A140362 -id:A140362 - cp's OEIS frontend

A255245 Numbers that divide the average of the squares of their aliquot parts.

10, 65, 140, 420, 2100, 2210, 20737, 32045, 200725, 207370, 1204350, 1347905, 1762645, 16502850, 31427800, 37741340, 107671200, 130643100, 200728169, 239719720, 357491225, 417225900, 430085380, 766750575, 1088692500, 1132409168, 1328204850, 1788379460

Offset: 1

Paolo P. Lava, Feb 20 2015

nonn

Ratio: 1, 1, 5, 10, 78, 1, 109, 565,...

If the ratio is equal to 1 we have 10, 65, 20737 (A140362).

			Aliquot parts of 10 are 1, 2, 5. The average of their squares is (1^2 + 2^2 + 5^2) / 3 = (1 + 4 + 25) / 3 = 30 / 3 = 10 and 10 / 10 = 1.

Giovanni Resta, Table of n, a(n) for n = 1..59 (terms < 10^11)

Cf. A001065, A255244.

with(numtheory); P:=proc(q) local a,b,k,n;
for n from 2 to q do a:=sort([op(divisors(n))]);
b:=add(a[k]^2,k=1..nops(a)-1)/(nops(a)-1);
if type(b/n,integer) then lprint(n);
fi; od; end: P(10^6);

Select[Range[10^6],Mod[Mean[Most[Divisors[#]^2]],#]==0&] (* Ivan N. Ianakiev, Mar 03 2015 *)

isok(n) = (q=(sumdiv(n, d, (d!=n)*d^2)/(numdiv(n)-1))) && (type(q)=="t_INT") && ((q % n) == 0); \\ Michel Marcus, Feb 20 2015

from _future_ import division
from sympy import factorint
A255245_list = []
for n in range(2,10**9):
    s0 = s2 = 1
    for p,e in factorint(n).items():
        s0 *= e+1
        s2 *= (p**(2*(e+1))-1)//(p**2-1)
    q, r = divmod(s2-n**2,s0-1)
    if not (r or q % n):
        A255245_list.append(n) # Chai Wah Wu, Mar 08 2015

More terms from Michel Marcus, Feb 20 2015
a(17)-a(24) from Chai Wah Wu, Mar 08 2015
a(25)-a(28) from Giovanni Resta, May 30 2016

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

A350707 Numbers m such that all prime factors of m^2+1 are Fibonacci numbers.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 8, 18, 34, 57, 144, 239, 322, 610, 1134903170

Offset: 1

Michel Lagneau, Mar 27 2022

The Fibonacci numbers in the sequence include 1, 2, 3, 5, 8, 144, 610 and 1134903170.

The sequence includes terms of the form sqrt(f(n) - 1) and sqrt(5 * f(n) - 1), where f(n) = Fibonacci(A281087(n)) * Fibonacci(A281087(n)+2) = A140362(n). - Daniel Suteu, Mar 29 2022

			57 is in the sequence because 57^2+1 = 2*5^3*13 and 2, 5 and 13 are Fibonacci numbers;
1134903170 = Fibonacci(45) is in the sequence because 1134903170^2+1 = 433494437*2971215073 = Fibonacci(43)*Fibonacci(47).

The sequence contains A281618 and A285282.

Cf. A000032, A000045, A005478, A352290, A245236, A339461.

with(numtheory):
A005478:={2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497,1066340417491710595814572169, 19134702400093278081449423917}:
for n from 0 to 11000 do:
   y:=factorset(n^2+1):n0:=nops(y):
   if A005478 intersect y = y
       then
       print(n):
       else
     fi:
od:

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(m) = my(f=factor(m^2+1)); for (i=1, #f~, if (!isfib(f[i,1]), return(0))); return(1); \\ Michel Marcus, Mar 29 2022

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A255245 Numbers that divide the average of the squares of their aliquot parts.

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

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A350707 Numbers m such that all prime factors of m^2+1 are Fibonacci numbers.

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Maple

PARI