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 A283552 Numbers k == 33 (mod 60) such that 2k+1, 2k+5, 3k+2 and 3k+8 are all primes. 1

%I #17 Mar 22 2024 08:13:07

%S 33,153,453,1953,4773,19353,23253,36273,37413,38793,40773,50133,51693,

%T 70413,70833,83433,88893,108393,115233,117873,131193,136113,157773,

%U 161733,164793,170973,184533,221793,234813,238293,258453,271893,272313,287313,304953,307713,325533,327753,330393

%N Numbers k == 33 (mod 60) such that 2k+1, 2k+5, 3k+2 and 3k+8 are all primes.

%C Andreas Weingartner used the first 913685 terms of this sequence to prove that the equation sigma(x) = sigma(x+k) has at least one solution for every even k in the range 2 <= k <= 10^(10^7). The upper bound is just lower than the product of 2a(n)+1 of these terms which equals 3.222... * 10^10000007.

%H Amiram Eldar, <a href="/A283552/b283552.txt">Table of n, a(n) for n = 1..1000</a>

%H A. Weingartner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Weingartner/wein3.html">On the Solutions of sigma(n) = sigma(n+k)</a>, Journal of Integer Sequences, Vol. 14 (2011), #11.5.5.

%e a(2) = 153, 2*153 + 1 = 307, 2*153 + 5 = 311, 3*153 + 2 = 461 and 3*153 + 8 = 467 are all primes.

%t Select[33 + Range[0, 6*10^5]*60, PrimeQ[2 # + 1] && PrimeQ[2 # + 5] && PrimeQ[3 # + 2] && PrimeQ[3 # + 8] &]

%Y Cf. A000203, A007373.

%K nonn

%O 1,1

%A _Amiram Eldar_, Mar 10 2017

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Last modified August 10 21:39 EDT 2024. Contains 375058 sequences. (Running on oeis4.)