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

0,1

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.

Links

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

Example

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.

Prog

(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)

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.

Sequence in context: A022495 A238489 A002537 * A230150 A301109 A301015

Adjacent sequences: A294998 A294999 A295000 * A295002 A295003 A295004

Keyword

nonn,hard

Author

M. F. Hasler, Nov 23 2017

Status

approved