cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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

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Author

Michel Lagneau, Nov 27 2022

Keywords

Comments

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 |
............................................

Examples

			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.

Crossrefs

Cf. A193462, A194039, A194594.

Programs

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