A054377 -id:A054377 - cp's OEIS frontend

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

A233045 1^m + 2^m + ... + m^m (mod m) for primary pseudoperfect numbers m.

Original entry on oeis.org

1, 1, 1, 1, 5797, 272753965, 8749232767, 1045741078641946876220133713545

Offset: 1

Jonathan Sondow, Dec 10 2013

A031971(m) (mod m) for m in A054377 = 2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086. The known values of m for which 1^m + 2^m + ... + m^m == 1 (mod m) are m = 1, 2, 6, 42, 1806.

For any m and prime p | m, use Sum_{j=1..m} j^m == -m/p (mod p) if p-1 | m or == 0 (mod p) otherwise (see Lemma 3 in Grau et al.) and the Chinese Remainder Theorem.

			The 1st primary pseudoperfect number is 2, and 1^2 + 2^2 = 5 == 1 (mod 2), so a(1) = 1.

J. M. Grau, A. M. Oller-Marcen, and J. Sondow, On the congruence 1^m + 2^m + ... + m^m == n (mod m) with n|m, arXiv:1309.7941 [math.NT].

Cf. A031971, A054377, A231409.

ps={2, 6, 42, 1806, 47058, 2214502422,52495396602, 8490421583559688410706771261086}; fa = FactorInteger; VonStaudt[n_] := Mod[n - Sum[If[IntegerQ[n/(fa[n][[i, 1]] - 1)], n/fa[n][[i, 1]], 0], {i, Length[fa[n]]}], n]; Table[VonStaudt[ps[[i]]], {i, 1, 8}]

a(n) = 1 for n = 1, 2, 3, 4.

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 *)

A283423 Prime power pseudoperfect numbers: numbers m > 1 such that 1/m + Sum 1/p^k = 1, where the sum is over the prime powers p^k | m.

Original entry on oeis.org

2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 64, 100, 128, 162, 256, 272, 294, 342, 486, 500, 512, 1024, 1458, 1806, 2048, 2058, 2500, 4096, 4374, 4624, 6498, 8192, 10100, 12500, 13122, 14406, 16384, 23994, 26406, 32768, 34362, 39366, 47058

Offset: 1

John Machacek, May 27 2017

nonn

Since primary pseudoperfect numbers (A054377) must be squarefree, it follows that primary pseudoperfect numbers are contained in this sequence.
This sequence contains all powers of 2. With the exception of the powers of 2, every prime power pseudoperfect number is a pseudoperfect number (A005835).
Every number in A073935 is a prime power pseudoperfect number (note: this sequence and A073935 agree for many terms but eventually differ starting at 23994 the 38th term of this sequence).
The number 2^k(2^k+1) is the sequence whenever 2^k+1 is a Fermat prime (A019434).

			m = 18 is in the sequence because 1/18 + 1/2 + 1/3 + 1/9 = 1.
m = 12 is NOT in the sequence because 1/12 + 1/2 + 1/4 + 1/3 != 1.

Giovanni Resta, Table of n, a(n) for n = 1..131 (terms < 10^11, first 101 terms from Amiram Eldar)
John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.

Cf. A005835, A054377, A073935.

ok[n_] := Total[n/Flatten@ Table[e[[1]] ^ Range[e[[2]]], {e, FactorInteger[n]}]] + 1 == n; Select[ Range[10^5], ok] (* Giovanni Resta, May 27 2017 *)

A226944 Numbers k such that 1/k + Sum_{p|k} 1/p > 1.

Original entry on oeis.org

30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420, 450, 462, 480, 510, 540, 546, 570, 600, 630, 660, 690, 714, 720, 750, 780, 798, 810, 840, 858, 870, 900, 924, 930, 960, 966, 990, 1020, 1050, 1080, 1092, 1110, 1140, 1170, 1200, 1218, 1230

Offset: 1

José María Grau Ribas, Jun 23 2013

nonn

1/k + Sum_{p|k} 1/p = 1 when k is a primary pseudoperfect number (A054377).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A054377.

fa=FactorInteger; A[n_] := Sum[1/fa[n][[i,1]], {i,Length@fa@n}]; Select[1 + Range@1200, A[#] + 1/# > 1 &]

is(n)=my(t=1-1/n); forprime(p=2,97,if(n%p==0,n/=p^valuation(n,p);t-=1/p; if(t<0,return(1)))); if(n<101^ceil(101*t), return(0)); my(f=factor(n)[,1]); tCharles R Greathouse IV, Dec 27 2013

A270816 For each primary pseudoperfect number n, this sequence gives the sum of (n/p + 1)/p for every prime divisor p of n.

Original entry on oeis.org

1, 3, 17, 691, 17521, 824473683, 19579678305, 3161039281414579992004338982115

Offset: 1

Paolo P. Lava, Mar 23 2016

			Prime factors of 42 are 2, 3 and 7: (42/2 + 1)/2 + (42/3 + 1)/3 + (42/7 + 1)/7 = 11 + 5 + 1 = 17.

Cf. A054377, A270815.

with(numtheory): P:=proc(q) local a,b,k,n,x;
x:=[2, 6, 42, 1806, 47058, 2214502422, 52495396602, 8490421583559688410706771261086];
for n from 1 to nops(x) do a:=ifactors(x[n])[2];
b:=add((x[n]/a[k][1]+1)/a[k][1],k=1..nops(a)); print(b);
od; end: P(10^4);

a(k) = Sum_{prime p|n(k)} (n(k)/p + 1)/p, where n(k) = A054377(k).

A189803 Composite numbers n such that n'' = n'-1 where n' and n'' are the first and the second arithmetic derivative of n.

Original entry on oeis.org

9, 185, 341, 377, 437, 9005, 30413, 33953, 41009, 51533, 82673, 92909, 103073, 126509, 143009, 165773, 181793, 184973, 191309, 228653, 231713, 246893, 291233, 311309, 316973, 319793, 329357, 353009, 358433, 374513, 398093, 405809, 431009, 460193, 467309

Offset: 1

Giorgio Balzarotti, Apr 27 2011

nonn

The sequence A189710 (n"=n'-1) includes all prime numbers because p'=1 and p" = 0. Composite numbers are not very frequent.

Are all terms semiprimes? These terms appear to be p*q such that p+q is a term in A054377, which has solutions to the equation n' = n-1. - T. D. Noe, Apr 27 2011

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

Cf. A002808, A003415, A054377, A068346, A189710.

ader(n) = my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1])); \\ A003415
isok(k) = if (!isprime(k), my(d=ader(k)); ader(d) == d - 1); \\ Michel Marcus, Mar 13 2023

cp's OEIS Frontend

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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A233045 1^m + 2^m + ... + m^m (mod m) for primary pseudoperfect numbers m.

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

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A283423 Prime power pseudoperfect numbers: numbers m > 1 such that 1/m + Sum 1/p^k = 1, where the sum is over the prime powers p^k | m.

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Mathematica

A226944 Numbers k such that 1/k + Sum_{p|k} 1/p > 1.

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PARI

A270816 For each primary pseudoperfect number n, this sequence gives the sum of (n/p + 1)/p for every prime divisor p of n.

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Maple

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A189803 Composite numbers n such that n'' = n'-1 where n' and n'' are the first and the second arithmetic derivative of n.

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PARI