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%I #127 Nov 30 2021 05:11:25
%S 2,6,42,1806,47058,2214502422,52495396602,
%T 8490421583559688410706771261086
%N Primary pseudoperfect numbers: numbers n > 1 such that 1/n + sum 1/p = 1, where the sum is over the primes p | n.
%C Primary pseudoperfect numbers are the solutions of the "differential equation" n' = n-1, where n' is the arithmetic derivative of n. - _Paolo P. Lava_, Nov 16 2009
%C Same as n > 1 such that 1 + sum n/p = n (and the only known numbers n > 1 satisfying the weaker condition that 1 + sum n/p is divisible by n). Hence a(n) is squarefree, and is pseudoperfect if n > 1. Remarkably, a(n) has exactly n (distinct) prime factors for n < 9. - _Jonathan Sondow_, Apr 21 2013
%C From the Wikipedia article: it is unknown whether there are infinitely many primary pseudoperfect numbers, or whether there are any odd primary pseudoperfect numbers. - _Daniel Forgues_, May 27 2013
%C Since the arithmetic derivative of a prime p is p' = 1, 2 is obviously the only prime in the sequence. - _Daniel Forgues_, May 29 2013
%C Just as 1 is not a prime number, 1 is also not a primary pseudoperfect number, according to the original definition by Butske, Jaje, and Mayernik, as well as Wikipedia and MathWorld. - _Jonathan Sondow_, Dec 01 2013
%C Is it always true that if a primary pseudoperfect number N > 2 is adjacent to a prime N-1 or N+1, then in fact N lies between twin primes N-1, N+1? See A235139. - _Jonathan Sondow_, Jan 05 2014
%C Also, integers n > 1 such that A069359(n) = n - 1. - _Jonathan Sondow_, Apr 16 2014
%H M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:<a href="http://doi.org/10.1016/j.dam.2018.05.022">10.1016/j.dam.2018.05.022</a> arXiv:<a href="http://arxiv.org/abs/1602.02407">1602.02407</a> [math.NT]
%H W. Butske, L. M. Jaje, and D. R. Mayernik, <a href="http://dx.doi.org/10.1090/S0025-5718-99-01088-1">On the Equation Sum_{p|N} 1/p + 1/N = 1, Pseudoperfect numbers and partially weighted graphs</a>, Math. Comput., 69 (1999), 407-420. [Title corrected by _Jonathan Sondow_, Apr 11 2012]
%H J. M. Grau, A. M. Oller-Marcen, and J. Sondow, <a href="http://arxiv.org/abs/1309.7941">On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m</a>, arXiv:1309.7941 [math.NT], 2013.
%H J. M. Grau, A. M. Oller-Marcén, and D. Sadornil, <a href="https://arxiv.org/abs/2111.14211">On µ-Sondow Numbers</a>, arXiv:2111.14211 [math.NT], 2021.
%H John Machacek, <a href="https://arxiv.org/abs/1706.01008">Egyptian Fractions and Prime Power Divisors</a>, arXiv:1706.01008 [math.NT], 2017.
%H J. Sondow and K. MacMillan, <a href="http://www.integers-ejcnt.org/l34/l34.pdf">Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2</a>, Integers 11 (2011), #A34.
%H J. Sondow and K. MacMillan, <a href="https://dissem.in/p/93532810/primary-pseudoperfect-numbers-arithmetic-progressions-and-the-erdos-moser-equation?deposit=2161">Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation</a>, Amer. Math. Monthly, 124 (2017) 232-240; <a href="http://arxiv.org/abs/1812.06566">arXiv:math/1812.06566 [math.NT]</a>, 2018.
%H J. Sondow and E. Tsukerman, <a href="https://arxiv.org/abs/1401.0322">The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers</a>, arXiv:1401.0322 [math.NT], 2014; see section 4.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimaryPseudoperfectNumber.html">Primary pseudoperfect number.</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Primary_pseudoperfect_number">Primary pseudoperfect number.</a>
%H OEIS Wiki, <a href="/wiki/Primary_pseudoperfect_numbers">Primary pseudoperfect numbers.</a>
%F A031971(a(n)) (mod a(n)) = A233045(n). - _Jonathan Sondow_, Dec 11 2013
%F A069359(a(n)) = a(n) - 1. - _Jonathan Sondow_, Apr 16 2014
%F a(n) == 36*(n-2) + 6 (mod 288) for n = 2,3,..,8. - Kieren MacMillan and _Jonathan Sondow_, Sep 20 2017
%e From _Daniel Forgues_, May 24 2013: (Start)
%e With a(1) = 2, we have 1/2 + 1/2 = (1 + 1)/2 = 1;
%e with a(2) = 6 = 2 * 3, we have
%e 1/2 + 1/3 + 1/6 = (3 + 2 + 1)/6 = (1*3 + 3)/(2*3) = (1 + 1)/2 = 1;
%e with a(3) = 42 = 6 * 7, we have
%e 1/2 + 1/3 + 1/7 + 1/42 = (21 + 14 + 6 + 1)/42 =
%e (3*7 + 2*7 + 7)/(6*7) = (3 + 2 + 1)/6 = 1;
%e with a(4) = 1806 = 42 * 43, we have
%e 1/2 + 1/3 + 1/7 + 1/43 + 1/1806 = (903 + 602 + 258 + 42 + 1)/1806 =
%e (21*43 + 14*43 + 6*43 + 43)/(42*43) = (21 + 14 + 6 + 1)/42 = 1;
%e with a(5) = 47058 (not oblong number), we have
%e 1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/47058 =
%e (23529 + 15686 + 4278 + 2046 + 1518 + 1)/47058 = 1.
%e For n = 1 to 8, a(n) has n prime factors:
%e a(1) = 2
%e a(2) = 2 * 3
%e a(3) = 2 * 3 * 7
%e a(4) = 2 * 3 * 7 * 43
%e a(5) = 2 * 3 * 11 * 23 * 31
%e a(6) = 2 * 3 * 11 * 23 * 31 * 47059
%e a(7) = 2 * 3 * 11 * 17 * 101 * 149 * 3109
%e a(8) = 2 * 3 * 11 * 23 * 31 * 47059 * 2217342227 * 1729101023519
%e If a(n)+1 is prime, then a(n)*[a(n)+1] is also primary pseudoperfect. We have the chains: a(1) -> a(2) -> a(3) -> a(4); a(5) -> a(6). (End)
%e A primary pseudoperfect number (greater than 2) is oblong if and only if it is not the initial member of a chain. - _Daniel Forgues_, May 29 2013
%e If a(n)-1 is prime, then a(n)*(a(n)-1) is a Giuga number (A007850). This occurs for a(2), a(3), and a(5). See A235139 and the link "The p-adic order . . .", Theorem 8 and Example 1. - _Jonathan Sondow_, Jan 06 2014
%t pQ[n_] := (f = FactorInteger[n]; 1/n + Sum[1/f[[i]][[1]], {i, Length[f]}] == 1)
%t Select[Range[2, 10^6], pQ[#] &] (* _Robert Price_, Mar 14 2020 *)
%o (Python)
%o from sympy import primefactors
%o A054377 = [n for n in range(2,10**5) if sum([n/p for p in primefactors(n)]) +1 == n] # _Chai Wah Wu_, Aug 20 2014
%o (PARI) isok(n) = if (n > 1, my(f=factor(n)[,1]); 1/n + sum(k=1, #f, 1/f[k]) == 1); \\ _Michel Marcus_, Oct 05 2017
%Y Cf. A005835, A007850, A069359, A168036, A190272, A191975, A203618, A216825, A216826, A230311, A235137, A235138, A235139, A236433.
%K nonn,more,hard
%O 1,1
%A _Eric W. Weisstein_
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