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6, 8, 15, 26, 69, 134, 393, 1556, 4659, 9314, 27933, 921327, 85680249, 171360494, 2227686253, 17821489976, 124750429783, 19336316610785, 4544034403522255, 3567067006764843005, 203322819385596050031, 25008706784428314148401, 825287323886134366896771, 91606892951360914725537141, 1923744751978579209236279751
Any nonzero number other than 4 or a prime could be chosen for a(1) so as to generate a nontrivial sequence (because A056240(r)=r for r=4 or a prime). In this sequence a(1) is set to 6 because it is the smallest composite number which is the sum of prime divisors of a greater number (8), and is therefore the smallest starting value for a non-stationary sequence of this kind. Let m = A056240(a(n-1)-q), where q is the greatest (prime or 4) < a(n-1)-1. Then a(n) = m*q, since sopfr(m*q) = sopf(m)+sopf(q) = a(n-1). Each term represents a step up (from the previous term) in the number of repeated iterations of sopfr required to reach a prime; a(n) >= A048133(n).
Let m = A056240(a(n-1)-q), where q is the greatest (prime or 4) < a(n-1)-1. Then a(n) = m*q, since sopfr(m*q) = sopf(m)+sopf(q) = a(n-1). Each term represents a step up (from the previous term) in the number of repeated iterations of sopfr required to reach a prime; a(n) >= A048133(n).
nonn,more
Terms a(18) onward from Max Alekseyev, Sep 20 2024
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Starting with a(1) = 6, a(n) is the smallest number whose sum of prime divisors (taken with multiplicity) is a(n-1). In other words , a(n) = A056240(a(n-1)).
Any nonzero number other than 4 or a prime could be chosen for a(1) so as to generate a nontrivial sequence (because A056240(r)=r for r=4 or a prime). In this sequence a(1) is set to 6 because it is the smallest nonzero composite number which is the sum of prime divisors of a greater number (8), and is therefore the smallest starting value for a non-stationary sequence of this kind. Let m = A056240(a(n-1)-q), where q is the greatest (prime or 4) < a(n-1)-1. Then a(n) = m*q, since sopfr(m*q) = sopf(m)+sopf(q) = a(n-1). Each term represents a step up (from the previous term) in the number of repeated iterations of sopfr required to reach a prime; a(n) >= A048133(n).
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Any non zero nonzero number other than 4 or a prime could be chosen for a(1) so as to generate a non trivial nontrivial sequence (because A056240(r)=r for r=4 or any a prime). In this sequence a(1) is set to 6 because it is the smallest non zero nonzero composite number which is the sum of prime divisors of a greater number (8), and is therefore the smallest starting value for a non -stationary sequence of this kind. Let m = A056240(a(n-1)-q), where q is the greatest (prime or 4) < a(n-1)-1. Then a(n) = m*q, since sopfr(m*q) = sopf(m)+sopf(q) = a(n-1). Each term represents a step up (from the previous term) in the number of repeated iterations of sopfr required to reach a prime; a(n) >= A048133(n).