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.

1, 2, 5, 10, 18, 24, 32, 60, 71, 100, 512, 2990, 9910, 10031, 12618, 32674, 53586, 153878, 223500, 312608, 369119, 386110, 466569, 4491817, 7068356, 8765871, 65311881

Offset: 1

Author

Devansh Singh, Jan 20 2022

Keywords

• nonn
• more

Comments

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.

A007506 is the subsequence of the prime terms of this sequence. - Amiram Eldar, Jan 20 2022

a(28) > 10^8. - David A. Corneth, Jan 21 2022

Links

• David A. Corneth, PARI program
• Jon E. Schoenfield, Magma program

Crossrefs

Cf. A334800, A000040, A007506.

Programs

Mathematica

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 *)

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

PARI

\\ See Corneth link \\ David A. Corneth, Jan 21 2022

Python

import sympy
A350878=[]
for m in range(1,15001):
    sum=0
    primes_lessthan_m_by2 = list(sympy.primerange(2,-(m//-2)))
    primes_between_m_by2_and_m = list(sympy.primerange(m//2+1,m))
    divisors_of_m=sympy.divisors(m,generator=False)
    divisors_of_m.remove(m)
    if m%2==0:
        divisors_of_m.remove(m//2)
    for p in primes_between_m_by2_and_m:
        sum+=p
    for p in primes_lessthan_m_by2:
        for d in divisors_of_m:
            if p< m//d and m%(d*p)!=0:
                sum+=d*p
    if sum%m==0:
       A350878.append(m)
print(A350878)

Extensions

a(16)-a(20) from Amiram Eldar, Jan 21 2022

a(21)-a(27) from David A. Corneth, Jan 21 2022