A005935 -id:A005935 - cp's OEIS frontend

A278931 Semiprimes whose ternary representations are also semiprime when read as a decimal number.

25, 49, 65, 82, 106, 115, 118, 121, 142, 143, 155, 187, 209, 235, 254, 259, 262, 265, 274, 289, 299, 314, 319, 326, 334, 335, 341, 355, 361, 382, 398, 415, 445, 451, 454, 458, 469, 493, 511, 515, 538, 551, 562, 566, 583, 586, 589, 614, 622, 634, 649, 667, 679

Offset: 1

K. D. Bajpai, Dec 04 2016

			65 is in the sequence because 5*13 = 65 (semiprime) and its ternary representation, 2102 = 2*1051, when read as a decimal number, is also semiprime.
115 is in the sequence because 5*23 = 115 (semiprime) and its ternary representation, 11021 = 103*107, when read as a decimal number, is also semiprime.

K. D. Bajpai, Table of n, a(n) for n = 1..9646
Wikipedia, Ternary numeral system

Subsequence of A001358.

Cf. A001358, A005935, A007089, A065721, A089981, A236537.

Select[Range[5000], PrimeOmega[#] == 2 && PrimeOmega[FromDigits[ IntegerDigits[ #, 3]]] == 2 &]

A285300 Numbers k such that 3^(k-1) == 2^(k-1) !== 1 (mod k).

Original entry on oeis.org

65, 133, 529, 793, 1649, 2059, 2321, 4187, 5185, 6305, 6541, 6697, 6817, 7471, 7613, 8113, 10963, 11521, 13213, 13333, 13427, 14701, 14981, 19171, 19201, 19909, 21349, 21667, 22177, 26065, 26467, 32873, 35443, 36569, 37333, 38897, 42121, 42127, 44023, 47081

Offset: 1

Thomas Ordowski, Apr 16 2017

nonn

All terms are odd composite numbers. There are no pseudoprimes to bases 2 or 3 in this sequence.
Are there infinitely many numbers of this kind?
From Max Alekseyev, Apr 16 2017: (Start)
Also, Fermat pseudoprimes base 2/3 that are not Fermat pseudoprimes base 2.
Also, the set difference of A073631 and either of ({1} U A001567), ({1} U A005935), or ({1} U A052155). (End)

			2^64 = 18446744073709551616 = 65 * 283796062672454640 + 16 and 3^64 = 3433683820292512484657849089281 = 65 * 52825904927577115148582293681 + 16. Therefore 65 is in the sequence.
Note: a(3) = 529 = 23^2 and a(40) = 47081 = 23^2 * 89.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A001567, A005935, A052155, A073631.

filter:= proc(n) local t;
  t:= 3 &^(n-1) mod n;
  if t = 1 then return false fi;
  t = 2 &^(n-1) mod n;
end proc:
select(filter, [seq(i,i=3..10^5,2)]); # Robert Israel, Apr 27 2017

Select[Range[2, 10^5], PowerMod[2, # - 1, #] == PowerMod[3, # - 1, #] != 1 &] (* Giovanni Resta, Apr 16 2017 *)

is(n) = Mod(3, n)^(n-1)==2^(n-1) && Mod(2, n)^(n-1)!=1 \\ Felix Fröhlich, Apr 27 2017

More terms from Giovanni Resta, Apr 16 2017

A308079 Pseudoprimes to base 3 that divide a Mersenne number.

Original entry on oeis.org

10974881, 193949641, 717653129, 8762386393, 19683169273, 24802217129, 78618861353, 121271968201, 146050578391, 169905267617, 188684740591, 232153956569, 290762221753, 306091598201, 336675266287, 394233108121, 592050558553

Offset: 1

Jeppe Stig Nielsen, May 11 2019

nonn

Members of A005935 that divide a member of A001348.
Odd members k of A005935 such that the multiplicative order of 2 modulo k is a prime. Odd members k of A005935 such that A002326((k-1)/2) is prime.
The known entries are proper divisors of a Mersenne number. It is not known if the Mersenne number itself can belong to this sequence.

			10974881 is in the sequence because it divides 2^239 - 1 (and 239 is prime), it is not a prime, but 3^10974880 === 1 (mod 10974881).

Amiram Eldar, Table of n, a(n) for n = 1..202 (terms below 10^15)
Mersenne Forum, Composite PRP (discussion thread).

Intersection of A005935 and A122094.
Subsequence of A052155.
Cf. A001348, A002326.

forstep(n=3,+oo,2,Mod(3,n)^(n-1)==1&&!ispseudoprime(n)&&ispseudoprime(znorder(Mod(2,n)))&&print1(n,", "))

A318055 Numbers k such that gcd(k, 2^k - 2) = 1 and gcd(k, 3^k - 3) > 1.

Original entry on oeis.org

247, 403, 559, 715, 871, 1027, 1339, 1495, 1651, 1807, 1963, 2009, 2035, 2119, 2587, 2743, 2899, 2993, 3055, 3211, 3523, 3649, 3679, 3835, 3977, 3991, 4147, 4303, 4331, 4453, 4615, 4633, 4699, 4771, 4927, 5239, 5395, 5617, 5707, 5863, 5995, 6019, 6031, 6161, 6331, 6487, 6799, 6929, 6955, 7081, 7111

Offset: 1

Thomas Ordowski, Aug 14 2018

nonn

Odd numbers k such that gcd(k,2^(k-1)-1) = 1 and gcd(k,3^(k-1)-1) > 1.
It seems that a(n) == 91 (mod 156) for infinitely many n.
Fermat pseudoprimes to base 3 (A005935) in this sequence are 16531, 49051, 72041, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A267999 and probably of A121707.
Cf. A139613(2n+1): it gives many terms of the sequence.
Cf. A005935.

Filtered([1..10000],k->Gcd(k,2^k-2) = 1 and Gcd(k,3^k-3) > 1);  # Muniru A Asiru, Oct 07 2018

select(k->gcd(k,2^k-2) = 1 and gcd(k,3^k-3) > 1,[$1..10000]); # Muniru A Asiru, Oct 07 2018

Select[Range[8000], GCD[#, 2^# - 2] == 1 && GCD[#, 3^# - 3] > 1 &] (* Amiram Eldar, Mar 31 2024 *)

isok(k) = (gcd(k,2^k-2) == 1) && (gcd(k,3^k-3) != 1); \\ Michel Marcus, Aug 14 2018

More terms from Michel Marcus, Aug 14 2018

A351336 Odd pseudoprimes to base 3; composite terms of A271116.

Original entry on oeis.org

91, 121, 671, 703, 949, 1105, 1541, 1729, 1891, 2465, 2665, 2701, 2821, 3281, 3367, 3751, 4961, 5551, 6601, 7381, 8401, 8911, 10585, 11011, 12403, 14383, 15203, 15457, 15841, 16471, 16531, 18721, 19345, 23521, 24661, 24727, 28009, 29161, 29341, 30857, 31621

Offset: 1

Bill McEachen, Feb 07 2022

nonn

Only 2760 of the first 10 million terms of A271116 are nonprimes (0.03%). These composite terms are heavily skewed to rightmost digit = 1.

These odd composites all appear in A005935 (A005935 having an additional 27 terms (all parity 0) through the same term = 179146207).

Bill McEachen, Table of n, a(n) for n = 1..5767

Intersection of A002808 and A271116; intersection of A005935 and A005408; subsequence of A038509.

q[n_] := CompositeQ[n] && Divisible[Round[3^n/12], n]; Select[Range[32000], q] (* Amiram Eldar, Feb 09 2022 *)

is(n) = (n>1) && !isprime(n) && (lift(Mod(3, 4*n)^(n-1))==1); \\ Michel Marcus, Feb 09 2022; after A271116

list(lim)=my(v=List()); forcomposite(n=91,lim\1, if(bittest(34,n%6) && Mod(3,n)^(n-1)==1, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Feb 09 2022

a(n) ~ (a(n-1)+a(n-2))/2 (conjectured). Bill McEachen, Nov 24 2024

A363215 Integers p > 1 such that 3^d == 1 (mod p) where d = A000265(p-1).

Original entry on oeis.org

2, 11, 13, 23, 47, 59, 71, 83, 107, 109, 121, 131, 167, 179, 181, 191, 227, 229, 239, 251, 263, 277, 286, 311, 313, 347, 359, 383, 419, 421, 431, 433, 443, 467, 479, 491, 503, 541, 563, 587, 599, 601, 647, 659, 683, 709, 719, 733, 743, 757, 827, 829, 839, 863

Offset: 1

Jeppe Stig Nielsen, May 21 2023

nonn

Inspired by an incorrect definition of strong pseudoprime to base 3.

As is obvious from the data, it fails to include all primes. Does include some composite numbers (pseudoprimes), namely 121, 286, 24046, 47197, 82513, ...

Jeppe Stig Nielsen, Table of n, a(n) for n = 1..10000
Wikipedia, Strong pseudoprime

Cf. A000265, A005935, A020229, A130433.

is(p)=my(d=p-1);d/=2^valuation(d,2);Mod(3,p)^d==1

from itertools import count, islice
def inA363215(n): return pow(3,n-1>>(~(n-1)&n-2).bit_length(),n)==1
def A363215_gen(startvalue=2): # generator of terms >= startvalue
    return filter(inA363215,count(max(startvalue,2)))
A363215_list = list(islice(A363215_gen(),20)) # Chai Wah Wu, May 22 2023

A374976 Odd k with p^k mod k != p for all primes p.

Original entry on oeis.org

1, 9, 27, 63, 75, 81, 115, 119, 125, 189, 207, 209, 215, 235, 243, 279, 299, 319, 323, 387, 407, 413, 423, 515, 517, 531, 535, 551, 567, 575, 583, 611, 621, 623, 667, 675, 707, 713, 729, 731, 747, 767, 779, 783, 799, 815, 835, 851, 869, 893, 899, 917, 923, 927

Offset: 1

Francois R. Grieu, Jul 26 2024

nonn

Alternatively: 1, and odd composites not a pseudoprime to any prime base.

The sequence contains no primes, no pseudoprimes to any prime base (A001567, A005935, A005936, A005938, A020139, A020141...), and no Carmichael numbers (A002997).

			k=3 (resp. 5, 7) is not in the sequence because for prime p=2 it holds p^k mod k = 2 which is p.
k=9 is in the sequence because for prime p=2 (resp. 3, 5, 7) it holds p^k mod k = 8 (resp. 0, 8, 1) which is not p, and for all other primes p it holds p>=k therefore p^k mod k can't be p.

Francois R. Grieu, Table of n, a(n) for n = 1..10000

Cf. A000040, A001567, A005935, A005936, A005938, A020139, A020141, A020145, A002997.

Cases[Range[1, 930, 2], k_/; (For[p=2, p=k)]

A185084 Number of Fermat pseudoprimes to base 3 less than 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 2, 3, 6, 10, 17, 21, 30, 44, 61, 87, 124, 175, 254, 362, 511, 696, 955, 1313, 1802, 2462, 3321, 4422, 5969, 8089, 10785, 14513, 19333, 25774, 34259, 45522

Offset: 1

Washington Bomfim, Mar 02 2012

			a(1) = a(2) = ... = a(6) = 0 because A005935(1) = 91 > 2^6.
a(7) = 2 since A005935(1) = 91, A005935(2) = 121, A005935(3) = 286, and 121 < 2^7 < 286.

Cf. A005935, A114245.

cnt = 0; Table[Do[If[! PrimeQ[i] && PowerMod[3, i-1, i] == 1, cnt++], {i, 2^(n-1) + 1, 2^n}]; cnt, {n, 20}] (* T. D. Noe, Mar 02 2012 *)

a(35)-a(37) from Amiram Eldar, Jul 18 2021

A247906 a(n) = n-th pseudoprime to base n.

Original entry on oeis.org

561, 286, 341, 781, 1105, 1105, 133, 364, 703, 793, 1105, 1099, 1891, 6541, 1271, 3991, 1649, 1849, 3059, 7363, 2047, 1738, 4537, 1128, 3145, 2993, 5365, 4069, 4097, 7421, 2465, 11305, 2937, 16589, 4495, 2044, 6601, 26885, 13073, 6892, 22945, 3885, 8695, 10879

Offset: 2

Felix Fröhlich, Sep 26 2014

nonn

			a(2) = A001567(2) = 561.
a(3) = A005935(3) = 286.

Cf. A007535, A098650.

Cf. Pseudoprimes to base b: A001567 (b=2), A005935 (b=3), A020136 (b=4), A005936 (b=5), A005937 (b=6), A005938 (b=7), A020137 (b=8), A020138 (b=9).

for(n=2, 20, i=0; forcomposite(c=2, 1e9, if(Mod(n, c)^(c-1)==1, i++; if(i==n, print1(c, ", "); i=0; break({1}))); if(c==1e9, print1(">1e9, "))))

A254519 Largest n-digit pseudoprime to base 3.

Original entry on oeis.org

91, 949, 8911, 97567, 997633, 9959413, 99971821, 999271891, 9999326731, 99997244929, 999989423051, 9999899578441, 99999695823301, 999999050050321, 9999997295187859, 99999997019370001

Offset: 2

Felix Fröhlich, Jan 31 2015

Cf. A005935, A067845.

for(n=2, 20, k=10^n; while(ispseudoprime(k) || Mod(3, k)^(k-1)!=1, k--); print1(k, ", "))

cp's OEIS Frontend

A278931 Semiprimes whose ternary representations are also semiprime when read as a decimal number.

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A285300 Numbers k such that 3^(k-1) == 2^(k-1) !== 1 (mod k).

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A308079 Pseudoprimes to base 3 that divide a Mersenne number.

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A318055 Numbers k such that gcd(k, 2^k - 2) = 1 and gcd(k, 3^k - 3) > 1.

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A351336 Odd pseudoprimes to base 3; composite terms of A271116.

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A363215 Integers p > 1 such that 3^d == 1 (mod p) where d = A000265(p-1).

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A374976 Odd k with p^k mod k != p for all primes p.

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Mathematica

A185084 Number of Fermat pseudoprimes to base 3 less than 2^n.

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