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A336547
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Numbers k such that for 1 <= i < j <= h, all sigma(p_i^e_i), sigma(p_j^e_j) are pairwise coprime, when k = p_1^e_1 * ... * p_h^e_h, where each p_i^e_i is the maximal power of prime p_i dividing k.
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21
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1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 28, 29, 31, 32, 36, 37, 38, 41, 43, 44, 45, 47, 48, 49, 50, 53, 54, 56, 59, 61, 62, 63, 64, 67, 68, 71, 72, 73, 74, 75, 76, 79, 80, 81, 83, 86, 89, 92, 96, 97, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 116, 117, 121, 122, 124, 125
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OFFSET
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1,2
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COMMENTS
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Numbers k such that A051027(k) = Product_{p^e||k} A051027(p^e) = A353802(n). Here each p^e is the maximal prime power divisor of k, and A051027(k) = sigma(sigma(k)). Numbers at which points A051027 appears to be multiplicative.
Proof that this interpretation is equal to the main definition:
(1) If none of sigma(p_1^e_1), ..., sigma(p_k^e_k) share prime factors, then A051027(k) = sigma(sigma(p_1^e_1) * ... * sigma(p_k^e_k)) = A051027(p_1^e_1) * ... * A051027(p_k^e_k), by multiplicativity of sigma.
(2) On the other hand, if say, gcd(sigma(p_i^e_i), sigma(p_j^e_j)) = c > 1 for some distinct i, j, then that c has at least one prime factor q, with product t = sigma(p_1^e_1) * ... * sigma(p_k^e_k) having a divisor of the form q^v (where v = valuation(t,q)), and the same prime factor q occurs as a divisor in more than one of the sigma(p_i^e_j), in the form q^k, with the exponents summing to v, then it is impossible to form sigma(q^v) = (1 + q + q^2 + ... + q^v) as a product of some sigma(q^k_1) * ... * sigma(q^k_z), i.e., as a product of (1 + q + ... + q^k_1) * ... * (1 + q + ... + q^k_z), with v = k_1 + ... + k_z, because such a product is always larger than (1 + q + ... + q^v). And if there are more such cases of "split primes", then each of them brings its own share to this monotonic inequivalence, thus Product_{p^e|n} A051027(p^e) = A353802(n) >= A051027(n), for all n.
Proof that also this interpretation is equal to the main definition:
(1) like in (1) above, if none of sigma(p_1^e_1), ..., sigma(p_k^e_k) share prime factors, then by the multiplicativity of phi.
(2) On the other hand, if say, gcd(sigma(p_i^e_i), sigma(p_j^e_j)) = c > 1 for some distinct i, j, then that c must have a prime factor q occurring in both sigma(p_i^e_i) and sigma(p_j^e_j), with say q^x being the highest power of q in the former, and q^y in the latter. Then phi(q^x)*phi(q^y) < phi(q^(x+y)), i.e. here the inequivalence acts to the opposite direction than with sigma(sigma(...)), so we have A353752(n) <= A062401(n) for all n.
(End)
All even perfect numbers (even terms of A000396) are included in this sequence. In general, for any perfect number n in this sequence, map k -> A026741(sigma(k)) induces on its unitary prime power divisors (p^e||n) a permutation that is a single cycle, mapping each one of them to the next larger one, except that the largest is mapped to the smallest one. Therefore, for a hypothetical odd perfect number n = x*a*b*c*d*e*f*g*h to be included in this sequence, where x is Euler's special factor of the form (4k+1)^(4h+1), and a .. h are even powers of odd primes (of which there are at least eight distinct ones, see P. P. Nielsen reference in A228058), further constraints are imposed on it: (1) that h < x < 2*a (here assuming that a, b, c, ..., g, h have already been sorted by their size, thus we have a < b < ... < g < h < x < 2*a, and (2), that we must also have sigma(a) = b, sigma(b) = c, ..., sigma(f) = g, sigma(h) = x, and sigma(x) = 2*a. Note that of the 400 initial terms of A008848, only its second term 81 is a prime power, so empirically this seems highly unlikely to ever happen.
(End)
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FORMULA
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EXAMPLE
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28 = 2^2 * 7 is present, as sigma(2^2) = 7 and sigma(7) = 8, and 7 and 8 are relatively prime (do not share prime factors). Likewise for all even terms of A000396. - Antti Karttunen, May 09 2022
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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The old definition moved to comments and replaced with an alternative definition from the comment section by Antti Karttunen, May 07 2022
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STATUS
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approved
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