Marius Coman - cp's OEIS frontend

A218483 Fermat pseudoprimes to base 2 which are congruent to 1 (mod 8).

561, 1105, 1729, 1905, 2465, 4033, 4369, 4681, 6601, 8321, 8481, 10585, 11305, 12801, 15841, 16705, 18705, 18721, 23001, 23377, 25761, 30121, 30889, 31417, 31609, 33153, 34945, 39865, 41041, 41665, 46657, 52633, 62745, 65281, 74665, 75361, 83665, 85489

Offset: 1

Marius Coman, Oct 30 2012

nonn

Old name was: Fermat pseudoprimes to base 2 of the form 8*p*n + p^2, where p is prime and n natural.
For p = 5 the formula becomes 40*n + 25. From the first 15 pseudoprimes divisible by 5, 12 are of the form 40*n + 25 (beside 3 of them which are of the form 40*n + 5). Conjecture: there are no pseudoprimes to base 2 of the form 40*n + 15.
Note: it can be seen that a pseudoprime can be written in this formula in more than one way: e.g., 561 = 8*3*23 + 3^2 = 8*11*5 + 11^2 = 8*17*2 + 17^2 or 1905 = 8*3*79 + 3^2 = 8*5*47 + 5^2.
Conjecture: If a Fermat pseudoprime to base 2 can be written as 8*p*n + p^2, where n is an integer and p one of its prime factors, then it can be written this way for any of its prime factors. Checked for all pseudoprimes from the sequence above.
Conjecture: If a Fermat pseudoprime to base 2 with two prime factors can be written as 8*p1*n + p1^2, where n is a natural number and p1 one of its two prime factors, then it can also be written as 8*p2*(-n) + p2^2, where p2 is the other prime factor. Checked for 4033 = 37*109(n = 9), 4369 = 17*257(n = 30), 4681 = 31*151(n = 15), 8321 = 53*157(n = 13), 18721 = 97*193(n = 12), 23377 = 97*241(n = 18), 31417 = 89*353(n = 33), 31609 = 73*433 (n = 45), 65281 = 97*673(n = 72), 85489 = 53*1613 (n = 195).
Conjecture: If a Fermat pseudoprime to base 2 cannot be written as 8*p*n + p^2, where n is an integer and p one of its prime factors, then it cannot be written this way for any of its prime factors. Checked for the following pseudoprimes: 341, 645, 1387, 2047, 2701, 2821, 3277, 4371, 5461, 7957, 10261, 13741, 13747, 13981, 14491, 15709, 19951, 29341, 31621, 42799, 49141, 49981, 55245, 60701, 60787, 63973, 65077, 68101, 72885, 80581, 83333.
Note: from the first 72 pseudoprimes, 39 can be written this way.
All three conjectures are true (obvious from new characterization). - Charles R Greathouse IV, Dec 07 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Pseudoprime.
Eric Weisstein's World of Mathematics, Poulet Number.

Cf. A001567, A216023, A214305.

select(t -> 2 &^ t mod t = 2 and not isprime(t), [seq(1+8*j,j=0..10^5)]); # Robert Israel, Dec 07 2014

Select[8 * Range[10^4] + 1, PowerMod[2, # - 1, #] == 1 && CompositeQ[#] &] (* Amiram Eldar, Mar 30 2021 *)

is(n)=n%8==1 && Mod(2,n)^n==2 && !isprime(n) \\ Charles R Greathouse IV, Dec 07 2014

Corrected by Charles R Greathouse IV, Dec 07 2014

New name from Charles R Greathouse IV, Dec 07 2014

A218010 Primes of the form (24*p + 1)/5, where p is a Fermat pseudoprime to base 2.

Original entry on oeis.org

1637, 2693, 20981, 22469, 40709, 42773, 49253, 65957, 69557, 123653, 140837, 235877, 451013, 623621, 626693, 716549, 1095557, 1370597, 1634693, 1761989, 2289461, 2459813, 2548229, 2563493, 2821733, 3414533, 4091909, 4093637, 4910981, 5530901, 5727461

Offset: 1

Marius Coman, Oct 18 2012

nonn

This is a subsequence of A107003.
The corresponding values of p: 341, 561, 4371, 4681, 8481, 8911, 10261, 13741, 14491, 25761, 29341, 49141, 93961, 129921, 130561, 149281, 228241, 285541, 340561, 439291, 512461, 530881, 532171, 534061, 597871, 736291, 764491, 782341, 852841, 903631, 951481.
From the first 128 natural solutions of this equation ((24*p + 1)/5, where p is Fermat pseudoprime to base 2), 31 are primes (the ones from the sequence above), 51 are products (not necessarily squarefree) of two prime factors and 41 are products of three prime factors; only 5 of them are products of four prime factors.
Conjecture: There is no absolute Fermat pseudoprime m for which n = (5*m - 1)/24 is a natural number (checked for the first 300 Carmichael numbers; if true, then the formula is a criterion to separate pseudoprimes at least from a subset of primes, because there are 37 primes m from the first 300 primes for which n = (5*m - 1)/24 is a natural number).
3380740301 is a counterexample to the conjecture. - Charles R Greathouse IV, Dec 07 2014

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Pseudoprime

Cf. A107003, A142399.

is(n)=my(t); n%48==5 && isprime(n) && !isprime(t=(5*n-1)/24) && Mod(2,t)^t==2 \\ Charles R Greathouse IV, Dec 07 2014

Corrected by Charles R Greathouse IV, Dec 07 2014

A217841 Fermat pseudoprimes n to base 3 for which sqrt(8*n + 1) is an integer.

Original entry on oeis.org

91, 703, 1891, 2701, 7381, 8911, 10585, 12403, 16471, 18721, 29161, 38503, 41041, 49141, 79003, 88831, 93961, 104653, 115921, 146611, 188191, 218791, 226801, 269011, 286903, 314821, 334153, 364231, 385003, 497503, 534061, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1237951

Offset: 1

Marius Coman, Oct 12 2012

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fermat Pseudoprime

Cf. A005935, A210461 (subsequence).

list(lim)=my(v=List(),n); lim\=1; forstep(k=27,sqrtint(8*lim+1),2, n=k^2>>3; if(Mod(3,n)^(n-1)==1, listput(v,n))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017

a(15)-a(18) and a(35) from Charles R Greathouse IV, Jun 30 2017

A217853 Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8.

Original entry on oeis.org

91, 7381, 597871, 48427561, 3922632451, 317733228541, 25736391511831, 2084647712458321, 168856464709124011, 1107867264956562636991, 588766087155780604365200461, 47690053059618228953581237351, 25344449488056571213320166359119221, 166284933091139163730593611482181209801

Offset: 1

Marius Coman, Oct 12 2012

nonn

These numbers were obtained for values of k from 1 to 20, with the following exceptions: k = 10, 12, 13, 16, 17, 19, for which were obtained 3^n mod n = 3^7, 3^31, 3^37, 3^25, 3^31, 3^13.
Conjecture: There are infinitely many Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8, where k is a natural number.
It is true: for example, when 2k+1 is a prime number (see A210461). - Bruno Berselli, Jan 22 2013

Charles R Greathouse IV, Table of n, a(n) for n = 1..190
Eric Weisstein's World of Mathematics, Fermat Pseudoprime

Cf. A005935, A210461 (subsequence), A217841.

Select[Table[(3^(4k + 2) - 1)/8, {k, 80}], PowerMod[3, # - 1, #] == 1 &] (* Alonso del Arte, May 14 2019 *)

list(lim)=my(v=List(),t); lim\=1; for(k=1,(logint(8*lim+1,3)-2)\4, t=3^(4*k + 2)>>3; if(Mod(3,t)^t==3, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Jun 30 2017

A217835 Fermat pseudoprimes to base 2 that can be written as p^2n - pn + p, where p is also a Fermat pseudoprime to base 2 and n is a positive integer.

Original entry on oeis.org

348161, 831405, 1246785, 1275681, 2077545, 2513841, 5977153, 9613297, 13333441, 13823601, 18137505, 19523505, 21474181, 21880801, 37695505, 38171953, 44521301, 47734141, 54448153, 72887585, 75151441, 95423329

Offset: 1

Marius Coman, Oct 12 2012

nonn

After a(22) = 95423329, no more terms through 10^8.
The corresponding (p,n): (341,3), (645,2), (645,3), (341,11), (645,5), (561,8), (1729,2), (1387,5), (341,120), (561,44), (1905,5), (645,47), (3277,2), (2701,3), (2047,9), (4369,2), (341,384), (2821,6), (2047,13), (2465,12), (3277,7), (4369,5).
Conjecture: For any Fermat pseudoprime p to base 2 there are infinitely many Fermat pseudoprimes to base 2 equal to p^2*n - p*n + p, where n is a positive integer.
See the sequence A215343: the generalized formula from there is p^2*n - p*n + p^2, which suggests an extrapolated formula for obtaining some Fermat pseudoprime to base 2 from another: p^2*n - p*n + p^k.
Conjecture: For any Fermat pseudoprime p to base 2 and any positive integer k, there are infinitely many Fermat pseudoprimes to base 2 equal to p^2*n - p*n + p^k, where n is a positive integer.

Eric Weisstein's World of Mathematics, Poulet Number

Cf. A001567, A213812, A215343.

A216830 Prime factors of Carmichael numbers divisible by 7, taken just once each as it appears first time, in order of the size of the Carmichael number respectively in order of their size if they are prime factors of the same Carmichael number.

Original entry on oeis.org

7, 13, 19, 31, 23, 41, 67, 73, 11, 103, 37, 101, 61, 109, 199, 433, 5, 17, 151, 577, 307, 163, 139, 181, 271, 739, 229, 251, 853, 1321, 991, 241, 53, 397, 1783, 1171, 907, 2971, 353, 593, 4057, 661, 193, 619, 89, 653, 157, 2089, 313, 331, 373, 937, 2053, 443, 3877

Offset: 1

Marius Coman, Sep 19 2012

nonn

It is remarkable that, if we note with p the numbers from sequence, for every p was obtained a prime, a squarefree semiprime or a number divisible by 5 through the formula 3*p + 4.
Primes obtained and the corresponding p in the brackets: 43(13), 61(19), 97(31), 73(23), 127(41), 223(73), 37(11), 113(103), 307(101), 331(109), 601(199), 1303(433), 19(5), 457(151), 421(139), 547(181), 2221(739), 691(229), 757(251), 3967(1321), 727(241), 163(53), 3517(1171), 1063(353), 1783(593), 1987(661), 1861(619), 271(89), 6271(2089), 997(331), 1123(373), 6163(2053).
Semiprimes obtained and the corresponding p in the brackets: 5^2(7), 5*41(67), 5*23(37), 11*17(61), 5*11(17), 5*347(577), 17*29(163), 19*43(271), 11*233(853), 13*229(991), 5*239(397), 53*101(1783), 37*241(2971), 11*53(193), 13*151(653), 23*41(313), 5*563(937), 31*43(443).
Numbers divisible by 5 (not semiprimes) obtained and the corresponding p in the brackets: 5^2*37(307), 5^2*109(907), 5^2*487(4057), 5^2*19(157), 5*13*179(3877).
This formula produces 35 primes for the first 55 values of p!
The formula can be extrapolated for all Carmichael numbers and all their prime factors: primes of type 3*p + d - 3, where p is a prime factor of a Carmichael number divisible by d; for instance, were obtained the following primes of type 3*p + 10, where p is a prime factor of a Carmichael number divisible by 13: 61, 31, 67, 103, 193, 43, 229, 337, 1201, 79, 211, 823, 607, 463, 1741, 499, 643, 733, 97, 2029, 139, 349, 4129, 6421, 1381, 2731, 1069, 853, 1021, 9421, 5413, 10831, 223, 1933, 8269 (which means 35 primes) for the first 55 values of p!

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Carmichael Number

Cf. A002997.

A216880 Numbers of the form 3p - 2 where p and 6p + 1 are prime.

Original entry on oeis.org

4, 7, 13, 19, 31, 37, 49, 67, 109, 139, 181, 217, 247, 301, 307, 319, 391, 409, 451, 517, 541, 697, 721, 769, 787, 811, 829, 847, 877, 931, 937, 991, 1039, 1099, 1117, 1189, 1327, 1381, 1399, 1507, 1669, 1729, 1777, 1801, 1819, 1921, 1957, 1981, 2047, 2179, 2251, 2281

Offset: 1

Marius Coman, Sep 19 2012

nonn

This formula produces many primes and semiprimes.
Taken just the terms from the sequence above:
n is prime for the following values of p: 3, 5, 7, 11, 13, 23, 37, 47, 61, 103, 137, 181, 257, 263, 271, 277, 293, 313, 331, 347, 373, 443, 461, 467, 557, 593, 601, 727, 751, 761.
n is a semiprime of the form (6*m + 1 )*(6*n + 1) for the following values of p: 73, 83, 101, 241, 367, 653, 661.
n is a semiprime of the form (6*m - 1 )*(6*n - 1) for the following values of p: 107, 131, 151, 173, 397, 503, 607, 641, 683.
n is the square of a prime for the following values of p: 2, 17.
n is an absolute Fermat pseudoprime for the following value of p: 577.
n is a product, not squarefree, of two primes for the following values of p: 283, 311.
Note: any number from the sequence is a term of one of the categories above.
This sequence is infinite under Dickson's conjecture. - Charles R Greathouse IV, Sep 20 2012

Marius A. Burtea, Table of n, a(n) for n = 1..18020

p=primes(10000);
m=1;
for  u=1:1000
    if  isprime(6*p(u)+1)==1
        sol(m)=3*p(u)-2;
        m=m+1;
    end
end
sol % Marius A. Burtea, Apr 10 2019

[3*p-2:p in PrimesUpTo(1000)| IsPrime(6*p+1)]; // Marius A. Burtea, Apr 10 2019

3#-2&/@Select[Prime[Range[200]],PrimeQ[6#+1]&] (* Harvey P. Dale, Mar 04 2023 *)

is(n)=n%3==1 && isprime(n\3+1) && isprime(2*n+5) \\ Charles R Greathouse IV, Dec 07 2014

a(1) added, comment corrected by Paolo P. Lava, Dec 18 2012

Missing term 697 added by Marius A. Burtea, Apr 10 2019

A216911 Numbers that are a prime factor of a Carmichael number divisible by 3, listed in the order in which they appear in the prime factorizations of such Carmichael numbers.

Original entry on oeis.org

3, 11, 17, 5, 47, 89, 101, 197, 29, 263, 521, 1559, 173, 3011, 71, 641, 1277, 53, 317, 4583, 617, 4019, 401, 3041, 41, 479, 3347, 131, 10427, 4643, 1301, 419, 6689, 5531, 281, 55217, 251, 2417, 4001, 491, 1601, 3137, 449, 3617, 107, 2969, 4211, 6737, 1061

Offset: 1

Marius Coman, Sep 20 2012

nonn

Every term, other than a(1)=3, is a prime of the form 6*k - 1.

			The first few Carmichael numbers that are divisible by 3 (A258801), their prime factorizations, and the terms of this sequence are as follows:
.
  m | A258801(m) & prime factorization | prime factors not appearing earlier
  --+----------------------------------+----------------------------------------
  1 |        561 = 3 * 11 *  17        | a(1)  =    3,  a(2)  = 11,   a(3) = 17;
  2 |      62745 = 3 *  5 *  47 *   89 | a(4)  =    5,  a(5)  = 47,   a(6) = 89;
  3 |     656601 = 3 * 11 * 101 *  197 | a(7)  =  101,  a(8)  = 197; ...

Charles R Greathouse IV, Table of n, a(n) for n = 1..3744
Eric Weisstein's World of Mathematics, Carmichael Number

Cf. A002997 (Carmichael numbers), A258801 (Carmichael numbers divisible by 3).

a(26), a(34), a(43) corrected by Charles R Greathouse IV, Sep 20 2012

A216944 The least odd prime p for which the n-th prime can be written as 3p + q - 3 for some odd prime q.

Original entry on oeis.org

3, 3, 3, 3, 3, 3, 5, 3, 5, 3, 3, 3, 3, 7, 3, 5, 3, 3, 5, 3, 7, 5, 3, 3, 3, 3, 7, 7, 3, 5, 5, 5, 3, 3, 7, 3, 3, 7, 5, 5, 3, 3, 5, 5, 11, 3, 3, 3, 5, 5, 3, 3, 3, 11, 3, 5, 3, 5, 11, 7, 3, 3, 7, 3, 11, 5, 3, 3, 7, 3, 3, 11, 3, 7, 5, 5, 7, 5, 5, 5, 3, 5, 3, 7, 5, 3, 3

Offset: 5

Marius Coman, Sep 20 2012

nonn

Conjecture: Any prime >= 11 can be written this way.

			The corresponding q and prime(n): (5,11), (7,13), (11,17), (13,19), (17,23), (23,29), (19,31), (31,37), (29,41), (37,43), (41,47), (47,53), (53,59), (43,61), (61,67), (67,73), (73,79), (71,83), (83,89), (79,97), (89,101), (97,103), (101,107), (103,109), (107,113), (109,127), (113,131), (131,137), (127,139), (137,149), (139,151), (151,157), (157,163), (149,167), (167,173), (173,179), (163,181), (179,191), (181,193), (191,197), (193,199), (199,211), (211,223), (197,227), (223,229), (227,233), (233,239), (229,241), (239,251), (251,257), (257,263), (263,269), (241,271),  (271,277), (269,281), (277,283), (281,293), (277,307), (293,311), (307,313), (311,317), (313,331), (331,337), (317,347), (337,349), (347,353), (353,359), (349,367), (367,373), (373,379), (353,383), (383,389), (379,397), (389,401), (397,409), (401,419), (409,421), (419,431), (421,433), (433,439), (431,443), (443,449), (439,457), (449,461), (457,463), (461,467), (467,479).

Robert Israel, Table of n, a(n) for n = 5..10000

f:= proc(n) local p,q;
  p:= 2;
  while 3*pRobert Israel, Mar 08 2018

a(n)=my(r=prime(n)+3); forprime(p=3,r\3-1,if(isprime(r-3*p), return(p))) \\ Charles R Greathouse IV, Dec 07 2014

New name from Charles R Greathouse IV, Dec 07 2014

Corrected by Charles R Greathouse IV, Dec 07 2014

A216646 a(n) = 1+2(d1 + 1)(d2 + 1)* … *(dk + 1), where d1, d2, ..., dk are the prime factors of the n-th Fermat pseudoprime to base 2 A001567(n).

Original entry on oeis.org

769, 1729, 2113, 3025, 2961, 4481, 6145, 4321, 6481, 5625, 7169, 6841, 8361, 9289, 12289, 9729, 11265, 16129, 16281, 17065, 24769, 21761, 21249, 26641, 34561, 36289, 34049, 28081, 32257, 29745, 32833, 37889, 43345, 63361, 38025, 40609, 72577, 47433, 71169

Offset: 1

Marius Coman, Sep 12 2012

nonn

It is notable how many primes, semiprimes, pseudoprimes, squares and multiples of 3 are in this sequence.
Primes obtained and the corresponding Fermat pseudoprime in the brackets: 769 (341), 2113 (645), 4481 (1729), 6481 (2465), 6841 (3277), 12289 (4371), 26641 (10585), 28081 (13747), 32257 (13981), 32833 (15709), 37889 (15841), 63361 (18705), 40609 (19951), 72577 (23001).
Semiprimes obtained and the corresponding Fermat pseudoprime in the brackets: 6145 (1905), 4321 (2047), 7169 (2821), 9289 (4369), 17065 (8321), 21761 (8911), 36289 (12801), 34049 (13741), 43345 (16705).
Pseudoprimes obtained and the corresponding Fermat pseudoprime in the brackets: 1729 (561).
Squares obtained and the corresponding Fermat pseudoprime in the brackets: 3025 = 5^2*11^2 (1105), 5625 = 3^2*5^4 (2701), 16129 = 127^2 (6601), 38025 = 3^2*5^2*13^2 (18721).
Multiples of 3 obtained and the corresponding Fermat pseudoprime in the brackets: 2961 = 3^2*329 (1387), 5625 = 3^2*625 (2701), 8361 = 3^2*929 (4033), 9729 = 3^2*1081 (4681), 3*3755 (5461), 16281 = 3^5*67 (7957), 21249 = 3^3*787 (10261), 29745 = 3^2*3305 (14491), 38025 = 3^2*4225 (18721), 47433 = 3*15811 (23377), 71169 = 3*23723 (25761).
The only numbers from the sequence above that are not into at least one of these categories (and the corresponding Fermat pseudoprime in the brackets) are 24769 = 17*31*47 (8481) and 34561 = 17*19*107 (11305).
An interesting correspondence with the function from the sequence A216404: with that one we obtain the pseudoprime 561 from the pseudoprime 1729 (2*a(n) + 1); with this one we obtain 1729 from 561 (a(n)). Another type of correspondence with that function: 2*a(n) + 1 = 769 for a(n) = 384 for that function (corresponding to pseudoprime 1905) while a(n) = 769 for this function (corresponding to pseudoprime 341).

Eric Weisstein's World of Mathematics, Poulet Number

Cf. A001567, A216404.

cp's OEIS Frontend

User: Marius Coman

A218483 Fermat pseudoprimes to base 2 which are congruent to 1 (mod 8).

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A218010 Primes of the form (24*p + 1)/5, where p is a Fermat pseudoprime to base 2.

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A217841 Fermat pseudoprimes n to base 3 for which sqrt(8*n + 1) is an integer.

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A217853 Fermat pseudoprimes to base 3 of the form (3^(4*k + 2) - 1)/8.

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A217835 Fermat pseudoprimes to base 2 that can be written as p^2*n - p*n + p, where p is also a Fermat pseudoprime to base 2 and n is a positive integer.

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A216830 Prime factors of Carmichael numbers divisible by 7, taken just once each as it appears first time, in order of the size of the Carmichael number respectively in order of their size if they are prime factors of the same Carmichael number.

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A216880 Numbers of the form 3p - 2 where p and 6p + 1 are prime.

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A216911 Numbers that are a prime factor of a Carmichael number divisible by 3, listed in the order in which they appear in the prime factorizations of such Carmichael numbers.

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A216944 The least odd prime p for which the n-th prime can be written as 3p + q - 3 for some odd prime q.

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A217835 Fermat pseudoprimes to base 2 that can be written as p^2n - pn + p, where p is also a Fermat pseudoprime to base 2 and n is a positive integer.

A216646 a(n) = 1+2(d1 + 1)(d2 + 1)* … *(dk + 1), where d1, d2, ..., dk are the prime factors of the n-th Fermat pseudoprime to base 2 A001567(n).