A007850 -id:A007850 - cp's OEIS frontend

A069819 Numbers k such that 1/(Sum_{p|k} (1/p) - 1), where p are the prime divisors of k, is a positive integer.

30, 60, 90, 120, 150, 180, 240, 270, 300, 360, 450, 480, 540, 600, 720, 750, 810, 858, 900, 960, 1080, 1200, 1350, 1440, 1500, 1620, 1716, 1722, 1800, 1920, 2160, 2250, 2400, 2430, 2574, 2700, 2880, 3000, 3240, 3432, 3444, 3600, 3750, 3840, 4050, 4320, 4500

Offset: 1

Benoit Cloitre, Apr 28 2002

Sequence is generated by A007850(n). For example: 30, 858, 1722 (30 = 2*3*5, 858 = 2*3*11*13, 1722 = 2*3*11*13) generate numbers of the form 2^a*3^b*5^c (A143207), 2^a*3^b*7^c*41^d, 2^a*3^b*11^c*13^d, (a,b,c,d => 1), which are in the sequence.

Equivalently, numbers k such that Sum_{p|k} 1/p - Product_{p|k} 1/p, where p are the prime divisors of k, is a positive integer. All these terms have at least 3 prime factors. When k is a term and p is a prime divisor of k, then p*k is another term (see Diophante link). - Bernard Schott, Dec 19 2021

			For k = 30 = 2*3*5, 1/(Sum_{p|n} (1/p) - 1) = 1/(1/2 + 1/3 + 1/5 - 1) = 30 hence 30 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..4233 (terms below 10^10)
Diophante, A1862 - Inversons les facteurs (in French).

Cf. A007850.

A143207 is a subsequence.

Select[Range[4320], (sum = Plus @@ (1/FactorInteger[#][[;;,1]])) > 1 && IntegerQ[1/(sum - 1)] &] (* Amiram Eldar, Feb 03 2020 *)

isok(k) = my(f=factor(k), x=1/(sum(i=1, #f~, 1/f[i,1]) -1)); (x>1) && (denominator(x)==1); \\ Michel Marcus, Dec 19 2021

from sympy import factorint
from fractions import Fraction
def ok(n):
    s = sum(Fraction(1, p) for p in factorint(n))
    return s > 1 and (s - 1).numerator == 1
print([k for k in range(1, 4501) if ok(k)]) # Michael S. Branicky, Dec 19 2021

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 */

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.

A369470 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, 2, 35, 455, 13624, 1176579

Offset: 1

Max Alekseyev, Jan 23 2024

For any n, a(n) >= A369469(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. A002966, A007850, A054377, A085098, A118086, A158649, A164014, A342267, A348625, A369469, A369607.

A075461 List of solutions to the Znám problem sorted first by length, then lexicographically.

Original entry on oeis.org

2, 3, 7, 47, 395, 2, 3, 11, 23, 31, 2, 3, 7, 43, 1823, 193667, 2, 3, 7, 47, 403, 19403, 2, 3, 7, 47, 415, 8111, 2, 3, 7, 47, 583, 1223, 2, 3, 7, 55, 179, 24323, 2, 3, 7, 43, 1807, 3263447, 2130014000915, 2, 3, 7, 43, 1807, 3263591, 71480133827, 2, 3, 7, 43

Offset: 1

Eric W. Weisstein, Sep 16 2002

			Starts with A075441(5)=2 5-term solutions 2,3,7,47,395; 2,3,11,23,31, followed by A075441(6)=5 6-term solutions, etc.

Max Alekseyev, Table of n, a(n) for n = 1..934 (flattened list of the n-term solutions for n = 5..8)
J. Janák and L. Skula, On the integers xi for which xi | x1 . . . xi-1 xi . . . xn + 1 holds, Mathematica Slovaca, Vol. 28 (1978), No. 3, 305--310.
J. Sondow and K. MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240; arXiv:math/1812.06566 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Znám's Problem
Wikipedia, Znám's problem.

Cf. A006585, A007850, A054377, A075441, A085098, A118086, A158649, A164014, A343074, A369469.

Edited by Max Alekseyev, Jan 25 2024

A187929 Odd numbers k such that 1^((k-1)/2) + 2^((k-1)/2) + .... + (k-1)^((k-1)/2) == 0 (mod k).

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 27, 29, 31, 35, 37, 39, 41, 43, 47, 51, 53, 55, 59, 61, 63, 67, 71, 73, 75, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 111, 113, 115, 119, 123, 125, 127, 131, 135, 137, 139, 143, 147, 149, 151, 155, 157, 159

Offset: 1

José María Grau Ribas, Mar 16 2011

nonn

Asymptotic density is 0.379...

Amiram Eldar, Table of n, a(n) for n = 1..10000
José María Grau, Florian Luca and Antonio M. Oller-Marcén, On a variant of Giuga numbers, Acta Mathematica Sinica, English Series, Vol. 28 No. 4 (2012), pp 653-660; preprint, arXiv:1103.3428 [math.NT], 2011.

Cf. A007850.

gi[n_]:=Mod[Sum[PowerMod[j,(n-1)/2,n],{j,n-1}],n]; Select[ Range[1,300,2], gi[#]==0&]

is(n)=my(e=(n-1)/2);sum(k=1,n-1,Mod(k,n)^e)==0;
select(is,vector(1000,i,2*i-1)) \\ on older versions, switch the arguments
\\ Charles R Greathouse IV, Mar 19 2011

A189941 Numbers n such that n''' = n''+ 1 where n'' and n''' are respectively the second and the third arithmetic derivative of n.

Original entry on oeis.org

186, 258, 322, 338, 3866, 4326, 4775, 18830, 19122, 27586, 34330, 34538, 41626, 46762, 49858, 49922, 54298, 55810, 70510, 82122, 86938, 89102, 101042, 101706, 106442, 110510, 112910, 118586, 120822, 129722, 133430, 134714, 150742, 157362, 158235, 163410

Offset: 1

Giorgio Balzarotti, May 01 2011

nonn

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

			186'= 161; 186"=161' = 30; 186"'=30'= 31-> 186'''= 186" +1 -> a(1)=186.

Donovan Johnson, Table of n, a(n) for n = 1..673

Cf. A003415, A007850.

Using 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(der(i)))-der(der(i))-1: if a=0 then j:=j+1; A[j]:=i: end if od

A190014 a(0)=0, a(1)=1, if n = (n-1)', then a(n)=0 otherwise a(n)=2*a((n-1)'), where n' is the arithmetic derivative of n.

Original entry on oeis.org

0, 1, 0, 2, 2, 4, 2, 8, 2, 4, 4, 16, 2, 8, 2, 8, 4, 4, 2, 8, 2, 4, 8, 16, 2, 4, 8, 16, 32, 4, 2, 0, 2, 4, 4, 16, 4, 4, 2, 8, 8, 4, 2, 8, 2, 4, 16, 8, 2, 32, 4, 8, 4, 16, 2, 512, 8, 8, 16, 0, 2, 8, 2, 8, 16, 4, 4, 16, 2, 4, 16, 0, 2, 16, 2, 16, 1024, 4, 4, 0, 2, 256, 4, 16, 2, 8, 16, 8, 4, 4, 2, 32, 4, 8, 8, 64, 4, 4, 2, 8, 32, 4

Offset: 0

Paolo P. Lava, May 04 2011

nonn

Only power of 2 and zeros. If p is prime than a(p+1)=2.
If n’>n+1 than a(n+1) is not immediately available. It is necessary to find a(n’)=2*a((n’-1)’) and, if necessary, to repeat the process until a term can be calculate. For instance:
a(9)=2*a(12) -> a(12)=2 and therefore a(9)=4.
Again: a(55)=2*a(81) -> a(81)=2*a(176) -> a(176)=8*a(112) -> a(112)=16 and therefore a(176)=128 -> a(81)=256 -> a(55)=512.
First zero at a(31)=2*a(31) and for all Giuga numbers plus one (31, 859, 1723, 66199, etc.). This because the so far known Giuga numbers satisfy the equation n’=n+1. Other zeros for a(59)=8*a(31), a(71)=16*a(31), a(79)=32*a(31),  a(106)=32*a(31), etc.
The general equation a(n+1)=k*a(n’), with k integer and |k|>1, a(0)=0, a(1)=1, leads to the following sequence:  0, 1, 0, k, k, k^2, k, k^3,k, k^2, k^2, k^4, k, k^3, k, k^3, k^2, k^2, k, k^3, etc.
For k=1 or k=-1 we get and incongruence because of a(31)=a(31).

			a(0)=0
a(1)=1
a(2)=a(1+1)=2*a(1')=2*a(0)=0
a(3)=a(2+1)=2*a(2')=2*a(1)=2
a(4)=a(3+1)=2*a(3')=2*a(1)=2
a(5)=a(4+1)=2*a(4')=2*a(4)=4
a(6)=a(5+1)=2*a(5')=2*a(1)=2
a(7)=a(6+1)=2*a(6')=2*a(5)=8  etc.

Paolo P. Lava, Table of n, a(n) for n = 0..5000
Paolo P. Lava, Plot of the first 5000 terms of the sequence

Cf. A003415, A007850,

with(numtheory);
P:=proc(i)
local a,f,n,p,pfs,t;
a:=array(0..100000); a[0]:=0; a[1]:=1; t:=2; lprint(0,a[0]); lprint(1,a[1]);
for n from 1 by 1 to i do
    pfs:=ifactors(n)[2]; f:=n*add(op(2,p)/op(1,p),p=pfs);
    a[n+1]:=t*a[f]; lprint(n+1,a[n+1]);
od;
end:

dn[0] = 0; dn[1] = 0; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; a[0] = 0; a[1] = 1; a[n_] := a[n] = Module[{d = dn[n - 1]}, If[d == n, 0, 2 a[d]]]; Array[a,100,0] (* T. D. Noe, May 05 2011 *)

A203617 Numbers m such that (m'-1)' = m+1, where m' denotes the arithmetic derivative of m.

Original entry on oeis.org

30, 210, 246, 858, 1722, 66198, 235290, 282342, 1929378, 1976394, 2214408306

Offset: 1

Paolo P. Lava, Jan 20 2012

The differential equation whose solutions are the Giuga 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 Giuga numbers plus pairs of numbers (x,y) for which x' = y+1 and y' = x+1.
A007850 is a subsequence of this sequence.
a(11) > 10^9. - Michel Marcus, Nov 05 2014
a(12) > 10^10. - Giovanni Resta, Jun 04 2016

			235290' = 282343; (282343 - 1)' = 282342' = 235291 = 235290 + 1, so 235290 is a term.
282342' = 235291; (235291 - 1)' = 235290' = 282343 = 282342 + 1, so 282342 is a term.

Cf. A007850, A185222, A203618.

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);

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

a(11) from Giovanni Resta, Jun 04 2016

A235139 Twin primes p, p+2 such that p+1 is a primary pseudoperfect number.

Original entry on oeis.org

5, 7, 41, 43, 47057, 47059

Offset: 1

Jonathan Sondow and Emmanuel Tsukerman, Jan 04 2014

Same as twin primes p, p+2 such that (p+1)*(p+2) is a primary pseudoperfect number (A054377). Appears also to be same as twin primes p, p+2 such that p*(p+1) is a Giuga number (A007850). See the link "The p-adic order of power sums...": Theorem 8, Example 1, and Question 1.
Is it always true that if a primary pseudoperfect number N > 2 is adjacent to a prime N-1 or N+1, then in fact N lies between twin primes N-1, N+1? For all 7 known primary pseudoperfect numbers N > 2, either both N-1 and N+1 are prime or neither is prime.
See A235364 for a similar property of Giuga numbers.

			For the twin primes (p,p+2) = (5, 7), (41, 43), (47057, 47059), the numbers p+1 = 6, 42, 47058 and (p+1)*(p+2) = 42, 1806, 2214502422 are primary pseudoperfect numbers, and p*(p+1) = 30, 1722, 2214408306 are Giuga numbers.

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.
MathWorld, Giuga Number
Wikipedia, Giuga number
Wikipedia, Primary pseudoperfect number

Cf. A007850, A054377.

A054377 = Cases[Import["https://oeis.org/A054377/b054377.txt", "Table"], {, }][[All, 2]];
lst = {}; For[i = 1, i <= Length[A054377], i++, n = A054377[[i]];
If[PrimeQ[n + 1] && PrimeQ[n - 1], AppendTo[lst, n - 1]; AppendTo[lst, n + 1]]]; lst (* Robert Price, Mar 14 2020 *)

cp's OEIS Frontend

A069819 Numbers k such that 1/(Sum_{p|k} (1/p) - 1), where p are the prime divisors of k, is a positive integer.

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

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A369470 a(n) = number of integer solutions to 1 <= x1 <= x2 <= ... <= xn to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn).

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A075461 List of solutions to the Znám problem sorted first by length, then lexicographically.

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A187929 Odd numbers k such that 1^((k-1)/2) + 2^((k-1)/2) + .... + (k-1)^((k-1)/2) == 0 (mod k).

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A189941 Numbers n such that n''' = n''+ 1 where n'' and n''' are respectively the second and the third arithmetic derivative of n.

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Maple

A190014 a(0)=0, a(1)=1, if n = (n-1)', then a(n)=0 otherwise a(n)=2*a((n-1)'), where n' is the arithmetic derivative of n.

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A203617 Numbers m such that (m'-1)' = m+1, where m' denotes the arithmetic derivative of m.

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