1,1

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

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

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

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

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

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

Also, integers n > 1 such that A069359(n) = n - 1. - Jonathan Sondow, Apr 16 2014

Max Alekseyev, Jose Maria Grau, and Antonio Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT], 2016-2018.

Thomas Bloom, Problem 313, Erdős Problems.

William Butske, Lynda M. Jaje and Daniel R. Mayernik, On the Equation Sum_{p|N} 1/p + 1/N = 1, Pseudoperfect numbers and partially weighted graphs, Math. Comput., 69 (1999), 407-420. [Title corrected by Jonathan Sondow, Apr 11 2012]

Erdős problems database contributors, Erdős problem database, see no. 313.

José María Grau, Antonio M. Oller-Marcén, and Jonathan Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m, arXiv:1309.7941 [math.NT], 2013.

José María Grau, Antonio M. Oller-Marcén, and Daniel Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.

John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.

Jonathan Sondow and Kieren MacMillan, Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n modulo k and k^2, Integers 11 (2011), #A34.

Jonathan Sondow and Kieren MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240; arXiv:math/1812.06566 [math.NT], 2018.

Jonathan Sondow and Emmanuel Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.

Eric Weisstein's World of Mathematics, Primary pseudoperfect number.

Wikipedia, Primary pseudoperfect number.

OEIS Wiki, Primary pseudoperfect numbers.

A031971(a(n)) (mod a(n)) = A233045(n). - Jonathan Sondow, Dec 11 2013

A069359(a(n)) = a(n) - 1. - Jonathan Sondow, Apr 16 2014

a(n) == 36*(n-2) + 6 (mod 288) for n = 2,3,..,8. - Kieren MacMillan and Jonathan Sondow, Sep 20 2017

From Daniel Forgues, May 24 2013: (Start)

With a(1) = 2, we have 1/2 + 1/2 = (1 + 1)/2 = 1;

with a(2) = 6 = 2 * 3, we have

1/2 + 1/3 + 1/6 = (3 + 2 + 1)/6 = (1*3 + 3)/(2*3) = (1 + 1)/2 = 1;

with a(3) = 42 = 6 * 7, we have

1/2 + 1/3 + 1/7 + 1/42 = (21 + 14 + 6 + 1)/42 =

(3*7 + 2*7 + 7)/(6*7) = (3 + 2 + 1)/6 = 1;

with a(4) = 1806 = 42 * 43, we have

1/2 + 1/3 + 1/7 + 1/43 + 1/1806 = (903 + 602 + 258 + 42 + 1)/1806 =

(21*43 + 14*43 + 6*43 + 43)/(42*43) = (21 + 14 + 6 + 1)/42 = 1;

with a(5) = 47058 (not oblong number), we have

1/2 + 1/3 + 1/11 + 1/23 + 1/31 + 1/47058 =

(23529 + 15686 + 4278 + 2046 + 1518 + 1)/47058 = 1.

For n = 1 to 8, a(n) has n prime factors:

a(1) = 2

a(2) = 2 * 3

a(3) = 2 * 3 * 7

a(4) = 2 * 3 * 7 * 43

a(5) = 2 * 3 * 11 * 23 * 31

a(6) = 2 * 3 * 11 * 23 * 31 * 47059

a(7) = 2 * 3 * 11 * 17 * 101 * 149 * 3109

a(8) = 2 * 3 * 11 * 23 * 31 * 47059 * 2217342227 * 1729101023519

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)

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

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

pQ[n_] := (f = FactorInteger[n]; 1/n + Sum[1/f[[i]][[1]], {i, Length[f]}] == 1)

Select[Range[2, 10^6], pQ[#] &] (* Robert Price, Mar 14 2020 *)

(Python)

from sympy import primefactors

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

(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

Cf. A005835, A007850, A069359, A168036, A190272, A191975, A203618, A216825, A216826, A230311, A235137, A235138, A235139, A236433.

Sequence in context: A386436 A123137 A014117 * A349193 A230311 A379686

Adjacent sequences: A054374 A054375 A054376 * A054378 A054379 A054380

nonn,more,hard

approved