A007850 -id:A007850 - cp's OEIS frontend

A326715 Values of n for which the denominator of (Sum_{prime p | n} 1/p - 1/n) is 1.

1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 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, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241

Offset: 1

Jonathan Sondow, Jul 20 2019

nonn

n is in the sequence iff either n = 1 or n is a prime or n is a Giuga number, by one definition of Giuga numbers A007850.

			a(30) = denominator(Sum_{prime p | 30} 1/p - 1/30) = denominator(1/2 + 1/3 + 1/5 - 1/30) = denominator(1/1) = 1, and 30 is a Giuga number.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Giuga number

Cf. A007850, A326689, A326690, A326691, A326692.

filter:= proc(n) local p;
   denom(add(1/p, p = numtheory:-factorset(n))-1/n)=1
end proc:
select(filter, [$1..300]); # Robert Israel, Dec 15 2020

PrimeFactors[n_] := Select[Divisors[n], PrimeQ];
f[n_] := Denominator[Sum[1/p, {p, PrimeFactors[n]}] - 1/n];
Select[Range[148], f[#] == 1 &]

n such that A326690(n) = 1.

A094960 Positive integers k such that the derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 28, 30, 36, 60

Offset: 1

Benoit Cloitre, Jun 19 2004

From Max Alekseyev, Dec 08 2011: (Start)
There are no other terms below 10^9.
k belongs to this sequence if k*binomial(k-1,m)*Bernoulli(m) is an integer for each m in 0..k-1. (End)
From Max Alekseyev, Jun 04 2012: (Start)
If for a prime p >= 3, k ends with base-p digits a,b with a+b >= p, then for m = (a+1)*(p-1), the number k*binomial(k-1,m)*Bernoulli(m) is not an integer (it contains p in the denominator). For p=3, this implies that k == 5, 7, or 8 (mod 9) are not in this sequence; for p=5, this implies that k == 9, 13, 14, 17, 18, 19, 21, 22, 23, or 24 (mod 25) are not in this sequence; and so on.
Conjecture: for every integer k > 78, there exists a prime p >= 3 such that the sum of last two base-p digits of k is at least p. This conjecture would imply that this sequence is finite and 60 is the last term. (End)
The conjecture is true for all k such that k+1 is not a prime, a power of 2, or a Giuga number (A007850). In this case, there exists a prime p >= 3 such that the base-p representation of k ends in a,p-1 with a > 0. - Max Alekseyev, Feb 16 2021
The sequence is finite and is a subsequence of A366169. The terms are those numbers k where A324370(k) = 1. It remains to show that 60 is the last term. This is very likely, since the terms depend on the estimation of a product of primes satisfying certain p-adic conditions that is connected with A324370. A proven asymptotic formula related to that product implies that this sequence is finite. See Kellner 2017, 2023, and BLMS 2018. - Bernd C. Kellner, Oct 02 2023

			B(6,x) = x^6 - 3*x^5 + (5/2)*x^4 - (1/2)*x^2 + 1/42 so B'(6,x) contains only integer coefficients and 6 is in the sequence.

Olivier Bordellès, Florian Luca, Pieter Moree, and Igor E. Shparlinski, Denominators of Bernoulli polynomials, Mathematika 64 (2018), 519-541.
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.;
arXiv:2310.01325 [math.NT], 2023.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A094960, A144845, A160014, A195441, A324370, A366168, A366169.

p := n -> if denom(diff(bernoulli(n, x), x)) = 1 then n else fi:
seq(p(n), n=1..100); # Emeric Deutsch

(* From Bernd C. Kellner, Oct 02 2023. (Start) *)
(* k-th derivative of BP: *)
k = 1; Select[Range[1000], Denominator[Together[D[BernoulliB[#, x],{x, k}]]] == 1&]
(* Exact denominator formula: *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
DBP[n_, k_] := Module[{m = n-k+1, fac = FactorialPower[n, k]}, If[n < 1 || k < 1 || n <= k, Return[1]]; Times@@Select[Prime[Range[PrimePi[(m+1)/(2 + Mod[m+1, 2])]]], !Divisible[fac, #] && SD[m, #] >= #&]];
k = 1; Select[Range[1000], DBP[#, k] == 1&]
(* End *)

is_A094960(k) = !#select(x->(denominator(x)!=1), Vec(deriv(bernpol(k)))); \\ Michel Marcus, Feb 15 2021

from itertools import count, islice
from sympy import Poly, diff, bernoulli
from sympy.abc import x
def A094960_gen(): # generator of terms
    return filter(lambda k:k<=1 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x)).coeffs()),count(1))
A094960_list = list(islice(A094960_gen(),10)) # Chai Wah Wu, Oct 03 2023

k is a term if A324370(k) = 1. - Bernd C. Kellner, Oct 02 2023

k is a term <=> 0 = Sum_{j=0..k-1} k*binomial(k - 1, j) mod Clausen(j), where Clausen(n) = A160014(n, 1). - Peter Luschny, Oct 04 2023

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

A198391 Numbers m such that Sum_{i=1..k} (1-1/p_i) + Product_{i=1..k} (1-1/p_i) is an integer, where p_i are the k prime factors of m (with multiplicity).

Original entry on oeis.org

2, 15, 20, 272, 476, 19024, 47425, 65792, 125172, 216900, 539280, 1222976, 1372736, 2770496, 3494336, 5321808, 5844528, 6177168, 7032528, 8885808, 20670768, 60727876, 69081344, 82724356, 95579136, 544382208, 907440192, 1657497600, 4295032832, 5048574976

Offset: 1

Paolo P. Lava, Oct 24 2011

nonn

The numbers of the sequence are solutions of the differential equation m’=(k-a)*m+b, which can be written as A003415(m)=(k-a)*m+A003958(m), where k is the number of prime factors of m, and a is the integer Sum_{i=1..k} (1-1/p_i) + Product_{i=1..k} (1-1/p_i).
If k = a we have m’ = b or A003415(m) = A003958(m). For instance 15 has prime factors 3, 5; its arithmetic derivative is 15’ = 8 and b = 3*5 - 3 - 5 + 1 = 8. The term 47425 has prime factors 5, 5, 7, 271. Its arithmetic derivative is 47425’ = 25920 and b = 5*5*7*271 - 5*5*7 - 5*5*271 - 5*7*271 - 5*7*271 + 5*5 + 5*7 + 5*271 + 5*7 + 5*271 + 7*271 - 5 - 5 - 7 - 271 + 1 = 25920.
The numbers of the sequence satisfy also Sum_{i=1..k} (1+1/p_i) - Product_{i=1..k} (1-1/p_i) = some integer.

			125172 has prime factors 2, 2, 3, 3, 3, 19, 61. 1 - 1/2 + 1 - 1/2 + 1 - 1/3 + 1 - 1/3 + 1 - 1/3 + 1 - 1/19 + 1 - 1/61 = 5715/1159 is the sum over the 1-1/p_i. (1-1/2) * (1-1/2) * (1-1/3) * (1-1/3) * (1-1/3) * (1-1/19) * (1-1/61) = 80/1159 is the product of the 1-1/p_i. The sum over sum and product is 5715/1159 + 80/1159 = 5, an integer.

Giovanni Resta, Table of n, a(n) for n = 1..48 (terms < 10^12)
J. M. Borwein and E. Wong, A survey of results relating to Giuga’s conjecture on primality, May 8, 1995.
Romeo Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

Cf. A003415, A003958, A007850, A199767.

isA198391 := proc(n)
    p := ifactors(n)[2] ;
    add(op(2,d)-op(2,d)/op(1,d),d=p) + mul((1-1/op(1,d))^op(2,d),d=p) ;
    type(%,'integer') ;
end proc:
for n from 2 to 20000000 do
    if isA198391(n) then
        printf("%d,\n",n);
    end if;
end do: # R. J. Mathar, Nov 26 2011

Select[Range[2, 10^5], IntegerQ[(Plus @@ # + Times @@ #) &@ (1 - 1/ Flatten[ Table[#1, {#2}] & @@@ FactorInteger@#])] &] (* Giovanni Resta, May 23 2016 *)

Missing a(23) and a(26)-a(30) from Giovanni Resta, May 23 2016

A342922 Numbers k such that A342925(k) = k + 2*A003415(k).

Original entry on oeis.org

6, 28, 29, 496, 857, 1721, 8128, 164284, 6511664, 33550336, 400902412, 8589869056

Offset: 1

Antti Karttunen, Apr 07 2021

Question: Are all odd terms in A001359?

Certainly any prime p such that A003415(p+1) = p + 2 satisfies the equation. See comments in A007850.

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Cf. A000396 (subsequence), A001359, A003415, A007850.

Cf. A203617, A342923, A342924, A342925.

Select[Range[2*10^5], #3 == #1 + 2 #2 & @@ Prepend[Map[If[# < 2, 0, # Total[#2/#1 & @@@ FactorInteger[#]]] &, {#, DivisorSigma[1, #]}], #] &] (* Michael De Vlieger, Feb 25 2022 *)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342925(n) = A003415(sigma(n));
isA342922(n) = (A342925(n)==(n+(2*A003415(n))));

Terms a(11) and a(12) from Antti Karttunen, Feb 25 2022

cp's OEIS Frontend

A326715 Values of n for which the denominator of (Sum_{prime p | n} 1/p - 1/n) is 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A094960 Positive integers k such that the derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

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.