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.

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

Amiram Eldar, Table of n, a(n) for n = 1..500

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

Cf. A000040, A023196, A005101, A001222.

Cf. A007684 (~ A007707), A007708, A007741.

Sequence in context: A019135 A069957 A091356 * A278690 A335838 A335837

Adjacent sequences: A107702 A107703 A107704 * A107706 A107707 A107708

Keyword

nonn

Author

Hugo van der Sanden, Jun 10 2005

Extensions

Data corrected by Amiram Eldar, Aug 08 2019

Status

approved