A326690 -id:A326690 - cp's OEIS frontend

A326692 Values of k for which the denominator of (Sum_{prime p | k} 1/p - 1/k) is k.

1, 4, 8, 9, 15, 16, 20, 24, 25, 27, 28, 32, 33, 35, 36, 40, 44, 45, 49, 51, 52, 60, 63, 64, 65, 68, 69, 72, 76, 77, 81, 85, 87, 88, 91, 92, 95, 96, 99, 100, 104, 108, 112, 115, 116, 117, 119, 121, 123, 124, 125, 128, 133, 135, 136, 140, 141, 143, 144, 145, 148

Offset: 1

Jonathan Sondow, Jul 20 2019

nonn

Any prime power p^k with k > 1 is a term, as 1/p - 1/p^k = (p^(k-1) - 1)/p^k which is in reduced form and has denominator p^k.

Are there infinitely many Carmichael numbers A002997 in the sequence?

			1/3 + 1/5 - 1/15 = 7/15 has denominator 15, so 15 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002997, A326689, A326690, A326691, A326715.

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Select[Range[148], f[#] == # &]

is(k) = {my(p = factor(k)[,1]); denominator(sum(i = 1, #p, 1/p[i]) - 1/k) == k;} \\ Amiram Eldar, Apr 26 2024

Solutions of A326690(x) = x. That is, fixed points of A326690.

A309268 Carmichael numbers m such that A309132(m) < m.

Original entry on oeis.org

561, 1105, 46657, 52633, 188461, 670033, 825265, 838201, 1082809, 2455921, 2628073, 4463641, 4767841, 5632705, 8830801, 11119105, 13187665, 16778881, 18307381, 18900973, 21584305, 22665505, 31146661, 31405501, 31692805, 34657141, 36765901, 38624041, 40280065

Offset: 1

Amiram Eldar and Thomas Ordowski, Jul 20 2019

nonn

A309132(m) divides m for all Carmichael numbers m, but apparently most of them equal A309132(m). Of the first 10000 Carmichael numbers, only 1341 are in this sequence.
The ratios a(n)/A309132(a(n)) are 3, 5, 13, 7, 133, 7, 133, 7, 7, 793, 7, 13, 13, ...
By Jonathan Sondow's theorem (cf. comments in A309132), these are Carmichael numbers m such that denominator(Sum_{prime p|m}1/p - 1/m) < m, i.e., A326690(m) < m.
Problem: are there Carmichael numbers m such that A309132(m) is prime? Equivalently, Carmichael numbers m such that A326690(m) is prime. None exist below 2^64. Conjecture: there are no such Carmichael numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A309132, A326690.

Subsequence of A002997 and A309235.

aQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]] && Denominator[ Total@(1/FactorInteger[n][[;; , 1]]) - 1/n] < n; Select[Range[10^6], aQ]

A309378 a(n) is the smallest squarefree number m with n prime factors such that Sum_{prime q|m} 1/q - 1/m = P/Q, where P <> Q are primes, for n > 1, or a(n) = 1 if no such m.

Original entry on oeis.org

1, 6, 105, 1330, 331230, 4081530, 127357230

Offset: 1

Amiram Eldar and Thomas Ordowski, Jul 26 2019

Associated fractions P/Q for n > 1 are 2/3, 2/3, 17/19, 191/181, 19/17, 5701/4241, .... Note that Q | m.

a(n) is the least m with Omega(m) = omega(m) = n such that A326689(m) is a prime P and A326690(m) is a prime Q, or a(n) = 1 if no such m.

			1/2 + 1/3 - 1/6 = 2/3,
1/3 + 1/5 + 1/7 - 1/105 = 2/3,
1/2 + 1/5 + 1/7 + 1/19 - 1/1330 = 17/19,
....
6 = 2*3, 105 = 3*5*7, 1330 = 2*5*7*19, 331230 = 2*3*5*61*181, 127357230 = 2*3*5*17*53*151, ... - _Jonathan Sondow_, Jul 27 2019

Cf. A000040, A120944, A190273, A190275, A326689, A326690.

m=2; s={}; Do[f = FactorInteger[n]; p = f[[;; , 1]]; e = f[[;; , 2]]; If[Max[e] > 1 || Length[e] < m, Continue[]]; frac = Total@(1/p) - 1/n; num = Numerator[frac]; den = Denominator[frac]; If[den != num && PrimeQ[num] && PrimeQ[den], AppendTo[s, n]; m++], {n, 1, 5*10^6}]; s

a(n) = {for(i = 2, oo, if(is(i, n), return(i)))}
is(m, qp) = {my(f = factor(m)); if(#f~ != qp, return(0)); if(Set(f[,2]) != Set([1]), return(0)); s = sum(i = 1, qp, 1/f[i, 1]) - 1/m; isprime(denominator(s)) && isprime(numerator(s))} \\ David A. Corneth, Jul 27 2019

A327033 N(p-1)/p + D(p-1)/p^2 with p the n-th prime and B(k) = N(k)/D(k) the k-th Bernoulli number.

Original entry on oeis.org

0, 1, 1, 1, 1, -37, -211, 2311, 37153, -818946931, 277930363757, -711223555487930419, -6367871182840222481, 35351107998094669831, 12690449182849194963361, -15116334304443206742413679091, 1431925649981017658678758915153153, -19921854762028779869513196624259348280501

Offset: 1

Jonathan Sondow, Aug 15 2019

sign

a(n) is an integer, as conjectured by Thomas Ordowski and proved by the author in A309132 and A326690.

Ordowski also conjectured that the sequence is a subsequence of A174341.

			Prime(6) = 13 and B(12) = -691/2730, so a(6) = -691/13 + 2730/13^2 = -37.

Wikipedia, Bernoulli number

Cf. A174341, A309132, A326690.

a[n_] := With[{p = Prime[n]}, With[{b = BernoulliB[p - 1]}, (p  Numerator[b] + Denominator[b])/p^2]];
Table[a[n], {n, 1, 18}]

a(n) = my(p = prime(n), b = bernfrac(p-1)); numerator(b)/p + denominator(b)/p^2; \\ Michel Marcus, Aug 16 2019

cp's OEIS Frontend

A326692 Values of k for which the denominator of (Sum_{prime p | k} 1/p - 1/k) is k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A309268 Carmichael numbers m such that A309132(m) < m.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A309378 a(n) is the smallest squarefree number m with n prime factors such that Sum_{prime q|m} 1/q - 1/m = P/Q, where P <> Q are primes, for n > 1, or a(n) = 1 if no such m.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

A327033 N(p-1)/p + D(p-1)/p^2 with p the n-th prime and B(k) = N(k)/D(k) the k-th Bernoulli number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI