A164643 -id:A164643 - cp's OEIS frontend

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

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

A175729 Numbers n such that the sum of the prime factors with multiplicity of n divides n-1.

Original entry on oeis.org

6, 21, 45, 52, 225, 301, 344, 441, 697, 1225, 1333, 1540, 1625, 1680, 1695, 1909, 2025, 2041, 2145, 2295, 2466, 2601, 2926, 3051, 3104, 3146, 3400, 3510, 3738, 3888, 3901, 4030, 4186, 4251, 4375, 4641, 4675, 4693, 4930, 5005, 5085, 5244, 5425, 6025, 6105

Offset: 1

K. T. Lee (7x3(AT)21cn.com), Aug 23 2010

			For example, 21=7x3, 7+3=10 which divides 21-1=20.

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

Disjoint from A130871 and A046346.
Cf. A001414.
Contains A164643.

[k:k in [2..6200]| IsIntegral((k-1)/( &+[m[1]*m[2]: m in Factorization(k)]))]; // Marius A. Burtea, Sep 16 2019

A001414 := proc(n) ifactors(n)[2] ; add( op(1,p)*op(2,p),p=%) ; end proc:
isA175729 := proc(n) if (n-1) mod A001414(n) = 0 then true; else false; end if; end proc:
for n from 2 to 10000 do if isA175729(n) then printf("%d,",n) ; end if; end do:
# R. J. Mathar, Aug 24 2010

fQ[n_] := Mod[n - 1, Plus @@ Flatten[ Table[ #1, {#2}] & @@@ FactorInteger@ n]] == 0; Select[ Range@ 6174, fQ] (* Robert G. Wilson v, Aug 25 2010 *)

from sympy import factorint
def ok(n): return n>1 and (n-1)%sum(p*e for p, e in factorint(n).items())==0
print([k for k in range(10**4) if ok(k)]) # Michael S. Branicky, Sep 30 2022

{n : A001414(n) | (n-1)}. [R. J. Mathar, Aug 24 2010]

Extended by R. J. Mathar and Robert G. Wilson v, Aug 24 2010

A340967 a(n) is the number of iterations of the map x -> n mod sopfr(x) starting with n to reach 0 or 1, where sopfr = A001414.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 4, 2, 1, 1, 2, 1, 2, 1, 4, 1, 3, 2, 4, 1, 3, 1, 1, 1, 2, 3, 3, 3, 2, 1, 4, 4, 3, 1, 3, 1, 4, 1, 3, 1, 2, 2, 2, 4, 1, 1, 5, 3, 2, 4, 4, 1, 1, 1, 4, 4, 2, 4, 2, 1, 4, 2, 1, 1, 1, 1, 3, 3, 3, 3, 3, 1, 2, 3, 3, 1, 1, 3, 4, 5, 2, 1, 3, 3, 3, 3, 5, 4, 2, 1, 2, 2

Offset: 1

J. M. Bergot and Robert Israel, Jan 31 2021

nonn

If n is prime, or n is in A164643, then a(n) = 1.

			a(12) = 3 because 12 mod (2+2+3) = 5, 12 mod 5 = 2 and 12 mod 2 = 0 (3 iterations).
a(54) = 5 because 54 mod (2+3+3+3) = 10, 54 mod (2+5) = 6, 54 mod 5 = 4, 54 mod (2+2) = 2, and 54 mod 2 = 0 (5 iterations).

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

Cf. A001414, A164643, A340969.

sopfr:= proc(n) local t;
  add(t[1]*t[2], t = ifactors(n)[2])
end proc:
f:= proc(n) local x,k;
  x:= n;
  for k from 1 do x:= n mod sopfr(x); if x <= 1 then return k fi od;
end proc:
f(1):= 0:
map(f, [$1..200]);

from sympy import factorint
def A340967(n):
    c, x = 0, n
    while x > 1:
        c += 1
        x = n % sum(p*e for p, e in factorint(x).items())
    return c # Chai Wah Wu, Feb 01 2021

A164698 Semiprimes pq such that pq - 1 divides p^2 + q^2 + 2.

Original entry on oeis.org

6, 21, 26, 51, 1157, 372101, 1288005205276048901

Offset: 1

Mohamed Bouhamida, Aug 22 2009

Semiprimes pq such that pq-1 divides (p+q)^2.
The third to fifth terms are Fib(3)*Fib(7), Fib(7)*Fib(11) and Fib(13)*Fib(17).
Products of two prime Fibonacci numbers F(k) and F(k+4) (see A001605 and A005478) are in the sequence.
6 and 26 are the only even terms. Odd terms contain products of pairs of consecutive terms from the following sequences: A005248, A001541, A033889, A033891. - Max Alekseyev, Aug 27 2009

			The semiprime 6 = 2*3 is in the sequence because 2*3 - 1 = 5 divides 2^2 + 3^2 + 2 = 15.

Cf. A001358, A000045, A140362, A164643.

isA001358 := proc(n) RETURN ( numtheory[bigomega](n) =2 ) ; end:
isA164698 := proc(n) if isA001358(n) then p := op(1,op(1,ifactors(n)[2]) ) ; q := n/p ; if (p^2+q^2+2) mod (p*q-1) = 0 then true; else false; fi; else false; fi; end:
for n from 4 to 3000000 do if isA164698(n) then print(n, ifactors(n)) ; fi; od: # R. J. Mathar, Aug 24 2009

Missing values added by R. J. Mathar, Aug 24 2009

a(7) from Max Alekseyev, Aug 27 2009

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

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A175729 Numbers n such that the sum of the prime factors with multiplicity of n divides n-1.

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A340967 a(n) is the number of iterations of the map x -> n mod sopfr(x) starting with n to reach 0 or 1, where sopfr = A001414.

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A164698 Semiprimes pq such that pq - 1 divides p^2 + q^2 + 2.

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