A235137 -id:A235137 - cp's OEIS frontend

A346557 9-Sondow numbers: numbers k such that p^s divides k/p + 9 for every prime power divisor p^s of k.

1, 2, 5, 22, 54, 378, 16254, 423522, 19930521798, 472458569418

Offset: 1

José María Grau Ribas, Feb 04 2022

Numbers k such that A235137(k) == 9 (mod k).
A positive integer k is a 9-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 9 for every prime power divisor p^s of k.
2) 9/k + Sum_{prime p|k} 1/p is an integer.
3) 9 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 9 (mod k).
Other numbers in the sequence: 19930521798, 472458569418, 76413794252037195696360941349774

Github, Jonathan Sondow (1943 - 2020)
J. M. Grau, A. M. Oller-Marcén and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
J. M. Grau, A. M. Oller-Marcen and J. Sondow, On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013-2014.

Cf. A054377, A007850, A235137, A348058, A348059.
(-1) and (-2) -Sondow numbers: A326715, A330069.
1-Sondow to 9-Sondow numbers: A349193, A330068, A346551, A346552, A346553, A346554, A346555, A346556, this sequence.

Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
Select[Range[1000000],Sondow[9][#]&]

a(9)-a(10) verified by Martin Ehrenstein, Feb 04 2022

A349193 1-Sondow numbers: numbers j such that p divides j/p + 1 for every prime divisor p of j.

Original entry on oeis.org

1, 2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086

Offset: 1

José María Grau Ribas, Nov 10 2021

nonn

These are the weak primary pseudoperfect numbers mentioned in Grau-Oller-Sondow (2013).
Includes the primary pseudoperfect numbers (A054377). Any weak primary pseudoperfect number which is not a primary pseudoperfect number must have more than 58 distinct prime factors, and therefore must be greater than 10^110; none are known.
A positive integer j is a k-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides j/p + k for every prime power divisor p^s of j.
2) k/j + Sum_{prime p|j} 1/p is an integer.
3) k + Sum_{prime p|j} j/p == 0 (mod j).
4) Sum_{i=1..j} i^A000010(j) == k (mod j).
Numbers m such that A235137(m) == 1 (mod m).

Github, Jonathan Sondow (1943 - 2020)
J. M. Grau, A. M. Oller-Marcén, and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
J. M. Grau, A. M. Oller-Marcen and J. Sondow, On the congruence 1^n + 2^n + ... + n^n = d (mod n), where d divides n, arXiv:1309.7941 [math.NT], 2013.

Cf. A000010, A054377, A007850, A235137, A348058, A348059.
(-1) and (-2)-Sondow numbers: A326715, A330069.
2-Sondow to 9-Sondow numbers: A330068, A346551, A346552, A346553, A346554, A346555, A346556, A346557.

Sondow[mu_][n_]:= Sondow[mu][n]= Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]];
Select[Range[100000],Sondow[1][#]&]

A235138 a(n) = Sum_{k=1..n} k^phi(n) (mod n) where phi(n) = A000010(n).

Original entry on oeis.org

0, 1, 2, 2, 4, 1, 6, 4, 6, 3, 10, 2, 12, 5, 7, 8, 16, 3, 18, 6, 11, 9, 22, 4, 20, 11, 18, 10, 28, 29, 30, 16, 19, 15, 23, 6, 36, 17, 23, 12, 40, 1, 42, 18, 21, 21, 46, 8, 42, 15, 31, 22, 52, 9, 39, 20, 35, 27, 58, 58, 60, 29, 33, 32, 47, 5, 66, 30, 43, 11, 70, 12, 72, 35, 35, 34, 59, 7, 78, 24, 54, 39, 82, 2, 63, 41, 55, 36, 88, 87, 71, 42, 59, 45, 71, 16, 96, 35, 57, 30

Offset: 1

Jonathan Sondow and Emmanuel Tsukerman, Jan 03 2014

a(n) = n-1 if and only if n is prime or is a Giuga number A007850.
a(n) = 1 if (and probably only if) n is a primary pseudoperfect number A054377.
a(2^k*p) = 2^(k-1)*p-2^k if p is an odd prime. - Robert Israel, Apr 25 2017

			a(4) = 30 (mod 4) = 2 since 1^(phi(4)) + 2^(phi(4)) + 3^(phi(4)) + 4^(phi(4))= 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = 30.

Robert Israel, Table of n, a(n) for n = 1..10000
Jonathan Sondow and K. 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 E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.
Wikipedia, Giuga number
Wikipedia, Primary pseudoperfect number

Cf. A007850, A054377, A235137.

f:= proc(n) local q; q:= numtheory:-phi(n);
   add(k&^q, k=1..n) mod n
end proc:
map(f, [$1..100]); # Robert Israel, Apr 25 2017

a[n_] := Mod[Sum[PowerMod[i, EulerPhi@n, n], {i, n}], n]

a(n)=my(p=eulerphi(n));sum(k=1,n,k^p) \\ Charles R Greathouse IV, Jan 04 2014

a(n) = A235137(n) (mod n).

Conjecture : a(n) = Sum_{d|n} phi(n/d)*d^phi(n) (mod n). - Ridouane Oudra, Feb 17 2024

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A346557 9-Sondow numbers: numbers k such that p^s divides k/p + 9 for every prime power divisor p^s of k.

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A349193 1-Sondow numbers: numbers j such that p divides j/p + 1 for every prime divisor p of j.

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A235138 a(n) = Sum_{k=1..n} k^phi(n) (mod n) where phi(n) = A000010(n).

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