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 A282092 Numbers m such that there exists at least one integer k < m such that m^2+1 and k^2+1 have the same prime factors. 0
 7, 18, 117, 239, 378, 843, 2207, 2943, 4443, 4662, 6072, 8307, 8708, 9872, 31561, 103682, 271443, 853932, 1021693, 3539232, 3699356, 6349657, 6907607, 7042807, 7249325, 9335094, 12623932, 12752043, 12813848, 22211431, 33385282, 42483057, 52374157, 105026693 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For the pairs (m, k), is k always unique? The pairs (m, k) are (7, 3), (18, 8), (117, 43), (239, 5), (378, 132), (843, 377), (2207, 987), (2943, 73), (4443, 53), (4662, 1568), (6072, 5118), (8307, 743), (8708, 2112), (9872, 2738), ... LINKS EXAMPLE 7 is in the sequence because of the pair (m, k) = (7, 3), 7^2+1 = 2*5^2 and 3^2+1 = 2*5 with the same prime factors 2 and 5. MAPLE with(numtheory): for n from 1 to 10000 do:   x:=factorset(n^2+1):    if issqrfree(n^2+1) = false    then     for m from 1 to n-1 do:     y:=factorset(m^2+1):        if x=y then printf (`%d %d \n`, n, m):         else         fi:      od:      fi:     od: MATHEMATICA Select[Range@ 5000, Function[m, Total@ Boole@ Table[Function[w, And[SameQ[First@ w, #], SameQ[Last@ w, #]] &@ Union@ Flatten@ w]@ Map[FactorInteger[#][[All, 1]] &, {m^2 + 1, k^2 + 1}], {k, m - 1}] > 0]] (* Michael De Vlieger, Feb 07 2017 *) PROG (Perl) use ntheory qw(:all); for (my (\$m, %t) = 1 ; ; ++\$m) { my \$k = vecprod(map{\$_->[0]}factor_exp(\$m**2+1)); push @{\$t{\$k}}, \$m; if (@{\$t{\$k}} >= 2) { print'('.join(', ', reverse(@{\$t{\$k}})).")\n"; } } # Daniel Suteu, Feb 08 2017 (PARI) isok(n)=ok = 0; vn = factor(n^2+1)[, 1]; for (k=1, n-1, if (factor(k^2+1)[, 1] == vn, ok = 1; break); ); ok; \\ Michel Marcus, Feb 09 2017 (PARI) squeeze(f)=factorback(f)\2 list(lim)=my(v=List(), m=Map(), t); for(n=1, lim, t=squeeze(factor(n^2+1)[, 1]); if(mapisdefined(m, t), listput(v, n), mapput(m, t, 0))); Vec(v) \\ Charles R Greathouse IV, Feb 12 2017 CROSSREFS Subsequence of A049532 (numbers n such that n^2 + 1 is not squarefree). Cf. A002522, A059591, A059592, A124809. Sequence in context: A203381 A207158 A304992 * A197938 A207153 A203222 Adjacent sequences:  A282089 A282090 A282091 * A282093 A282094 A282095 KEYWORD nonn AUTHOR Michel Lagneau, Feb 06 2017 EXTENSIONS a(15)-a(29) from Daniel Suteu, Feb 08 2017 a(30) from Daniel Suteu, Feb 10 2017 a(31)-a(34) from Joerg Arndt, Feb 11 2017 STATUS approved

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Last modified January 28 12:46 EST 2022. Contains 350656 sequences. (Running on oeis4.)