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

1,2

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

Table of n, a(n) for n=1..27.

David A. Corneth, PARI program

Jon E. Schoenfield, Magma program

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 *)

Prog

(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)
(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

Crossrefs

Cf. A334800, A000040, A007506.

Sequence in context: A172114 A146010 A077631 * A354246 A025223 A348919

Adjacent sequences: A350875 A350876 A350877 * A350879 A350880 A350881

Keyword

nonn,more

Author

Devansh Singh, Jan 20 2022

Extensions

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

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

Status

approved