A190275 -id:A190275 - cp's OEIS frontend

A326690 Denominator of the fraction (Sum_{prime p | n} 1/p - 1/n).

1, 1, 1, 4, 1, 3, 1, 8, 9, 5, 1, 4, 1, 7, 15, 16, 1, 9, 1, 20, 7, 11, 1, 24, 25, 13, 27, 28, 1, 1, 1, 32, 33, 17, 35, 36, 1, 19, 13, 40, 1, 21, 1, 44, 45, 23, 1, 16, 49, 25, 51, 52, 1, 27, 11, 8, 19, 29, 1, 60, 1, 31, 63, 64, 65, 11, 1, 68, 69, 35, 1, 72

Offset: 1

Jonathan Sondow, Jul 18 2019

Theorem. If n is a prime or a Carmichael number, then a(n) = A309132(n) = denominator of (N(n-1)/n + D(n-1)/n^2), where B(k) = N(k)/D(k) is the k-th Bernoulli number. This is a generalization of Theorem 1 in A309132 that A309132(p) = 1 if p is a prime. The proof generalizes that in A309132. As an application of Theorem, for n a prime or a Carmichael number one can compute A309132(n) without calculating Bernoulli numbers; see A309268.
A composite number n is a Giuga number A007850 if and only if a(n) = 1. (In fact, Sum_{prime p | n} 1/p - 1/n = 1 for all known Giuga numbers n.)
Semiprimes m = pq such that 1/p + 1/q - 1/m = p/q are exactly A190275. - Amiram Eldar and Thomas Ordowski, Jul 22 2019
The preceding comment may be rephrased as "Semiprimes m = pq such that A326689(m) = p and a(m) = q are exactly A190275." - Jonathan Sondow, Jul 22 2019
More generally, semiprimes m = pq such that 1/p + 1/q - 1/m = P/Q are exactly A190273, where P <> Q are primes. In other words, semiprimes m such that A326689(m) is prime and a(m) is prime are exactly A190273. - Amiram Eldar and Thomas Ordowski, Jul 25 2019

			-1/1, 0/1, 0/1, 1/4, 0/1, 2/3, 0/1, 3/8, 2/9, 3/5, 0/1, 3/4, 0/1, 4/7, 7/15, 7/16, 0/1, 7/9, 0/1, 13/20, 3/7, 6/11, 0/1, 19/24, 4/25, 7/13, 8/27, 17/28, 0/1, 1/1
a(12) = denominator of (Sum_{prime p | 12} 1/p - 1/12) = denominator of (1/2 + 1/3 - 1/12) = denominator of 3/4 = 4.
Computing A309132(561) involves numerator(B(560)) which has 865 digits. But 561 is a Carmichael number, so Theorem implies A309132(561) = a(561) = denominator(1/3 + 1/11 + 1/17 - 1/561) = denominator(90/187) = 187.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Wikipedia, Bernoulli number
Wikipedia, Carmichael number
Wikipedia, Giuga number

Numerators are A326689. Quotients n/a(n) are A326691.
Cf. A069359, A007947 (denominator of Sum_{prime p | n} 1/p).
Cf. A002997, A007850, A190275, A309132, A309235, A309268, A309378, A326692.

[1] cat [Denominator(&+[1/p:p in PrimeDivisors(k)]-1/k):k in [2..72]]; // Marius A. Burtea, Jul 27 2019

A326690 := n -> denom((A069359(n)-1)/n):
seq(A326690(n), n=1..72); # Peter Luschny, Jul 22 2019

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Table[ f[n], {n, 100}]

a(n) = denominator(sumdiv(n, d, isprime(d)/d) - 1/n); \\ Michel Marcus, Jul 19 2019

p = lambda n: [n//f[0] for f in factor(n)]
A326690 = lambda n: ((sum(p(n)) - 1)/n).denominator()
[A326690(n) for n in (1..72)] # Peter Luschny, Jul 22 2019

a(n) = 1 if n is a prime or a Giuga number A007850.
a(n) = denominator of (N(n-1)/n + D(n-1)/n^2) if n is a Carmichael number A002997.
a(n) = denominator((A069359(n) - 1)/n). - Peter Luschny, Jul 22 2019

A190273 Numbers n such that n' = m+1, with n and m semiprimes and gcd(m,n)>1, where n' is the arithmetic derivative of n.

Original entry on oeis.org

6, 10, 21, 26, 39, 55, 57, 74, 93, 111, 122, 146, 155, 201, 203, 253, 301, 305, 314, 327, 381, 386, 417, 471, 497, 543, 554, 597, 626, 633, 689, 737, 755, 791, 794, 842, 889, 905, 914, 921, 1011, 1027, 1055, 1081, 1082, 1137, 1226, 1227, 1322, 1346, 1379, 1461, 1466, 1477, 1497, 1514, 1623, 1655, 1703, 1711, 1713, 1731, 1751, 1754, 1893, 1967, 1994

Offset: 1

Giorgio Balzarotti, May 07 2011

nonn

The sequence is related to the Rassias Conjecture ("for any prime p there are two primes p1 and p2 such that p*p1=p1+p2+1, p>2, p2>p1", see A190272-A190275), because n = p1*p2, m=p1*p -> p1*p = p1+p2-1. The sequence includes the cases with p=p1 (or p2). Generalization can be achieved by removing semiprimarity condition or accepting gcd(n,m)=1. The differential equation in its general form n'=m+1 includes Giuga Numbers, i.e., n'=b*n+1, or n'=n+1 (A007850).

These are semiprimes n = p*q such that 1/p + 1/q - 1/n = P/Q, where P <> Q are primes. Cf. A326690. - Amiram Eldar and Thomas Ordowski, Jul 25 2019

			n=6, 6'=5, m=5+1=6, gcd(6,6)=6 -> a(1)=6

For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.

Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A007850 (Giuga numbers), A190272 (n'=m-1), A190273, A190274, A190275.

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
seq(`if`(bigomega(i)=2 and bigomega(der(i)-1)=2 and gcd(i,der(i)-1)>1,i,NULL),i=1..2000);

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

A306270 Composite numbers k such that b^(k(k-1)) == 1 (mod k^2) for every b coprime to k.

Original entry on oeis.org

4, 6, 12, 20, 21, 28, 42, 52, 60, 66, 84, 105, 156, 165, 186, 220, 231, 273, 276, 301, 364, 385, 420, 465, 506, 532, 561, 609, 645, 651, 660, 780, 804, 903, 946, 1036, 1045, 1065, 1092, 1105, 1204, 1265, 1281, 1365, 1491, 1540, 1705, 1716, 1729, 1771, 1806, 1860

Offset: 1

Amiram Eldar and Thomas Ordowski, Feb 01 2019

nonn

These are composites k such that lambda(k^2) divides k(k-1), where lambda is the Carmichael function A002322.
Since lambda(p^2) = phi(p^2) = p(p-1), where p is a prime, then by Euler's theorem b^(p(p-1)) == 1 (mod p^2) for every b indivisible by p.
This sequence includes all Carmichael numbers A002997.
Conjecture: all semiprimes > 4 in this sequence are in A190275. - Thomas Ordowski, Jul 19 2020
The conjecture was verified up to 1063290841. - Amiram Eldar, Jul 19 2020

Cf. A002322, A002997, A306259.

A190275 is a subsequence. - Thomas Ordowski, Jul 19 2020

Select[Range[2000], CompositeQ[#] && Divisible[#(#-1), CarmichaelLambda[#^2]] &]

isok(k) = (k!=1) && !isprime(k) && !(k*(k-1) % lcm(znstar(k^2)[2])); \\ Michel Marcus, Mar 12 2019

A341274 Composite numbers k that are divisible by (k mod sopfr(k))+floor(k/sopfr(k)), where sopfr = A001414.

Original entry on oeis.org

4, 6, 12, 16, 20, 21, 24, 27, 30, 36, 40, 45, 48, 52, 56, 60, 66, 70, 72, 75, 80, 84, 88, 90, 96, 105, 108, 112, 117, 126, 140, 150, 152, 160, 180, 182, 192, 195, 198, 200, 220, 224, 225, 231, 240, 252, 255, 256, 270, 286, 288, 290, 301, 306, 308, 320, 330, 344, 345, 352, 360, 378, 384, 396, 429

Offset: 1

J. M. Bergot and Robert Israel, Feb 08 2021

nonn

			a(5) = 20 is a term because sopfr(20) = 2*2+5 = 9, and 20 is divisible by (20 mod 9)+floor(20/9) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Contains A190275.

Cf. A001414.

spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:
filter:= proc(n) local s, m;
  if isprime(n) then return false fi;
  s:= spf(n);
  m:= n mod s;
  n mod (m + (n-m)/s) = 0
end proc:
select(filter, [$4..500]);

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A326690 Denominator of the fraction (Sum_{prime p | n} 1/p - 1/n).

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A190273 Numbers n such that n' = m+1, with n and m semiprimes and gcd(m,n)>1, where n' is the arithmetic derivative of n.

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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.

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A306270 Composite numbers k such that b^(k(k-1)) == 1 (mod k^2) for every b coprime to k.

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A341274 Composite numbers k that are divisible by (k mod sopfr(k))+floor(k/sopfr(k)), where sopfr = A001414.

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