A054377 -id:A054377 - cp's OEIS frontend

A168036 Difference between n' and n, where n' is the arithmetic derivative of n (A003415).

0, -1, -1, -2, 0, -4, -1, -6, 4, -3, -3, -10, 4, -12, -5, -7, 16, -16, 3, -18, 4, -11, -9, -22, 20, -15, -11, 0, 4, -28, 1, -30, 48, -19, -15, -23, 24, -36, -17, -23, 28, -40, -1, -42, 4, -6, -21, -46, 64, -35, -5, -31, 4, -52, 27, -39, 36, -35, -27, -58, 32, -60, -29

Offset: 0

Paolo P. Lava, Nov 17 2009

Let k = n'-n. For k = -1 n is a primary pseudoperfect number (A054377), apart from n=1; For k=0 n is p^p, being p a prime number (A051674); For k = 1 n is a Giuga number (A007850).

T. D. Noe, Table of n, a(n) for n = 0..10000

Cf. A007850, A051674, A054377, A136141.

Cf. A003415, A051674, A083347, A083348.

a168036 n = a003415 n - n  -- Reinhard Zumkeller, May 22 2015

with(numtheory);
A168036:=proc(q)
local n,p;
for n from 0 to q do
  print(n*add(op(2,p)/op(1,p),p=ifactors(n)[2])-n); od; end:
A168036(1000); # Paolo P. Lava, Nov 05 2012

np[k_] := Module[{f, n, m, p}, If[k < 2, np[k] = 0; Return[0], If[PrimeQ[k], np[k] = 1; Return[1], f = FactorInteger[k, 2]; m = f[[1, 1]]; n = k/m; p = m np[n] + n np[m]; np[k] = p; Return[p]]]];
Table[np[n] - n, {n, 0, 100}] (* Robert Price, Mar 14 2020 *)

a(A083347(n)) < 0; a(A051674(n)) = 0; a(A083348(n)) > 0. - Reinhard Zumkeller, May 22 2015

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = -1 + Sum_{p prime} 1/(p*(p-1)) = A136141 - 1 = -0.226843... . - Amiram Eldar, Dec 08 2023

A229300 Numbers n such that A031971(1806n) == n (mod 1806n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 53, 54, 56, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 81

Offset: 1

José María Grau Ribas, Sep 19 2013

nonn

Complement of A229304.
The asymptotic density is in [0.7747,0.812570].
The numbers k = 1, 2, 6, 42, 1806, 47058, 2214502422, 8490421583559688410706771261086 = A230311 are the only values of k such that the set {n: A031971(k*n) == n (mod k*n)} is nonempty. Its smallest element is n = 1, 1, 1, 1, 1, 5, 5, 39607528021345872635 = A231409. (Comment corrected and expanded by Jonathan Sondow, Dec 10 2013.)

Jose María Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^n + 2^n +... + n^n = d (mod n), where d divides n

Cf. A031971, A231409.
Cf. A231562 (numbers n such that A031971(8490421583559688410706771261086*n) == n (mod 8490421583559688410706771261086*n)).
Cf. A229312 (numbers n such that A031971(47058*n) == n (mod 47058*n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229313 (numbers n such that A031971(47058*n) <> n (mod 47058*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).
Cf. A054377 (primary pseudoperfect numbers).

g[n_] := Mod[Sum[PowerMod[i, n, n], {i, n}], n]; Select[Range[100], g[1806*#] == # &]

A235137 a(n) = Sum_{k = 1..n} k^phi(n), where phi(n) = A000010(n).

Original entry on oeis.org

1, 3, 14, 30, 979, 91, 184820, 8772, 978405, 25333, 40851766526, 60710, 36720042483591, 19092295, 5666482312, 9961449608, 76762718946972480009, 105409929, 164309788542828686799730, 70540730666, 15909231318568907, 67403375450475, 1433191209985108404653810959324, 351625763020, 15975648280734359596251725645

Offset: 1

Jonathan Sondow and Emmanuel Tsukerman, Jan 03 2014

nonn

a(n) == -1 (mod n) if and only if n is prime or is a Giuga number A007850.

a(n) == 1 (mod n) if (and probably only if) n is a primary pseudoperfect number A054377.

			a(4) = 30 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.
a(5) = 979, since phi(5) = 4 and 1^4 + 2^4 + 3^4 + 4^4 + 5^4 = 1 + 16 + 81 + 256 + 625 = 979.
a(6) = 91, since phi(6) = 2 and 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36 = 91.

Seiichi Manyama, Table of n, a(n) for n = 1..388
J. Sondow and K. MacMillan, Reducing the Erdős-Moser equation 1^n + 2^n + ... + k^n = (k+1)^n modulo k and k^2, Integers 11 (2011), #A34.
J. 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. A000010, A007850, A054377, A235138.

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

a(n) = sum(k=1, n , k^eulerphi(n)); \\ Michel Marcus, Oct 21 2015

a(n) (mod n) = A235138(n).

A230311 Numbers n such that 1^(kn) + 2^(kn) + ... + (kn)^(kn) == k (mod kn) for some k; that is, numbers n such that A031971(kn) == k (mod k*n) for some k.

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Nov 06 2013

Least such k is A231409. No other terms for n < 10^110 (see Grau, Oller-Marcen, Sondow (2015) p. 428). - Jonathan Sondow, Nov 30 2013

Same as quotients Q = m/n of solutions to the congruence 1^m + 2^m + . . . + m^m == n (mod m) with n|m. For Q > 1, a necessary condition is that Q be a primary pseudoperfect number A054377. The condition is not sufficient since the primary pseudoperfect number 52495396602 is not a member. - Jonathan Sondow, Jul 13 2014

Jose María Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^m + 2^m + . . . + m^m == n (mod m) with n|m, Monatshefte für Mathematik 177 (2015) 421-436, DOI 10.1007/s00605-014-0660-0

Cf. A031971, A231409.
Cf. A231562 (numbers n such that A031971(8490421583559688410706771261086*n) == n (mod 8490421583559688410706771261086*n)).
Cf. A229312 (numbers n such that A031971(47058*n) == n (mod 47058*n)).
Cf. A229300 (numbers n such that A031971(1806*n)== n (mod n*1806)).
Cf. A229301 (numbers n such that A031971(42*n) == n (mod 42*n)).
Cf. A229302 (numbers n such that A031971(6*n) == n (mod 6*n)).
Cf. A229303 (numbers n such that A031971(2*n) == n (mod 2*n)).
Cf. A229313 (numbers n such that A031971(47058*n) <> n (mod 47058*n)).
Cf. A229304 (numbers n such that A031971(1806*n) <> n (mod n*1806)).
Cf. A229305 (numbers n such that A031971(42*n) <> n (mod 42*n)).
Cf. A229306 (numbers n such that A031971(6*n) <> n (mod 6*n)).
Cf. A229307 (numbers n such that A031971(2*n) <> n (mod 2*n)).
Cf. A229308 (primitive numbers in A229304).
Cf. A229309 (primitive numbers in A229305).
Cf. A229310 (primitive numbers in A229306).
Cf. A229311 (primitive numbers in A229307).
Cf. A054377 (primary pseudoperfect numbers).

a(n) = A054377(n-1) for n = 2, 3, 4, 5, 6, 7, but a(8) = A054377(8). - Jonathan Sondow, Jul 13 2014

Definition corrected by Jonathan Sondow, Nov 30 2013

A330068 Numbers k such that Sum_{i=1..k} i^A000010(k) == 2 (mod k).

Original entry on oeis.org

1, 3, 4, 12, 84, 3612, 94116, 4429004844, 104990793204

Offset: 1

José María Grau Ribas, Nov 30 2019

Apparently includes the sequence 2*A054377.
Additional terms include 4429004844, 104990793204, and 16980843167119376821413542522172.
a(10) > 10^13. - Giovanni Resta, Feb 27 2020

Cf. A000010, A054377, A007850, A330069.

G[n_, k_] := G[n, k] = Mod[Sum[PowerMod[i, k, n], {i, 1, n}], n];
Select[Range[2000], G[#, EulerPhi[#]] == 2 &]
fa=FactorInteger;
se[n_, k_] := Select[Transpose[fa[n]][[1]], IntegerQ[k/(# - 1)] &];
sumlis[li_] := Sum[li[[i]], {i, 1, Length[li]}]
Table[If[Mod[-n/se[n, EulerPhi[n]] // sumlis, n] == 2, n], {n, 1, 1000000}] // Union

isok(n) = sumdiv(n, d, eulerphi(n/d) * Mod(d, n)^eulerphi(n)) == 2; \\ Daniel Suteu, Jan 13 2020

a(8)-a(9) from Giovanni Resta, Feb 27 2020

A330069 Numbers k such that Sum_{i=1..k} i^A000010(k) == -2 (mod k).

Original entry on oeis.org

1, 4, 60, 1716, 3444, 132396, 4428816612, 48846257124

Offset: 1

José María Grau Ribas, Nov 30 2019

Apparently includes the sequence 2*A007850.
Additional terms include 4428816612, 48846257124, 865498410347676, 29474266940021148, 1101686782618260636, 488394001964999430175732692, 1108159829234141602577157118356, 3821334362841015969111519832677012.
a(9) > 10^13. - Giovanni Resta, Feb 27 2020

Cf. A000010, A054377, A007850, A330068.

G[n_, k_] := G[n, k] = Mod[Sum[PowerMod[i, k, n], {i, 1, n}], n];
Select[Range[2000], G[#, EulerPhi[#]] == n-2 &]
fa=FactorInteger;
se[n_, k_] := Select[Transpose[fa[n]][[1]], IntegerQ[k/(# - 1)] &];
sumlis[li_] := Sum[li[[i]], {i, 1, Length[li]}]
Table[If[Mod[-n/se[n, EulerPhi[n]] // sumlis, n] == n-2, n], {n, 1,
   1000000}] // Union

isok(n) = sumdiv(n, d, eulerphi(n/d) * Mod(d, n)^eulerphi(n)) == -2; \\ Daniel Suteu, Jan 13 2020

a(7)-a(8) from Giovanni Resta, Feb 27 2020

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

cp's OEIS Frontend

A168036 Difference between n' and n, where n' is the arithmetic derivative of n (A003415).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

A229300 Numbers n such that A031971(1806*n) == n (mod 1806*n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A235137 a(n) = Sum_{k = 1..n} k^phi(n), where phi(n) = A000010(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A230311 Numbers n such that 1^(k*n) + 2^(k*n) + ... + (k*n)^(k*n) == k (mod k*n) for some k; that is, numbers n such that A031971(k*n) == k (mod k*n) for some k.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

A330068 Numbers k such that Sum_{i=1..k} i^A000010(k) == 2 (mod k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A330069 Numbers k such that Sum_{i=1..k} i^A000010(k) == -2 (mod k).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

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

A229300 Numbers n such that A031971(1806n) == n (mod 1806n).

A230311 Numbers n such that 1^(kn) + 2^(kn) + ... + (kn)^(kn) == k (mod kn) for some k; that is, numbers n such that A031971(kn) == k (mod k*n) for some k.