A086551 -id:A086551 - cp's OEIS frontend

A080371 a(n) is the smallest x such that the quotient d(x+1)/d(x) equals n, where d = A000005.

2, 1, 11, 23, 47, 59, 191, 167, 179, 239, 5119, 359, 20479, 2111, 719, 839, 983039, 1259, 786431, 3023, 2879, 15359, 62914559, 3359, 22031, 266239, 6299, 6719, 13690208255, 5039, 22548578303, 7559, 156671, 6881279, 25919, 10079, 1168231104511, 5505023, 479231, 21839

Offset: 1

Labos Elemer, Feb 24 2003

nonn

a(37) > 10^12. - Donovan Johnson, Sep 02 2013

Conjecture: a(p) = 2^(p-1) * k - 1 for some k > 0 and prime p. - David A. Corneth, Jan 26 2021

			n = 49: a(49) = 233279 = m, d(m+1) = 98, d(m) = 2, quotient = 49.

David A. Corneth, Table of n, a(n) for n = 1..196
Donovan Johnson, All 435 terms <= 10^12

Cf. A000005, A080372.

t = Table[ 0, {50}]; Do[ s = DivisorSigma[0, n+1] / DivisorSigma[0, n]; If[ s < 51 && t[[s]] == 0, t[[s]] = n], {n, 1, 10^8}]; t

{a(n) = my(k=1); while(numdiv(k+1)!=n*numdiv(k), k++); k} \\ Seiichi Manyama, Jan 17 2021

a(n) = Min_{x : d[x+1]/d[x] = n}.

a(n) = A086551(n) - 1. - Hugo Pfoertner, Jan 26 2021

More terms from Robert G. Wilson v, Feb 27 2003
a(29) and a(31) from Donovan Johnson, Dec 26 2012
More terms from David A. Corneth, Jan 27 2021

A086552 Numbers x such that tau(x)/tau(x-1) is an integer, where tau() is the number of divisors function.

Original entry on oeis.org

2, 3, 6, 8, 12, 14, 15, 18, 20, 22, 24, 27, 30, 32, 34, 35, 38, 39, 40, 42, 44, 45, 48, 50, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 84, 86, 87, 88, 90, 94, 95, 96, 98, 99, 102, 104, 105, 108, 110, 114, 117, 119, 120, 123, 126, 128, 130, 132, 134, 135, 136, 138

Offset: 1

Amarnath Murthy, Aug 28 2003

nonn

Conjecture: (1) tau(x)/tau(x-1) = n has solutions for every n. (2) If x is the smallest number for a given n such that tau(x)/tau(x-1) = n > 1, then x-1 is a prime.

			12 is a member as tau(12)/tau(11) = 3, 15 is a member as tau(15)/tau(14) = 1.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A057922, A086550, A086551.

with(numtheory): a:=proc(n) if type(tau(n)/tau(n-1), integer)=true then n else fi end: seq(a(n),n=2..150); # Emeric Deutsch, Mar 25 2005

isok(n) = denominator(numdiv(n)/numdiv(n-1)) == 1; \\ Michel Marcus, Apr 12 2018

More terms from David Wasserman and Emeric Deutsch, Mar 25 2005

cp's OEIS Frontend

A080371 a(n) is the smallest x such that the quotient d(x+1)/d(x) equals n, where d = A000005.

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