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.

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

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<r} prime(n+i) for some r, which turns out to be r = a(n). - M. F. Hasler, May 08 2017

From Jianing Song, Apr 21 2021: (Start)

Let x_1 < x_2 < ... < x_k < ... be the numbers of the form p of p^2 + p, where p is a prime >= 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<a(n)} prime(n+i). - M. F. Hasler, Jun 15 2017

Example

a(2) = 5 since 945 = 3^3*5*7 is an abundant number with p_2 = 3 as its smallest prime factor, and no such number exists with fewer than 5 prime factors.

Prog

(PARI) A108227(n, s=1+1/prime(n))=for(a=1, 9e9, if(2<s*=1+1/prime(n+a), return(a+1))) \\ M. F. Hasler, Jun 15 2017
(PARI) isform(k, q) = my(p=prime(k)); if(isprime(q) && (q>=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