A348059 -id:A348059 - cp's OEIS frontend

A348058 a(n) = Min {k > n : A235137(k) == n (mod k)}, or -1 if no such minimum exists.

2, 3, 10, 5, 14, 7, 15, 16, 22, 11, 21, 13, 114, 156, 34, 17, 38, 19, 33, 25, 45, 23, 35, 80, 186, 228, 58, 29, 30, 31, 51, 64, 63, 76, 57, 37, 258, 2244, 55, 41, 86, 43, 69, 104, 94, 47, 65, 160, 1518, 372, 106, 53, 354, 81, 87, 624, 99, 59, 77, 61, 402

Offset: 1

José María Grau Ribas, Sep 26 2021

nonn

Conjecture: For all n, a(n) > 0.

If a(673) > 0 then a(673) > 10^10.

José María Grau Ribas, Table of n, a(n) for n = 1..672

Cf. A235137, A348059.

Giuga1[mu_][n_] := Giuga1[mu][n] =
Mod[Sum[PowerMod[i, EulerPhi[n], n], {i, 1, n}] - mu, n] == 0;
A348058[n_] := A348058[n] =
{Clear[ww];  Do[If[Giuga1[n][i], ww = i; Break[]], {i, n + 1, 20000000}]; ww}[[1]];
Table[A348058[n],{n,61}]

a(n) = my(k=n+1); while (sum(i=1, k , Mod(i, k)^eulerphi(k)) != n, k++); k; \\ Michel Marcus, Sep 28 2021

A346551 3-Sondow numbers: numbers k such that p^s divides k/p + 3 for every prime power divisor p^s of k.

Original entry on oeis.org

1, 2, 10, 18, 126, 5418, 141174, 6643507266, 157486189806

Offset: 1

José María Grau Ribas, Dec 04 2021

Numbers k such that A235137(k) == 3 (mod k).
A positive integer k is a 3-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 3 for every prime power divisor p^s of k.
2) 3/k + Sum_{prime p|k} 1/p is an integer.
3) 3 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 3 (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, 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[1000000],Sondow[3][#]&]

a(8)-a(9) from Martin Ehrenstein, Dec 31 2021

A346552 4-Sondow numbers: numbers k such that p^s divides k/p + 4 for every prime power divisor p^s of k.

Original entry on oeis.org

1, 5, 8, 24, 168, 7224, 188232, 8858009688, 209981586408

Offset: 1

José María Grau Ribas, Jan 11 2022

Numbers k such that A235137(k) == 4 (mod k).
A positive integer k is a 4-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 4 for every prime power divisor p^s of k.
2) 4/k + Sum_{prime p|k} 1/p is an integer.
3) 4 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 4 (mod k).
Other numbers in the sequence: 8858009688, 209981586408, 33961686334238753642827085044344

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, 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[4][#]&]

isok(k) = {my(f=factor(k)); for (i=1, #f~, my(p=f[i,1]); for (j=1, f[i,2], if ((k/p + 4) % p^j, return(0)));); return(1);} \\ Michel Marcus, Jan 17 2022

a(8)-a(9) verified by Martin Ehrenstein, Jan 21 2022

A346553 5-Sondow numbers: numbers k such that p^s divides k/p + 5 for every prime power divisor p^s of k.

Original entry on oeis.org

1, 2, 3, 14, 66, 1974, 307146, 3270666, 42404405538, 318501038226

Offset: 1

José María Grau Ribas, Jan 16 2022

Numbers k such that A235137(k) == 5 (mod k).
A positive integer k is a 5-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 5 for every prime power divisor p^s of k.
2) 5/k + Sum_{prime p|k} 1/p is an integer.
3) 5 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 5 (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, 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[10^7], Sondow[5][#]&]

isok(k) = {my(f=factor(k)); for (i=1, #f~, my(p=f[i,1]); for (j=1, f[i,2], if ((k/p + 5) % p^j, return(0)));); return(1);} \\ Michel Marcus, Jan 17 2022

a(9)-a(10) from Martin Ehrenstein, Jan 19 2022

A346554 6-Sondow numbers: numbers k such that p^s divides k/p + 6 for every prime power divisor p^s of k.

Original entry on oeis.org

1, 4, 7, 9, 20, 36, 252, 10836, 282348, 13287014532, 314972379612

Offset: 1

José María Grau Ribas, Jan 16 2022

Numbers k such that A235137(k) == 6 (mod k).
A positive integer k is a 6-Sondow number if satisfies any of the following equivalent properties:
1) p^s divides k/p + 6 for every prime power divisor p^s of k.
2) 6/k + Sum_{prime p|k} 1/p is an integer.
3) 6 + Sum_{prime p|k} k/p == 0 (mod k).
4) Sum_{i=1..k} i^phi(k) == 6 (mod k).
Other numbers in the sequence: 13287014532, 314972379612, 50942529501358130464240627566516

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, 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[6][#]&]

isok(k) = {my(f=factor(k)); for (i=1, #f~, my(p=f[i,1]); for (j=1, f[i,2], if ((k/p + 6) % p^j, return(0)));); return(1);} \\ Michel Marcus, Jan 17 2022

a(10)-a(11) verified by Martin Ehrenstein, Jan 21 2022

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

Original entry on oeis.org

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][#]&]

cp's OEIS Frontend

A348058 a(n) = Min {k > n : A235137(k) == n (mod k)}, or -1 if no such minimum exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A346551 3-Sondow numbers: numbers k such that p^s divides k/p + 3 for every prime power divisor p^s of k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A346552 4-Sondow numbers: numbers k such that p^s divides k/p + 4 for every prime power divisor p^s of k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A346553 5-Sondow numbers: numbers k such that p^s divides k/p + 5 for every prime power divisor p^s of k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A346554 6-Sondow numbers: numbers k such that p^s divides k/p + 6 for every prime power divisor p^s of k.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Links