.

Formula

a(1) = 1; for n>1, a(n) = number of values k in range 1 .. n-1 such that A056239(a(k)) = A056239(a(n-1)).