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
Examples
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.
Programs
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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
Comments