A089212
Primes p such that p-1 and p+1 are divisible by a fifth power.
Original entry on oeis.org
13121, 20897, 25759, 75329, 80191, 106433, 118751, 137537, 153089, 157951, 176417, 191969, 196831, 207521, 212383, 215297, 230849, 243487, 251263, 274591, 281249, 285281, 313471, 318751, 321247, 324161, 331937, 336799, 347489, 378593
Offset: 1
13121 is a term since 13121 - 1 = 2^6 * 5 * 41, 13121 + 1 = 2 * 3^8.
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f[n_]:=Max[Last/@FactorInteger[n]]; lst={};Do[p=Prime[n];If[f[p-1]>=5&&f[p+1]>=5,AppendTo[lst,p]],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 03 2009 *)
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\\ Input no. of iterations n, power p and number to subtract and add k.
ispowerfree(m,p1) = { flag=1; y=component(factor(m),2); for(i=1,length(y), if(y[i] >= p1,flag=0;break); ); return(flag) }
powerfreep4(n,p,k) = { c=0; pc=0; forprime(x=2,n, pc++; if(!ispowerfree(x-k,p) && !ispowerfree(x+k,p), c++; print1(x","); ) ); print(); print(c","pc","c/pc+.0) }
A166003
Primes p such that p+-1, p+-2 and p+-3 are not squarefree.
Original entry on oeis.org
47527, 186247, 218527, 245149, 269953, 377543, 390449, 432277, 447823, 453053, 469649, 518123, 568177, 584911, 589273, 606323, 632347, 661547, 761347, 831751, 848213, 897577, 913327, 925949, 952253, 1172351, 1205647, 1220347, 1241477
Offset: 1
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f[n_]:=Max[Last/@FactorInteger[n]]; q=2;lst={};Do[p=Prime[n];If[f[p-3]>=q&&f[p-2]>=q&&f[p-1]>=q&&f[p+1]>=q&&f[p+2]>=q&&f[p+3]>=q,AppendTo[lst,p]],{n,6*8!}];lst
Select[Prime[Range[100000]],NoneTrue[#+{-3,-2,-1,1,2,3},SquareFreeQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 17 2018 *)
A166000
Primes p such that p-5, p-3, p+3, and p+5 are divisible by cubes.
Original entry on oeis.org
12253, 14747, 65173, 83003, 93253, 95747, 109139, 147253, 176747, 213349, 255253, 282253, 284747, 287437, 305267, 311747, 315517, 336253, 338747, 364699, 365747, 444253, 452579, 471253, 525253, 554747, 583789, 633253, 716747, 741253, 743747
Offset: 1
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filter:= proc(p) local d;
if not isprime(p) then return false fi;
for d in [-5,-3,3,5] do
if max(map(t -> t[2], ifactors(p+d)[2])) < 3 then return false fi;
od;
true
end proc:
select(filter, [seq(t,t=7..10^6,2)]); # Robert Israel, Apr 21 2016
# alternative
isA166000 := proc(n)
if isprime(n) then
isA046099(n-3) and isA046099(n+3) and isA046099(n-5) and isA046099(n+5) ;
else
false;
end if;
end proc: # R. J. Mathar, Aug 14 2024
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f[n_]:=Max[Last/@FactorInteger[n]]; q=3;lst={};Do[p=Prime[n];If[f[p-5]>=q&&f[p-3]>=q&&f[p+3]>=q&&f[p+5]>=q,AppendTo[lst,p]],{n,4*8!}];lst
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ncf(n)={vecmax(factor(n)[,2])>2};forprime(p=5,1e7,if(ncf(p+5)&&ncf(p+3)&&ncf(p-3)&&ncf(p-5),print1(p","))) /* Charles R Greathouse IV, Oct 05 2009 */
A166001
Primes p such that p-5, p-4, p+4, and p+5 are each divisible by a cube > 1.
Original entry on oeis.org
751379, 2414507, 2839621, 3170371, 4469629, 5736371, 21154909, 22556371, 22991629, 23313371, 23748629, 24338371, 28372621, 31628371, 32079757, 33009629, 41078371, 42270629, 43465307, 44446621, 49746667, 50579339
Offset: 1
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f[n_]:=Max[Last/@FactorInteger[n]]; q=3;lst={};Do[p=Prime[n];If[f[p-5]>=q&&f[p-4]>=q&&f[p+4]>=q&&f[p+5]>=q,AppendTo[lst,p]],{n,4*8!}];lst
A166002
Primes p such that p-6, p-5, p+5, and p+6 are each divisible by a cube greater than 1.
Original entry on oeis.org
1934869, 6136619, 11195869, 11845499, 12385381, 33919619, 39139381, 39790381, 52937869, 53209381, 53631131, 54601619, 58690381, 62892131, 67951381, 77212381, 80224619, 88874869, 94544869, 95734381, 99936131, 103805869, 108827869
Offset: 1
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f[n_]:=Max[Last/@FactorInteger[n]]; q=3;lst={};Do[p=Prime[n];If[f[p-6]>=q&&f[p-5]>=q&&f[p+5]>=q&&f[p+6]>=q,AppendTo[lst,p]],{n,5*9!}];lst
A086534
Smallest prime p sandwiched between two numbers that are divisible by n-th powers.
Original entry on oeis.org
2, 17, 271, 1249, 13121, 13121, 153089, 1272833, 28146689, 193562623, 652963841, 1378557953, 29096394751, 316431663103, 2191221587969, 15356401156097, 128200797454337, 314394051346433, 314394051346433, 28344942091829249, 201993039632138239, 267803891553271807
Offset: 1
a(3) = 271, 270 = 3^3*10 and 272 = 2^3*34. 271 is the smallest such number.
a(4) = 1249, 1248 =2^4*78, 1250 = 5^4*2. 1250 is the smallest such number.
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PrimeExponents[n_] := Max[ Flatten[ Table[ # [[2]], {1}] & /@ FactorInteger[n]]]; NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; a = Table[0, {15}]; p = 1; Do[p = NextPrim[p]; b = Min[ PrimeExponents[p - 1], PrimeExponents[p + 1]]; If[ a[[b]] == 0, a[[b]] = p; Print[b, " ", p]], {n, 1, 70000000}]; a
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