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A107705
a(n) is the least number of prime factors in any non-deficient number that has the n-th prime as its least prime factor.
5
2, 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, 5053, 5293
OFFSET
1,1
COMMENTS
Barring unforeseen odd perfect numbers (which it has been proved must have at least 29 prime factors if they exist at all), if we replace "non-deficient" in the description with "abundant", the value of a(1) becomes 3 and all other values stay the same.
The above mentioned sequence is A108227, see there for a comment on the relation of this sequence to that of primitive abundant numbers (A006038) which are products of consecutive primes, i.e., of the form N = Product_{0<=i<r} prime(n+i) for some r. The corresponding non-deficient products are A007702. - M. F. Hasler, Jun 15 2017
LINKS
FORMULA
a(n) = A007684(n)-n+1. A007702(n) = Product_{0<=i<a(n)} prime(n+i). - M. F. Hasler, Jun 15 2017
EXAMPLE
a(2) is 5 since 1) there are abundant numbers with a(2)=5 prime factors of which p_2=3 is the least prime factor (such as 945 = 3^3.5.7); 2) there are no non-deficient numbers with fewer than 5 prime factors, of which 3 is the least prime factor.
PROG
(PARI) A107705(n, s=1+1/prime(n))=for(a=1, 9e9, 2>(s*=1+1/prime(n+a))||return(a+1)) \\ M. F. Hasler, Jun 15 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Hugo van der Sanden, Jun 10 2005
EXTENSIONS
Data corrected by Amiram Eldar, Aug 08 2019
STATUS
approved