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 "nondeficient" 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 nondeficient products are A007702. - M. F. Hasler, Jun 15 2017
An additional distinct prime factor q adds a factor 1 + 1/q to the abundancy sigma(N, -1) = sigma(N)/N, while raising a prime factor p already present in N to the square contributes a factor 1 + 1/(p^2 + p). Therefore, when there are many prime factors, the second option (or a variant) may become more interesting. See EXAMPLE n = 10. - M. F. Hasler, Jun 05 2026
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..500 (corrected by Amiram Eldar, May 09 2026)
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 nondeficient numbers with fewer than 5 prime factors, of which 3 is the least prime factor.
For n = 10, we have abundancy sigma(A002110(10+155)/A002110(9),-1) ~ 2 - 0.00228687, where A002110 = primorial numbers. Adding a factor prime(10+156) = 983 would still yield a deficient number, but adding a second factor prime(10) = 29 instead (with (1 + 1/29 + 1/29^2)/(1 + 1/29) = 1 + 1/870) gives abundancy 2 + 9.36e-6. Hence, a(10) = A001222(prime(10)*A002110(10+155)/A002110(9)) = 157. - M. F. Hasler, Jun 05 2026
MATHEMATICA
formQ[p_, q_] := If[q >= p && PrimeQ[q], True, If[IntegerQ[Sqrt[4*q + 1]], Module[{r = (Floor[Sqrt[4*q + 1]] - 1)/2}, r >= p && PrimeQ[r]], False]];
a[n_] := Module[{prod = 1, sum = 0, p = Prime[n], i}, i = p; While[prod < 2, If[formQ[p, i], prod *= (1 + 1/i); sum++]; i++]; sum];
PROG
(PARI) apply( {A107705(n)=my(p=prime(n), s=1+1/p); while(2>s*=1+1/p=nextprime(p+1), ); n=#p=primes([prime(n), p]); p=Mat([p~, [1|p<-p]~]); for(i=1, n, while(p[i, 1]^(p[i, 2]+2)\(p[i, 1]-1) <= p[n, 1], p[i, 2]++; n--); sigma(p[1..n-1, ], -1)>2 && n--; (i<n && p[i, 2]>1)||break); vecsum(p[1..n, 2])}, [1..20]) \\ M. F. Hasler, Jun 05 2026, replacing earlier code from Jun 15 2017
CROSSREFS
Cf. A002110 (primorial numbers).
KEYWORD
nonn
AUTHOR
Hugo van der Sanden, Jun 10 2005
EXTENSIONS
Data aligned with 2017 PARI program output by Amiram Eldar, Aug 08 2019
Original data restored by Amiram Eldar, May 09 2026
STATUS
approved