A054377 -id:A054377 - cp's OEIS frontend

A346555 7-Sondow numbers: numbers k such that p^s divides k/p + 7 for every prime power divisor p^s of k.

1, 2, 6, 15, 78, 294, 12642, 539026980558

Offset: 1

José María Grau Ribas, Feb 04 2022

Numbers k such that A235137(k) == 7 (mod k).
A positive integer k is a 7-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 7 for every prime power divisor p^s of k.
2) 7/k + Sum_{prime p|k} 1/p is an integer.
3) 7 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 7 (mod k).

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, this sequence, 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[10000000],Sondow[7][#]&]

a(8) from Martin Ehrenstein, Feb 04 2022

A346556 8-Sondow numbers: numbers k such that p^s divides k/p + 8 for every prime power divisor p^s of k.

Original entry on oeis.org

1, 3, 16, 48, 336, 14448, 376464, 17716019376, 419963172816

Offset: 1

José María Grau Ribas, Feb 04 2022

Numbers k such that A235137(k) == 8 (mod k).
A positive integer k is a 8-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 8 for every prime power divisor p^s of k.
2) 8/k + Sum_{prime p|k} 1/p is an integer.
3) 8 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 8 (mod k).
Other numbers in the  sequence: 17716019376, 419963172816, 67923372668477507285654170088688

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, this sequence, A346557.

Sondow[mu_][n_]:=Sondow[mu][n]=Module[{fa=FactorInteger[n]},IntegerQ[mu/n+Sum[1/fa[[i,1]],{i,Length[fa]}]]]
Select[Range[400000],Sondow[8][#]&]

a(8)-a(9) verified by Martin Ehrenstein, Feb 04 2022

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

Original entry on oeis.org

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][#]&]

A189710 Numbers n such that n'' = n'-1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 185, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263

Offset: 1

Giorgio Balzarotti, Apr 25 2011

nonn

The composite numbers in the sequences are: 9, 185, 341, 377, 437, 9005, 30413, 33953, 41009, 51533, 82673, 92909,....

			9' = 6, 9''= 6'= 5, 9" = 9'- 1 -> 9 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A003415, A007850, A054377, A189639.

import Data.List (elemIndices)
a189710 n = a189710_list !! (n-1)
a189710_list = elemIndices 0 $
   zipWith (-) (map a003415 a003415_list) (map pred a003415_list)
-- Reinhard Zumkeller, May 09 2011

#using Michael B. Porter's code from A003415
der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2])
for i from 1 to n do a:=der(der(i))-der(i)+1; if a=0 then j:=j+1; B[j]:=i; end if od

A190272 Numbers n such that n' = a -1, with n and a semiprimes and gcd(a,n) > 1, where n' is the arithmetic derivative of n.

Original entry on oeis.org

6, 14, 15, 22, 33, 38, 46, 51, 62, 86, 87, 91, 95, 118, 141, 142, 145, 158, 159, 166, 206, 249, 262, 267, 278, 287, 295, 321, 326, 382, 395, 398, 411, 422, 445, 446, 473, 502, 519, 537, 542, 545, 581, 591, 622, 662, 695, 699, 703, 718, 745, 758, 766, 789, 838, 886, 895, 926, 951, 958, 995, 998, 1046, 1126, 1139, 1145, 1167, 1199, 1238, 1262, 1318, 1329, 1347, 1382, 1401, 1441, 1486, 1678, 1707, 1717, 1718, 1726, 1745, 1757, 1761, 1766

Offset: 1

Giorgio Balzarotti, May 07 2011

nonn

This sequence is infinite, assuming Dickson's conjecture. In fact, the conjecture implies that there are infinitely many terms of this sequence divisible by any fixed prime p. - Charles R Greathouse IV, May 08 2011

Related to the Rassias Conjecture ("for any odd prime p there are primes q < r such that p*q = q+r+1") setting n = q*r, a = q+r+1. The sequence includes the cases with p = q (or p = r). Generalization can be achieved by removing the semiprimality condition or accepting gcd(n,a)=1. The differential equation in its general form n' = a + 1 includes Primary Pseudoperfect numbers, i.e., n' = n-1 (A054377).

			For n=6, 6' = 5, a = 5-1 = 4, gcd(4,6)=2, so 6 is a term.

For Rassias conjecture: Preda Mihăilescu, Review of Problem Solving and Selected Topics in Number Theory, Newsletter of the European Mathematical Society, March 2011, p. 46.

Cf. A001358 (semiprimes), A003415 (arithmetic derivative), A054377 (Primary Pseudoperfect).

der:=n->n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
# for quick reference only
seq(`if`(bigomega(i)=2 and bigomega(der(i)+1)=2 and gcd(i,der(i)+1)>1,i,NULL),i=1..2000);

find(lim)=my(v=List());forprime(p=2,sqrtint(lim\2),forstep(q=2*p-1,lim\p,p+p,if(isprime(q\p+2)&isprime(q),listput(v,p*q))));vecsort(Vec(v)) \\ Charles R Greathouse IV, May 08 2011

Semiprimes pq with (p+q+1)/p prime. - Charles R Greathouse IV, May 08 2011

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

A189639 Numbers n such that n'' = n'+1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

Original entry on oeis.org

161, 209, 221, 1935, 4265, 15941, 22217, 24041, 25637, 30377, 38117, 39077, 48617, 49097, 55877, 68441, 73817, 76457, 80357, 88457, 95237, 98117, 99941, 105641, 110057, 115397, 122537, 130217, 131141, 136517, 143237, 147941, 148697, 152357, 154457, 159077

Offset: 1

Giorgio Balzarotti, Apr 24 2011

The arithmetic derivative of a(n) is a Giuga's number A007850 (solution of n' = n+1).

			161' = 30, 161'' = 30' = 31 ==> 161'' = 161'+1 so 161 is a term.

Reinhard Zumkeller and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 100 terms from Reinhard Zumkeller)

Cf. A003415, A007850, A054377, A132632, A189639.

import Data.List (elemIndices)
a189710 n = a189710_list !! (n-1)
a189710_list = elemIndices 0 $
   zipWith (-) (map a003415 a003415_list) (map pred a003415_list)
-- Reinhard Zumkeller, May 09 2011

/* using Michael B. Porter's code from A003415: */
A003415(n) = {local(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))} /* arithmetic derivative */
for(n=1,10^6,d1=A003415(n);d2=A003415(d1);if(d2==d1+1,print1(n,", "))); /* show terms */
/* Joerg Arndt, Apr 25 2011 */

A203618 Numbers m such that (m'+1)' = m-1, where m' is the arithmetic derivative of m.

Original entry on oeis.org

1, 2, 6, 42, 104, 120, 165, 245, 272, 561, 1806, 47058, 765625, 1137501, 3874128, 9131793, 2214502422, 52495396602

Offset: 1

Paolo P. Lava, Jan 20 2012

The differential equation whose solutions are the primary pseudoperfect numbers is m' = k*m-1, with k a positive integer. Let us rewrite the equation as m'+1 = k*m and then take the derivative: (m'+1)' = (k*m)' = k'*m + k*m' = k'*m + k*(k*m-1) = (k'+k^2)*m-k. Let k=1: (m'+1)' = m-1. The solutions of this equation are the primary pseudoperfect numbers plus pairs of numbers (x,y) for which x' = y-1 and y' = x-1.
A054377 is a subsequence of this sequence.
a(17) > 10^9. - Michel Marcus, Nov 05 2014
a(19) > 10^11. - Giovanni Resta, Jun 04 2016

			765625' = 1137500; (1137500 + 1)' = 1137501' = 765624 = 765625 - 1, so 765625 is a term.
1137501' = 765624; (765624 + 1)' = 765625' = 1137500 = 1137501 - 1, so 1137501 is a term.

Cf. A054377, A203617.

with(numtheory);
P:=proc(i)
local a,n,p,pfs;
for n from 1 to i do
  pfs:=ifactors(n)[2]; a:=n*add(op(2,p)/op(1,p),p=pfs);
  pfs:=ifactors(a+1)[2]; a:=(a+1)*add(op(2,p)/op(1,p),p=pfs);
  if a=n-1 then print(n); fi;
od;
end:
P(10000000);

A003415[n_]:=If[Abs[n]<2,0,n*Total[#2/#1&@@@FactorInteger[Abs[n]]]];
Select[Range[1,100000],A003415[A003415[#]+1]==#-1&] (* Julien Kluge, Jul 08 2016 *)

ad(n) = sum(i=1, #f=factor(n)~, n/f[1, i]*f[2, i]);
isok(n) = ad(ad(n)+1) == n-1; \\ Michel Marcus, Nov 05 2014

a(17)-a(18) from Giovanni Resta, Jun 04 2016

A369469 a(n) = number of integer solutions to 1 <= x1 < x2 < ... < xn to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn).

Original entry on oeis.org

1, 1, 1, 24, 293, 9219, 787444

Offset: 1

Max Alekseyev, Jan 23 2024

For any n, A369470(n) >= a(n) >= 1 (see A369607).

Max Muller et al., The diophantine equation \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right), MathOverflow, 2024.

Cf. A006585, A007850, A054377, A085098, A118086, A158649, A164014, A343074, A369470, A369607.

cp's OEIS Frontend

A346555 7-Sondow numbers: numbers k such that p^s divides k/p + 7 for every prime power divisor p^s of k.

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A346556 8-Sondow numbers: numbers k such that p^s divides k/p + 8 for every prime power divisor p^s of k.

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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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A189710 Numbers n such that n'' = n'-1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

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A190272 Numbers n such that n' = a -1, with n and a semiprimes and gcd(a,n) > 1, where n' is the arithmetic derivative of n.

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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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A189639 Numbers n such that n'' = n'+1 where n' and n'' are respectively the first and the second arithmetic derivative of n (A003415).

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