A194594 -id:A194594 - cp's OEIS frontend

A194579 Numbers whose sum of the their nonprime divisors is prime.

4, 6, 8, 10, 12, 16, 22, 27, 28, 32, 40, 44, 46, 52, 58, 68, 80, 82, 88, 106, 112, 116, 124, 125, 136, 148, 164, 165, 166, 172, 176, 178, 192, 208, 226, 232, 236, 250, 256, 262, 284, 292, 304, 316, 328, 332, 346, 352, 358, 368, 382, 388, 420, 428, 435

Offset: 1

Michel Lagneau, Aug 29 2011

nonn

			The divisors of 28 are {1, 2, 4, 7, 14, 28} and the sum of its nonprime divisors is 1 + 4 + 14 + 28 = 47 is prime, hence 28 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A194594.

with(numtheory): for n from 1 to 435 do x:=divisors(n): n1:=nops(x): s:=0: for k from 1 to n1 do if not isprime(x[k]) then s:=s+x[k] fi od: if isprime(s) then printf(`%d, `, n) fi od:

f[n_] := Plus @@ Select[Divisors[n], ! PrimeQ[#] &]; Select[Range[435], PrimeQ[f[#]] &] (* T. D. Noe, Aug 29 2011 *)

isok(n) = isprime(sumdiv(n, d, if (!isprime(d), d))); \\ Michel Marcus, Jan 07 2020

A358704 Numbers m such that the sum of the prime divisors and the sum of the nonprime divisors of m^2+1 are both prime.

Original entry on oeis.org

3, 9, 172, 309, 327, 392, 473, 483, 557, 578, 633, 657, 693, 699, 747, 767, 819, 820, 829, 909, 911, 1007, 1013, 1028, 1030, 1057, 1084, 1141, 1157, 1186, 1252, 1308, 1311, 1382, 1577, 1585, 1620, 1682, 1721, 1722, 1727, 1749, 1841, 1849, 1874, 1972, 2019, 2134

Offset: 1

Michel Lagneau, Nov 27 2022

nonn

The primes of the sequence are 3, 557, 829, 911, 1013, 1721, ...
The corresponding pairs (p, q) = (sum of prime divisors, sum of nonprime divisors) are (7, 11), (43, 83), (163, 36293), ...
There is a subsequence {b(n)} = {3, 9, 309, 699, 819, ...} such that q/p < 2 (see the following table).
Conjecture: when b(n) tends to infinity, q/p tends to 2.
+------+----------+----------+-------------+
| b(n) |        p |        q |      q/p    |
+------+----------+----------+-------------+
|    3 |        7 |       11 | 1.571428571 |
|    9 |       43 |       83 | 1.930232558 |
|  309 |    47743 |    95483 | 1.999937164 |
|  699 |   244303 |   488603 | 1.999987720 |
|  819 |   335383 |   670763 | 1.999991055 |
|  909 |   413143 |   826283 | 1.999992739 |
| 1311 |   859363 |  1718723 | 1.999996509 |
| 1749 |  1529503 |  3059003 | 1.999998039 |
| 3201 |  5123203 | 10246403 | 1.999999414 |
| 4809 | 11563243 | 23126483 | 1.999999741 |
............................................

			3 is in the sequence because the divisors of 3^2 + 1 = 10 are {1, 2, 5, 10} and 1 + 10 = 11 and 2 + 5 = 7 are prime numbers.

Cf. A193462, A194039, A194594.

f[n_]:=Plus@@Select[Divisors[n^2+1], !PrimeQ[#]&]; g[n_]:=Plus@@First/@FactorInteger[n^2+1]; Select[Range[2200], PrimeQ[f[#]&&PrimeQ[g[#]]]&]

isok(m) = my(f=factor(m^2+1), sp=vecsum(f[, 1])); isprime(sp) && isprime(sigma(f)-sp); \\ Michel Marcus, Nov 28 2022

cp's OEIS Frontend

A194579 Numbers whose sum of the their nonprime divisors is prime.

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