A057896 -id:A057896 - cp's OEIS frontend

A057895 Nonnegative numbers which can be written as m^k-m [with m and k nonnegative and 0^0 taken as 1].

0, 1, 2, 6, 12, 14, 20, 24, 30, 42, 56, 60, 62, 72, 78, 90, 110, 120, 126, 132, 156, 182, 210, 240, 252, 254, 272, 306, 336, 342, 380, 420, 462, 504, 506, 510, 552, 600, 620, 650, 702, 720, 726, 756, 812, 870, 930, 990, 992, 1020, 1022, 1056, 1122, 1190, 1260

Offset: 0

Henry Bottomley, Sep 26 2000

nonn

			a(8)=30 is in the sequence since 30=2^6-2 (and also =6^2-6).

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A000918, A057896, A308324.

N:= 1000: # to get all entries up to N
S:= {0,1}:
for m from 2 to floor((1+sqrt(1+4*N))/2) do
     S:= S union {seq(m^k - m,k=2 .. floor(log(N+m)/log(m)))}
end do:
S; # Robert Israel, Feb 12 2013

A306746 A Goldbug number is an even number 2m for which there exists a subset of the prime non-divisors, P={p1, p2, p3, ..., pk}, of 2m where (2m-p1)(2m-p2)(2m-p3)...(2m-pk) has only elements of P as factors and one of the pi is between m/2 and m for even m and between (m+1)/2 and m-1 for odd m.

Original entry on oeis.org

128, 1718, 1862, 1928, 6142

Offset: 1

Craig J. Beisel, Mar 07 2019

A Goldbug number is an even number 2m for which there exists some subset of the prime non-divisors (PNDs) of 2m, 2 < p1 < p2 < p3 < ... < pk < m, such that (2m-p1)*(2m-p2)*(2m-p3)*...*(2m-pk) has only p1,p2,p3,...,pk as factors and one of the pi is between n/2 and n for even n and between (n+1)/2 and n-1 for odd n. We do not need to consider the case where n is prime, since then n itself is a Goldbach pair. A Goldbug number is called order k if the maximal subset satisfying the property is of size k. These numbers arise from Goldbug's Algorithm which attempts to find a Goldbach pair for a particular even number by starting with a given PND p1 and successively adding the factors of the product (2m - p1)*...*(2m - pk) to the search until a pair is found. Goldbug numbers are those even numbers for which Goldbug's Algorithm is not guaranteed to find a Goldbach pair since it could reach a subset of the PNDs which does not contain new information about additional PNDs to add to the search.
Goldbug numbers are a special case of Basic Pipes as defined by Wu. It has been shown computationally a(7) > 5*10^8. See link.
Goldbug numbers serve as a link between Goldbach's conjecture and the Pillai conjecture since order 2 Goldbug numbers represent solutions to its generalized difference equation. For example, sequence A057896 demonstrates there are no order 2 Goldbugs less than 10^24 since it would imply additional solutions to the equation a^x-a = b^y-b. In fact, theorem 3 from Scott[1993] implies that no additional order 2 Goldbugs exist at all.

			Although 2200 and the prime non-divisors 3 and 13 might seem to satisfy the definition since (2200 - 13)*(2200 - 3) = 4804839 = 3^7*13^3, 2200 is not an order k=2 Goldbug since neither 3 or 13 is in the interval (n/2,n).
A higher-order example is the term 128, for which there exists a subset of the PNDs such that the corresponding product (128 - 3)*(128 - 5)*(128 - 7)*(128 - 11)*(128 - 13)*(128 - 17)*(128 - 23)*(128 - 29)*(128 - 37)*(128 - 41)*(128 - 43)*(128 - 47)*(128 - 53)*(128 - 59) = 8147166895749452778629296875 = (3^14)*(5^8)*(7^2)*(11^3)*(13^2)*17*(23^2)*29*37*41 and 37 and 41 are in the interval (32,64). Therefore, 128 is a Goldbug number of order k=14.

Craig J. Beisel, Maximal Sets of PNDs Satisfying Goldbug Property for First 5 Terms
Craig J. Beisel, Enumeration of all Goldbug subsets for the term 128.
Craig J. Beisel, Goldbug's Algorithm.
Andrzej Bożek, Exceptional autonomous components of Goldbach factorization graphs, arXiv:1909.09900 [math.NT], 2019.
Bert Dobbelaere, C++ Program.
Christian Goldbach, Letter to L. Euler, June 7, 1742.
Mathematics Stack Exchange, Searching for Goldbug Numbers.
Reese Scott, On the Equations p^x - b^y = c and a^x + b^y = c^z, Journal of Number Theory, Volume 44, Issue 2, June 1993.
Willie Wu, Pipe Theory.
Index entries for sequences related to Goldbach conjecture

Cf. A244408, A057896.

isgbk(n,k) = {if (n % 2, return (0)); f=factor(n) [, 1]; vp = setminus(primes([3, n/2]), f~); forsubset([#vp,k], s, w=vecextract(vp, s); if(#w>1 && setminus(factor(x=prod(i=1, #s, n-w[i]))[, 1]~, Set(w))==[], return(1)););return(0);} \\ tests if n is order k Goldbug;

A308394 Numbers which can be written in the form m^k - m with m prime and k a positive integer.

Original entry on oeis.org

0, 2, 6, 14, 20, 24, 30, 42, 62, 78, 110, 120, 126, 156, 240, 254, 272, 336, 342, 506, 510, 620, 726, 812, 930, 1022, 1320, 1332, 1640, 1806, 2046, 2162, 2184, 2394, 2756, 3120, 3422, 3660, 4094, 4422, 4896, 4970, 5256, 6162, 6558, 6806, 6840, 7832, 8190, 9312

Offset: 1

Craig J. Beisel, May 24 2019

The only known terms which have two representations where m is prime are 6 and 2184. It is conjectured by Bennett these are the only terms with this property.

			a(9) = 2^6 - 2 = 62.
For the two terms known to have two representations we have a(3) = 6 = 2^3 - 2 = 3^2 - 3 and a(33)= 2184 = 3^7 - 3 = 13^3 - 13.

Robert Israel, Table of n, a(n) for n = 1..10000
Michael Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.
Dana Mackenzie, 2184: An Absurd (and Adsurd) Tale, Integers (Electronic Journal of Combinatorial Number Theory), 18 (2018), A33.

Cf. A057895, A057896, A246068, A308324.

Subsequences: A000918 (2^n - 2), A036689 (p^2 - p), A058809 (3^n - 3), A178671 (5^n - 5).

N:= 10^6; # to get all terms <= N
P:= select(isprime,[2,seq(i,i=3..floor((1+sqrt(1+4*N))/2),2)]):
S:= {0,seq(seq(m^k-m,k=2..floor(log[m](N+m))),m=P)}:
sort(convert(S,list)); # Robert Israel, Aug 11 2019

x=List([]); lim=10000; forprime(m=2, lim, for(k=1, 100, y=(m^k-m); if(y>lim, break, i=setsearch(x, y, 1); if(i>0, listinsert(x, y, i))))); for(i=1, #x, print(x[i]));

isok(n) = {forprime(p=2, oo, my(keepk = 0); for (k=1, oo, if ((x=p^k - p) == n, return(1)); if (x > n, keepk = k; break);); if (keepk == 2, break););} \\ Michel Marcus, Aug 06 2019

cp's OEIS Frontend

A057895 Nonnegative numbers which can be written as m^k-m [with m and k nonnegative and 0^0 taken as 1].

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A306746 A Goldbug number is an even number 2m for which there exists a subset of the prime non-divisors, P={p1, p2, p3, ..., pk}, of 2m where (2m-p1)(2m-p2)(2m-p3)...(2m-pk) has only elements of P as factors and one of the pi is between m/2 and m for even m and between (m+1)/2 and m-1 for odd m.

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A308394 Numbers which can be written in the form m^k - m with m prime and k a positive integer.

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Author

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Maple

PARI

PARI