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 A108227 a(n) is the least number of prime factors for any abundant number with p_n (the n-th prime) as its least factor. 9
 3, 5, 9, 18, 31, 46, 67, 91, 122, 158, 194, 238, 284, 334, 392, 456, 522, 591, 668, 749, 835, 929, 1028, 1133, 1242, 1352, 1469, 1594, 1727, 1869, 2019, 2163, 2315, 2471, 2636, 2802, 2977, 3157, 3342, 3534, 3731, 3933, 4145, 4358, 4581, 4811 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS If we replace "abundant" in the definition with "non-deficient", we get the same sequence with an initial 2 instead of 3, barring an astronomically unlikely coincidence with some as-yet-undiscovered odd perfect number. [This is sequence A107705. - M. F. Hasler, Jun 14 2017] It appears that all terms >= 5 correspond to the odd primitive abundant numbers (A006038) which are products of consecutive primes (cf. A285993), i.e., of the form N = Product_{0<=i= prime(n). Then a(n) is the smallest N such that Product_{i=1..N} (1 + 1/x_i) > 2. See my link below for a proof. For example, for n = 3, we have {x_1, x_2, ..., x_k, ...} = {5, 7, 11, 13, 17, 19, 23, 29, 5^2 + 5, ...}, we have Product_{i=1..8} (1 + 1/x_i) < 2 and Product_{i=1..9} (1 + 1/x_i) > 2, so a(3) = 9. (End) LINKS Amiram Eldar, Table of n, a(n) for n = 1..500 Jianing Song, Notes for A108227 FORMULA a(n) = A007684(n)-n+1, for n>1. A007741(n) = Product_{0<=i=p), 1, if(issquare(4*q+1), my(r=(sqrtint(4*q+1)-1)/2); isprime(r) && (r>=p), 0)) a(n) = my(Prod=1, Sum=0); for(i=prime(n), oo, if(isform(n, i), Prod *= (1+1/i); Sum++); if(Prod>2, return(Sum))) \\ Jianing Song, Apr 21 2021 CROSSREFS Cf. A000040, A005101, A006038, A001222. Cf. A107705. Cf. A007707 (~ A007684), A007708, A007741. Cf. A001276 (least number of prime factors for a (p_n)-rough abundant number, counted without multiplicity). Sequence in context: A074861 A281852 A120941 * A289912 A289914 A251704 Adjacent sequences: A108224 A108225 A108226 * A108228 A108229 A108230 KEYWORD nonn AUTHOR Hugo van der Sanden, Jun 17 2005 EXTENSIONS Data corrected by Amiram Eldar, Aug 08 2019 STATUS approved

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Last modified May 26 13:25 EDT 2024. Contains 372826 sequences. (Running on oeis4.)