A005935 -id:A005935 - cp's OEIS frontend

A271393 a(1) = 3, a(n+1) = (3^a(n)-1)/2.

3, 13, 797161

Offset: 1

Thomas Ordowski, Apr 06 2016

nonn

The next term is too large to include.
The terms a(1), a(2), and a(3) are primes.
The next term a(4) is a composite number, a pseudoprime to base 3.
If a(n) is a pseudoprime to base 3, then a(n+1) is a pseudoprime to base 3.
Note that a(n) divides a(n+1)-1 for every n.
a(4) has 380343 digits. - Altug Alkan, Apr 09 2016

R. Steuerwald (1948), see A005935.

Cf. A005935 (see Steuerwald's theorem), A007013, A028491.

NestList[(3^# - 1)/2 &, 3, 3] (* Michael De Vlieger, Apr 06 2016 *)

A306144 Numbers k > 2 such that 3^(k-1) == 1 (mod k) and gcd(k, 2^(k-1)-1) = 1.

Original entry on oeis.org

286, 16531, 24046, 49051, 72041, 182527, 192713, 232726, 258017, 327781, 442471, 443713, 453259, 574397, 625873, 652879, 655051, 668431, 705091, 903631, 1236031, 1241143, 1250833, 1287091, 1304446, 1309111, 1351601, 1414639, 1563151, 1817743, 1899451, 1908397

Offset: 1

Thomas Ordowski, Aug 18 2018

nonn

The odd terms are "anti-Carmichael pseudoprimes (3,2)" defined as follows: numbers k > 1 such that 3^k == 3 (mod k) and gcd(k, 2^k-2) = 1. Cf. A300762 (2,3).

We impose k>2, since we want these to be pseudoprimes, thus composite numbers.

Subsequence of A005935.

Cf. A130433.

Select[Range[3, 2*10^6], PowerMod[3, #-1, #] == 1 && GCD[#, #-1 + PowerMod[2, #-1, #]] == 1 &] (* Giovanni Resta, Aug 18 2018 *)

isok(k) = (k>2) && (Mod(3, k)^(k-1) == Mod(1, k)) && (gcd(k, 2^(k-1)-1) == 1); \\ Michel Marcus, Aug 18 2018

More terms from Michel Marcus, Aug 18 2018

Further terms from Giovanni Resta, Aug 18 2018

A333316 Numbers k such that k^2 + 1 is a Fermat pseudoprime to base 3.

Original entry on oeis.org

216, 660, 1484, 1560, 8208, 52164, 544320, 592956, 649800, 4321800, 5103210, 6182220, 10621380, 21415680, 24471720, 135307008, 359624088, 535019100, 1071782250, 1113233520, 1227427740, 1527496740, 9462748008, 143935711920

Offset: 1

Amiram Eldar, Mar 14 2020

a(24) > 7*10^10.
The corresponding pseudoprimes a(n)^2 + 1 are 46657, 435601, 2202257, 2433601, 67371265, ...
a(25) > 7.5*10^11. - Giovanni Resta, Mar 15 2020

			216 is a term since 216^2 + 1 = 46657 is a Fermat pseudoprime to base 3.

Cf. A002522, A005574, A005935, A135590.

Select[Range[10^3], CompositeQ[#^2 + 1] && PowerMod[3, #^2, #^2 + 1] == 1 &]

a(24) from Giovanni Resta, Mar 15 2020

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A271393 a(1) = 3, a(n+1) = (3^a(n)-1)/2.

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A306144 Numbers k > 2 such that 3^(k-1) == 1 (mod k) and gcd(k, 2^(k-1)-1) = 1.

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A333316 Numbers k such that k^2 + 1 is a Fermat pseudoprime to base 3.

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