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A350878 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.

%I A350878 #60 Mar 09 2022 00:42:47
%S A350878 1,2,5,10,18,24,32,60,71,100,512,2990,9910,10031,12618,32674,53586,
%T A350878 153878,223500,312608,369119,386110,466569,4491817,7068356,8765871,
%U A350878 65311881
%N A350878 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 A350878 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 A350878 A007506 is the subsequence of the prime terms of this sequence. - _Amiram Eldar_, Jan 20 2022
%C A350878 a(28) > 10^8. - _David A. Corneth_, Jan 21 2022
%H A350878 David A. Corneth, <a href="/A350878/a350878.gp.txt">PARI program</a>
%H A350878 Jon E. Schoenfield, <a href="/A350878/a350878.txt">Magma program</a>
%t A350878 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 A350878 (Python)
%o A350878 import sympy
%o A350878 A350878=[]
%o A350878 for m in range(1,15001):
%o A350878     sum=0
%o A350878     primes_lessthan_m_by2 = list(sympy.primerange(2,-(m//-2)))
%o A350878     primes_between_m_by2_and_m = list(sympy.primerange(m//2+1,m))
%o A350878     divisors_of_m=sympy.divisors(m,generator=False)
%o A350878     divisors_of_m.remove(m)
%o A350878     if m%2==0:
%o A350878         divisors_of_m.remove(m//2)
%o A350878     for p in primes_between_m_by2_and_m:
%o A350878         sum+=p
%o A350878     for p in primes_lessthan_m_by2:
%o A350878         for d in divisors_of_m:
%o A350878             if p< m//d and m%(d*p)!=0:
%o A350878                 sum+=d*p
%o A350878     if sum%m==0:
%o A350878        A350878.append(m)
%o A350878 print(A350878)
%o A350878 (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 A350878 (PARI) \\ See Corneth link \\ _David A. Corneth_, Jan 21 2022
%Y A350878 Cf. A334800, A000040, A007506.
%K A350878 nonn,more
%O A350878 1,2
%A A350878 _Devansh Singh_, Jan 20 2022
%E A350878 a(16)-a(20) from _Amiram Eldar_, Jan 21 2022
%E A350878 a(21)-a(27) from _David A. Corneth_, Jan 21 2022