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A358704
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Numbers m such that the sum of the prime divisors and the sum of the nonprime divisors of m^2+1 are both prime.
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0
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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
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OFFSET
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1,1
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COMMENTS
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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 |
............................................
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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f[n_]:=Plus@@Select[Divisors[n^2+1], !PrimeQ[#]&]; g[n_]:=Plus@@First/@FactorInteger[n^2+1]; Select[Range[2200], PrimeQ[f[#]&&PrimeQ[g[#]]]&]
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PROG
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(PARI) isok(m) = my(f=factor(m^2+1), sp=vecsum(f[, 1])); isprime(sp) && isprime(sigma(f)-sp); \\ Michel Marcus, Nov 28 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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