A178443 Two numbers k and l we call equivalent if they have the same vector of exponents with positive components in prime power factorization. Let a(1)=1, a(2)=3. If n>=3 is prime, then a(n) is the smallest prime greater than a(n-1); otherwise, a(n)>a(n-1) is the smallest number equivalent to n such that prime power factorization of a(n) contains only primes which already appeared in the sequence.

1, 3, 5, 9, 11, 15, 17, 27, 121, 187, 191, 275, 277, 573, 831, 14641, 14653, 109443, 109451, 131877, 161183, 249101, 249103, 254221, 214710409, 1603785503, 3146151623077, 23500268975459, 23500268975497, 352504034632455, 352504034632459, 675511501766876508493, 8283939628810696270871857123

Offset

1,2

Comments

The sequence contains exactly 33 terms.

Links

Table of n, a(n) for n=1..33.

Example

By the condition, a(12) should be more than a(11)=191. Since 12 has vector of positive exponents (2,1), then we seek already constructed prime terms p<q in the sequence and choose the smallest number of the form p^2*q>191. It is 275=5^2*11. Thus a(12)=275. Further, a(13) should be the nearest prime more than 275. It is 277.

Prog

(Sage)
@CachedFunction
def A178443(n):
    if n <= 2: return {1:1, 2:3}[n]
    if is_prime(n): return next_prime(A178443(n-1))
    psig_n = list(m for p, m in factor(n))
    primes_seen = sorted(set(filter(is_prime, map(A178443, range(2, n)))))
    possibles = (prod(p**m for p, m in zip(pvec, psig_n)) for pvec in Combinations(primes_seen, len(psig_n)))
    return min(p for p in possibles if p > A178443(n-1))
# D. S. McNeil, Jan 01 2011

Crossrefs

Cf. A178442, A172980, A172999

Sequence in context: A064988 A166699 A191110 * A287421 A214214 A281505

Adjacent sequences: A178440 A178441 A178442 * A178444 A178445 A178446

Keyword

nonn,fini,full

Author

Vladimir Shevelev, Dec 22 2010

Status

approved