A190273 -id:A190273 - 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

A190275 Semiprimes of the form p*(p^2 - p + 1).

Original entry on oeis.org

6, 21, 301, 2041, 296341, 486877, 2666437, 3420301, 4304341, 7152001, 38159521, 42387097, 54296677, 95235601, 158048281, 229971241, 265434901, 383712781, 454166017, 775307917, 972261181, 1063290841, 1304557801, 1392422041, 1730882401, 1863895261, 2631883561, 2879450461, 3714274297, 3845297341, 4070454361, 4256780041, 4849695001, 5328809461, 5722533337, 5838483601, 7218898681, 7841065621

Offset: 1

Giorgio Balzarotti, May 07 2011

nonn

This sequence is infinite, assuming Schinzel's Hypothesis H.
Related to Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q + r + 1") setting p = q. Generalization can be achieved by removing semiprimality condition and accepting p^e, e >= 2.
These are semiprimes m = p*q such that 1/p + 1/q - 1/m = p/q. Cf. A326690. - Amiram Eldar and Thomas Ordowski, Jul 22 2019

			a(1) = 6 = 2*3 = 2*(2^2-2+1).
a(2) = 21 = 3*7 = 3*(3^2-3+1).
a(3) = 301 = 7*43 = 7*(7^2-7+1).

Giovanni Resta, Table of n, a(n) for n = 1..10000
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. A065508 (primes p such that p^2-p+1 is prime).

Cf. A001358 (semiprime), A003415 (arithmetic derivative), A164643, A190272 (n'=a-1), A190273 (n'=a+1), A190274 (n'=p^2-1).

seq(`if`(isprime((ithprime(i)^2-ithprime(i)+1))=true,(ithprime(i)^2-ithprime(i)+1)*ithprime(i),NULL),i=1..300);

p = Select[Prime@ Range@ 500, PrimeQ[#^2 - # + 1] &]; p (p^2 - p + 1) (* Giovanni Resta, Jul 22 2019 *)

forprime(p=2,1e4,if(isprime(k=p^2-p+1),print1(p*k", "))) \\ Charles R Greathouse IV, May 08 2011

A190274 Numbers n such that n' = p^2-1, with n = semiprime = pq, n' is the arithmetic derivative of n. Also: semiprimes of the form p(p^2-p-1).

Original entry on oeis.org

15, 95, 287, 1199, 4607, 23519, 28799, 101567, 223199, 296207, 352799, 903167, 1019999, 2032127, 2230799, 2666159, 3285599, 5896799, 7606367, 13939199, 19392479, 28839887, 36154799, 46139039, 54295919, 62412767, 68250239, 73384079, 74440799, 90316799, 95234687, 109672319, 115263647, 118129199, 214562399, 223279487, 234503807, 236792879, 262963199, 270420767, 309829727, 355897439, 422999999, 486823247, 589884959, 628687487

Offset: 1

Giorgio Balzarotti, May 07 2011

nonn

The sequence shows similarity with 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") with p1=p we have p*p=p+p2-1 (see A190272). Generalization can be achieved by removing semiprimarity condition and accepting p^e, e>=2.

			n=15, 15'=8, a=8+1=9=3^2 -> a(1)=15

European Mathematical Society, Newsletter (see book reviews), March 2011, page 46.

Cf. A001358 (semiprime), A003415 (arithmetic derivative), A190273 (n'=a-1), A190273 (n'=a+1).

seq(`if`(isprime((ithprime(i)^2-ithprime(i)-1))=true,(ithprime(i)^2-ithprime(i)-1)*ithprime(i),NULL),i=1..300);

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

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

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A190275 Semiprimes of the form p*(p^2 - p + 1).

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A190274 Numbers n such that n' = p^2-1, with n = semiprime = p*q, n' is the arithmetic derivative of n. Also: semiprimes of the form p*(p^2-p-1).

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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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A190274 Numbers n such that n' = p^2-1, with n = semiprime = pq, n' is the arithmetic derivative of n. Also: semiprimes of the form p(p^2-p-1).