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A262228 Deficiency sequence: a(0) = 1, a(n) is the smallest prime p > a(n-1) such that the product of a(i), 1 <= i < n, is deficient (belongs to A005100). 1
1, 2, 5, 11, 59, 653, 84761, 2763189059, 377406001499268899, 2638619515495963542360422694651593, 135435890329895562961039215198033899386421965445591860752412324961 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The product of the first n+1 terms is the smallest deficient multiple of the product of the first n terms.

The product of any finite number of distinct terms of this sequence is deficient.

LINKS

Table of n, a(n) for n=0..10.

FORMULA

a(n) = A151800(floor(1/(2*(Product_{i=2..n-1} a(i)/(a(i)+1))-1)), where A151800 is the "next larger prime" function.

lim_{n->inf} A001065(Product_{i=0..n} a(i))/(Product_{i=0..n} a(i)) = 1. [Corrected by M. F. Hasler, Dec 04 2017]

Conjecture: log(a(n)) ~ e^(an+b) where a and b are approximately 0.6 and -1.6 respectively.

EXAMPLE

a(3) = 11 because A001065(2*5*7) = A001065(70) = 74 > 70, and A001065(2*5*11) = A001065(110) = 106 < 110.

From M. F. Hasler, Dec 14 2017: (Start)

Let Q(x) = 1/(2x/sigma(x) - 1), P(n) = Product( a(k), k<n): P(0) = 1 (empty product). Then:

Q(P(0)) = 1, a(0) = nextprime(1) = 2 = P(1).

Q(P(1)) = 3, a(1) = 5. (2*3 is perfect, P(2) = 2*5 is deficient.)

Q(P(2)) = 9, a(2) = 11. (2*5*7 is weird, P(3) = 2*5*11 is deficient.)

Q(P(3)) = 54, a(3) = 59. (P(3)*53 is weird, P(4) = 2*5*11*59 is deficient.)

Q(P(4)) = 648, a(4) = 653. (P(4)*647 is weird, P(5) = 2*5*11*59*653 is deficient.)

Q(P(5)) = 84758.4, a(5) = 84761. (P(5)*84751 is abundant and semiperfect: sum of all proper divisors except {1, 2, 11, 22, 55, 59, 590}; P(6) = 2*5*11*59*653*84761 is deficient.) (End)

PROG

(PARI) lista(nn) = {print1(p=1, ", "); vp = [p]; for (n=2, nn, np = nextprime(1+floor(1/(2*prod(i=2, n-1, vp[i]/(vp[i]+1))-1))); vp = concat(vp, np); print1(np, ", "); ); } \\ Michel Marcus, Oct 16 2015

(PARI) a=List(); m=1; for(n=0, 13, listput(a, p=nextprime(1\(2/sigma(m, -1)-1)+1)); p>default(primelimit)&&addprimes(p); m*=p) \\ M. F. Hasler, Dec 14 2017

CROSSREFS

Cf. A001065, A005100, A151800 (nextprime).

Cf. A002975 (primitive weird numbers), A000203 (sigma), A295001 (same definition but a(0) = 4).

Sequence in context: A127010 A140547 A131480 * A213073 A267527 A286453

Adjacent sequences:  A262225 A262226 A262227 * A262229 A262230 A262231

KEYWORD

nonn

AUTHOR

Chayim Lowen, Sep 15 2015

STATUS

approved

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Last modified October 17 22:48 EDT 2018. Contains 316297 sequences. (Running on oeis4.)