A063528
Smallest number such that it and its successor are both divisible by an n-th power larger than 1.
Original entry on oeis.org
2, 8, 80, 80, 1215, 16767, 76544, 636416, 3995648, 24151040, 36315135, 689278976, 1487503359, 1487503359, 155240824832, 785129144319, 4857090670592, 45922887663615, 157197025673216, 1375916505694208, 2280241934368767, 2280241934368767, 2280241934368767
Offset: 1
a(4) = 80 since 2^4 = 16 divides 80 and 3^4 = 81 divides 81.
- J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 242, p. 67, Ellipses, Paris 2008.
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k = 4; Do[k = k - 2; a = b = 0; While[ b = Max[ Transpose[ FactorInteger[k]] [[2]]]; a <= n || b <= n, k++; a = b]; Print[k - 1], {n, 0, 19} ]
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b(n,p=2,q=3)=local(i);i=Mod(p,q^n)^-n; min(p^n*lift(i)-1,p^n*lift(-i))
a(n)=local(r);r=b(n);if(r>5^n,r=min(r,min(b(n,2,5),b(n,3,5))));r /* Franklin T. Adams-Watters, May 27 2011 */
A174113
Smallest number k such that k, k+1, and k+2 are all divisible by an n-th power.
Original entry on oeis.org
48, 1375, 33614, 2590623, 26890623, 2372890624, 70925781248, 2889212890624, 61938212890624, 4497636425781248, 8555081787109375, 2665760081787109375, 98325140081787109375, 198816740081787109374, 11776267480163574218750, 872710687480163574218750, 50783354512519836425781248
Offset: 2
a(3) = 1375 because
1375 = 11 * 5^3;
1376 = 172 * 2^3;
1377 = 51 * 3^3.
- J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 1375, p. 135, Ellipses, Paris 2008.
Cf.
A068780,
A068781,
A068140,
A068782,
A068783,
A068784,
A045330,
A059737,
A063528,
A051903,
A051903.
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with(numtheory):for n from 2 to 6 do: i:=0:for k from 1 to 3000000 while(i=0) do:j:=0:
for a from 0 to 2 do: ii:=0:for m from 1 to 4 while(ii=0) do:p:=ithprime(m)^n:if irem(k+a,p)=0 then j:=j+1:ii:=1:else fi:od:od:if j=3 then i:=1:print(k):else fi:od:od:
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a(n)=my(ch,t,best=30^n);forprime(a=2, 29, forprime(b=2, 29, if(a==b,next); ch=chinese(Mod(0,a^n), Mod(-1,b^n)); if(lift(ch)>=best, next); forprime(c=2, 29, if(a==c || b==c, next); t=lift(chinese(ch, Mod(-2, c^n))); if(tCharles R Greathouse IV, Jan 16 2012
A045330
Lesser of the smallest pair of consecutive numbers divisible by an n-th power, but not both divisible by an (n+1)-st power.
Original entry on oeis.org
2, 8, 135, 80, 1215, 16767, 76544, 636416, 3995648, 24151040, 36315135, 689278976, 11573190656, 1487503359, 155240824832, 785129144319, 4857090670592, 45922887663615, 157197025673216, 1375916505694208
Offset: 0
A358818
a(n) is the least number k such that A046660(k) = A046660(k+1) = n.
Original entry on oeis.org
1, 44, 135, 80, 8991, 29888, 123200, 2316032, 1043199, 24151040, 217713663, 689278976, 11573190656, 76876660736, 311969153024, 2035980763136, 2741258240000, 215189482110975
Offset: 0
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e[n_] := PrimeOmega[n] - PrimeNu[n]; a[n_] := Module[{k = 1}, While[e[k] != n || e[k + 1] != n, k++]; k]; Array[a, 10, 0]
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e(n) = {my(f = factor(n)); bigomega(f) - omega(f)};
a(n) = {my(k=1); while(e(k) != n || e(k+1) !=n , k++); k};
Showing 1-4 of 4 results.
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