|
%I #60 Mar 09 2022 00:42:47
%S 1,2,5,10,18,24,32,60,71,100,512,2990,9910,10031,12618,32674,53586,
%T 153878,223500,312608,369119,386110,466569,4491817,7068356,8765871,
%U 65311881
%N Integers m that divide the sum of values d*p < m, where d is a divisor of m, p is a prime, and d*p does not divide m.
%C Conjecture: The sum of values d*p < m in the definition of the sequence is equal to m for m = 5 only. True for m <= 15000.
%C A007506 is the subsequence of the prime terms of this sequence. - _Amiram Eldar_, Jan 20 2022
%C a(28) > 10^8. - _David A. Corneth_, Jan 21 2022
%H David A. Corneth, <a href="/A350878/a350878.gp.txt">PARI program</a>
%H Jon E. Schoenfield, <a href="/A350878/a350878.txt">Magma program</a>
%t q[n_] := Module[{ds = Divisors[n], s = 0, r}, Do[r = n/d; ps = Select[Range[2, r], PrimeQ[#] && ! Divisible[n, d*#] &]; s += Total[d*ps], {d, ds}]; Divisible[s, n]]; Select[Range[3000], q] (* _Amiram Eldar_, Jan 20 2022 *)
%o (Python)
%o import sympy
%o A350878=[]
%o for m in range(1,15001):
%o sum=0
%o primes_lessthan_m_by2 = list(sympy.primerange(2,-(m//-2)))
%o primes_between_m_by2_and_m = list(sympy.primerange(m//2+1,m))
%o divisors_of_m=sympy.divisors(m,generator=False)
%o divisors_of_m.remove(m)
%o if m%2==0:
%o divisors_of_m.remove(m//2)
%o for p in primes_between_m_by2_and_m:
%o sum+=p
%o for p in primes_lessthan_m_by2:
%o for d in divisors_of_m:
%o if p< m//d and m%(d*p)!=0:
%o sum+=d*p
%o if sum%m==0:
%o A350878.append(m)
%o print(A350878)
%o (PARI) isok(m) = {my(d=divisors(m), s=0); forprime(p=2, m, for(k=1, #d, my(x=d[k]*p); if ((x < m) && (m % x), s+=x););); (s % m) == 0;} \\ _Michel Marcus_, Jan 21 2022
%o (PARI) \\ See Corneth link \\ _David A. Corneth_, Jan 21 2022
%Y Cf. A334800, A000040, A007506.
%K nonn,more
%O 1,2
%A _Devansh Singh_, Jan 20 2022
%E a(16)-a(20) from _Amiram Eldar_, Jan 21 2022
%E a(21)-a(27) from _David A. Corneth_, Jan 21 2022
| 