A295001 a(n) = nextprime(1/(2/sigma[-1](P(n)) - 1)) where P(n) = Product_{0 <= k < n} a(k), sigma[-1](x) = sigma(x)/x, a(0) = 4.

4, 11, 23, 257, 13007, 44512049, 46880563785749, 125637016478802067649031191, 652182699863469019760217209096329987925268834143233, 1800254420479597976179975458181139131985404009703136640765845238082635790500153934999846722641241849

Offset: 0

Author

M. F. Hasler, Nov 23 2017

Keywords

• nonn
• hard

Comments

Here, nextprime(x) = min { p > x; p prime }, prevprime(x) = max { p < x; p prime }.

The next term, a(10) ~ 3.1*10^196, is too large to be displayed above.

From a(3) on, a(n+1) has roughly twice the number of digits of a(n).

For n >= 1, a(n) is the least prime such that Product_{k=0..n} a(k) is deficient. This implies that (Product_{k=0..n-1} a(k))*prevprime(a(n)) is perfect for n = 1, and a primitive weird number (A002975) for some but not all larger n.

Examples

			Let Q(x) = 1/(2/sigma[-1](x) - 1), P(n) = Product(a(k), k=0..n-1), and start with a(0) = 4 = P(1). Then:
Q(P(1)) = 7, a(1) = 11. (4*7 is perfect, P(2) = 4*11 is deficient.)
Q(P(2)) = 21, a(2) = 23. (4*11*19 is weird, P(3) = 4*11*23 is deficient.)
Q(P(3)) = 252, a(3) = 257. (P(3)*251 is weird, P(4) = 4*11*23*257 is deficient.)
Q(P(4)) = 13003.2, a(4) = 13007. (P(4)*13003 is weird, P(5) = 4*11*23*257*13007 is deficient.)
Q(P(5)) = 44512006.7..., a(5) = 44512049. (P(5)*44511949 is weird ; P(6) = 4*11*257*44512049 is deficient.)
P(6)*prevprime(a(6)) is semiperfect, i.e., no more weird.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 0..13

Crossrefs

Cf. A002975 (primitive weird numbers), A000203 (sigma).

The nextprime and prevprime functions are here used for possibly non-integral arguments, but rounding these down or up allows the use of the nextprime and prevprime functions for integer arguments, A151800 and A151799.

See A262228 for the variant starting with a(0) = 1.

Programs

PARI

A295001=List(m=4);for(n=1,13,listput(A295001,p=nextprime(1\(2/sigma(m,-1)-1)+1));p>default(primelimit)&&addprimes(p);m*=p)