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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. 2
4, 11, 23, 257, 13007, 44512049, 46880563785749, 125637016478802067649031191, 652182699863469019760217209096329987925268834143233, 1800254420479597976179975458181139131985404009703136640765845238082635790500153934999846722641241849 (list; graph; refs; listen; history; text; internal format)
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_{0 <= k <= n} a(k) is deficient. This implies that (Product_{0 <= k < n} 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 to use 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 A008184 A008185

Adjacent sequences:  A294998 A294999 A295000 * A295002 A295003 A295004

KEYWORD

nonn,hard

AUTHOR

M. F. Hasler, Nov 23 2017

STATUS

approved

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Last modified February 21 02:19 EST 2018. Contains 299388 sequences. (Running on oeis4.)