A093441
Lexicographically earliest sequence of primes such that a(n) - 1 == 0 (mod a(n - 1) - 1) where a(n) - 1 is a squarefree number; a(1) = 3.
Original entry on oeis.org
3, 7, 31, 211, 2311, 43891, 1272811, 16546531, 976245271, 36121074991, 1119753324691, 52628406260431, 3526103219448811, 186883470630786931, 7662222295862264131, 743235562698639620611, 54256196077000692304531, 6130950156701078230411891, 631487866140211057732424671
Offset: 1
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a[1] = 3; a[n_] := a[n] = Block[{k = m = a[n - 1] - 1}, k *= 2; While[ !PrimeQ[k + 1] || !SquareFreeQ[k], k += m]; k + 1]; Table[ a[n], {n, 17}] (* Robert G. Wilson v, Apr 30 2004 *)
A275969
Least k such that phi(k) has exactly n prime factors (counted with multiplicity).
Original entry on oeis.org
3, 5, 13, 17, 51, 85, 193, 257, 769, 1285, 3281, 4369, 12289, 21845, 49601, 65537, 196611, 327685, 786433, 1114129, 3158273, 5570645, 12648641, 16843009, 50397953, 84215045, 202113281, 286331153, 805384193, 1431655765, 3221225473, 8168859365, 12952273921
Offset: 1
a(2) = 5 because phi(5) = 4 has 2 prime factors (counted with multiplicity).
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Table[k = 1; While[PrimeOmega@ EulerPhi@ k != n, k++]; k, {n, 16}] (* Michael De Vlieger, Aug 15 2016 *)
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a(n) = {my(k = 1); while(bigomega(eulerphi(k)) != n, k++); k; }
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use ntheory ":all"; sub a275969 { my($k,$n)=(1,shift); $k++ while scalar(factor(euler_phi($k))) != $n; $k; } # Dana Jacobsen, Aug 16 2016
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use v5.16; use ntheory ":all";
my($s,$chunk,$lp,@done) = (1,2e6,0);
while (1) {
my @npf = map { scalar(factor($_)) } euler_phi($s, $s+$chunk-1);
if (vecany { $_>$lp } @npf) {
while (my($idx,$val) = each @npf) {
$done[$val] //= $s+$idx if $val > $lp;
}
while ($done[$lp+1]) { $lp++; say "$lp $done[$lp]"; }
}
$s += $chunk;
} # Dana Jacobsen, Aug 16 2016
A075591
Smallest squarefree number with n prime divisors which is the average of a twin prime pair.
Original entry on oeis.org
6, 30, 462, 2310, 43890, 1138830, 17160990, 300690390, 15651726090, 239378649510, 12234189897930, 568815710072610, 19835154277048110, 693386350578511590, 37508276737897976010, 3338236629672919864890
Offset: 2
a(4) = 462 because 462 = 2*3*7*11 and the twin primes are 461 and 463.
Cf.
A073918 (least prime p such that p-1 has exactly n distinct prime factors),
A098026 (least prime p such that p+1 has exactly n distinct prime factors).
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Generate[pIndex_, i_] := Module[{p2, t}, p2=pIndex; While[p2[[i]]++; Do[p2[[j]]=p2[[i]]+j-i, {j, i+1, Length[p2]}]; t=Times@@Prime[p2]; t
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