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A290340
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Numbers m such that each of the four consecutive integers m, m+1, m+2, m+3 has squarefree rank 1.
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2
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17, 241, 242, 1249, 4049, 4799, 17297, 120049, 206081, 281249, 388961, 470447, 538721, 1462049, 1566449, 1808801, 1916881, 3302449, 3302450, 3693761, 3959297, 5385761, 5664976, 6118001, 6986321, 9305297, 10479041, 14268481, 16831601, 20110481, 22997761, 27661922, 28140001
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OFFSET
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1,1
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COMMENTS
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A162642(k) is the squarefree rank of k.
Numbers that are the first of four consecutive terms of A228056 form a subsequence: 242, 3302450, 22997761, 27661922, 28140001, 64866050, ... consisting of those numbers m in this sequence such that m, m+1, m+2, and m+3 are all composite. - Charles R Greathouse IV, Sep 30 2021
One of for positive integer m, m+1, m+2, m+3 is of the form 4*k + 2 = 2*(2*k + 1). As 2 has an odd exponent the exponents in the prime factorization and 2*k + 1 is odd, the number of odd exponents in the prime factorization of 2*k + 1 must be 0 i.e., 2*k + 1 is a perfect square and so one of m, m+1, m+2, m+3 is of the form 2*t^2 where t is an odd square. - David A. Corneth, Nov 09 2023
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LINKS
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EXAMPLE
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m = 17 is in the sequence as the number of odd prime exponents of each of the numbers m = 17 through m + 3 = 20 is 1. - David A. Corneth, Nov 06 2023
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PROG
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(Magma)
A162642:=func<n|&+[Integers()|pe[2]mod 2:pe in Factorisation(n)]>;
c:=func<a|[n:n in a|n+1 in a]>;
c(c(c([n:n in[1..10^6]|A162642(n)eq 1])));
(PARI) list(lim)=my(u=vectorsmall(4), v=List(), s, i); forfactored(n=2, lim\1+3, if(i++>4, i=1); s-=u[i]; s+=u[i]=(vecsum(n[2][, 2]%2)==1); if(s==4, listput(v, n[1]-3))); Vec(v); \\ Charles R Greathouse IV, Sep 30 2021
(PARI) \\ See PARI link
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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