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 A358657 Numbers such that the three numbers before and the three numbers after are squarefree semiprimes. 2
 216, 143100, 194760, 206136, 273420, 684900, 807660, 1373940, 1391760, 1516536, 1591596, 1611000, 1774800, 1882980, 1891764, 2046456, 2051496, 2163420, 2163960, 2338056, 2359980, 2522520, 2913840, 3108204, 4221756, 4297320, 4334940, 4866120, 4988880, 5108796, 5247144, 5606244, 5996844 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS All numbers in this sequence are divisible by 36. Proof: Suppose k is odd and in this sequence; then either k-1 or k-3 is divisible by 4, creating a contradiction. Suppose k is even, but not divisible by k; then k-2 is divisible by 4, creating a contradiction. Suppose k is not divisible by 3. Then there exists a number j such that 3*j and 3*(j+1) are among squarefree semiprimes surrounding k; one of them is divisible by 6, creating a contradiction. Suppose k is divisible by 3, but not by 9; then one of the squarefree semiprimes k-3 or k+3 is divisible by 9, creating a contradiction. Since each term k is divisible by 36, it follows that (k-3)/3, (k-2)/2, (k+2)/2, and (k+3)/3 are primes. Additionally, none of the six integers nearest to k can be the cube of a prime: for any prime p > 3, p^3 == {+-1, +-17} (mod 36), so only k-1 or k+1 could be the cube of a prime, yet in either of those cases, that cube's two nearest neighbors, p^3 - 1 and p^3 + 1, would both be factorable (i.e., p^3 - 1 = (p^2 + p + 1)*(p - 1) and p^3 + 1 = (p^2 - p + 1)*(p + 1)), and neither would be a semiprime. Thus, since neither k-1 nor k+1 can be the cube of a prime, testing whether each has four divisors (see the Magma code below) is equivalent to testing whether each is a squarefree semiprime. - Jon E. Schoenfield, Nov 26 2023 LINKS Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 169 terms from Robert Israel) FORMULA a(n) = 2*(A158476(n) + 1). - Hugo Pfoertner, Dec 12 2022 EXAMPLE The following numbers are squarefree semiprimes: 213 = 3*71, 214 = 2*107, 215 = 5*43, 217 = 7*31, 218 = 2*109, and 219 = 3*73. Thus, 216 is in this sequence. MAPLE N:= 10^6: # for terms <= N P:= select(isprime, [2, seq(i, i=3..N/2, 2)]): S:= NULL: for i from 1 to nops(P) do p:= P[i]; r:= ListTools:-BinaryPlace(P, N/p); if r <= i then break fi; S:= S, op(p * P[i+1 .. r]); od: S:= sort([S]): J:= select(t -> S[t+5] = S[t]+6, [\$1..nops(S)-5]): map(t -> S[t+2]+1, J); # Robert Israel, Nov 26 2023 MATHEMATICA Select[Range[10000000], Transpose[FactorInteger[# - 3]][[2]] == {1, 1} && Transpose[FactorInteger[# - 2]][[2]] == {1, 1} && Transpose[FactorInteger[# - 1]][[2]] == {1, 1} && Transpose[FactorInteger[# + 3]][[2]] == {1, 1} && Transpose[FactorInteger[# + 2]][[2]] == {1, 1} && Transpose[FactorInteger[# + 1]][[2]] == {1, 1} &] 36*Flatten@Position[({1, 1}==Last@Transpose@FactorInteger@# &/@ {#-3, #-2, #-1, #+1, #+2, #+3}) & /@ (36*Range@(10^6)), {True ..}] (* Hans Rudolf Widmer, Aug 01 2024 *) PROG (Python) from itertools import count, islice from sympy import isprime, factorint def issfsemiprime(n): return list(factorint(n).values()) == [1, 1] if n&1 else isprime(n//2) def ok(n): return all(issfsemiprime(n+i) for i in (-2, 2, -3, -1, 1, 3)) def agen(): yield from (k for k in count(36, 36) if ok(k)) print(list(islice(agen(), 20))) # Michael S. Branicky, Nov 26 2022 (Magma) a:=[]; IsP:=IsPrime; Tau:=NumberOfDivisors; for m in [1..170000] do t:=36*m; if IsP((t-3) div 3) and IsP((t+3) div 3) and IsP((t-2) div 2) and IsP((t+2) div 2) and Tau(t-1) eq 4 and Tau(t+1) eq 4 then a:=a cat [t]; end if; end for; a; // Jon E. Schoenfield, Nov 26 2023 CROSSREFS Cf. A001358, A158476, A350101, A358666. Sequence in context: A167127 A269821 A268364 * A048100 A222337 A008697 Adjacent sequences: A358654 A358655 A358656 * A358658 A358659 A358660 KEYWORD nonn AUTHOR Tanya Khovanova and Massimo Kofler, Nov 25 2022 STATUS approved

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Last modified September 18 10:24 EDT 2024. Contains 375999 sequences. (Running on oeis4.)