A215672 -id:A215672 - cp's OEIS frontend

A215944 Fermat pseudoprimes to base 2 with three prime factors divisible by a smaller Fermat pseudoprime to base 2.

13981, 137149, 158369, 176149, 276013, 285541, 294409, 348161, 387731, 423793, 488881, 493697, 617093, 625921, 847261, 1052503, 1052929, 1104349, 1128121, 1152271, 1398101, 1461241, 1472353, 1507561, 1534541, 1549411, 1746289, 1840357, 1857241, 2299081

Offset: 1

Marius Coman, Aug 28 2012

nonn

These are members of A215150 having three divisors. - T. D. Noe, Aug 28 2012
Almost all the numbers from the sequence above can be written as p*((m + 1)*p - m)*((n + 1)*p - n), where m, n, p are natural numbers (in the brackets is written the Poulet number which every one of them is divisible by):
(1)  n*(2*n - 1)*(3*n - 2): the number 294409 (2701);
(2)  n*(2*n - 1)*(5*n - 4): the numbers 285541 (4681), 488881 (2701);
(3)  n*(2*n - 1)*(11*n - 10): the number 625921 (10261);
(4)  n*(2*n - 1)*(15*n - 14): the number 1461241 (2701);
(5)  n*(3*n - 2)*(4*n - 3): the numbers 13981 (341), 137149 (2047);
(6)  n*(3*n - 2)*(5*n - 4): the number 1152271 (5461);
(7)  n*(3*n - 2)*(8*n - 7): the number 1840357 (5461);
(8)  n*(3*n - 2)*(10*n - 9): the number 2299081 (5461);
(9)  n*(3*n - 2)*(12*n - 11): the number 1746289 (4033);
(10) n*(3*n - 2)*(31*n - 30): the number 1052503 (15709);
(11) n*(3*n - 2)*(102*n - 101): the number 348161 (341);
(12) n*(3*n - 2)*(442*n - 441): the number 1507561 (341);
(13) n*(4*n - 3)*(7*n - 6): the number 176149 (1387);
(14) n*(4*n - 3)*(11*n - 10): the number 276013 (1387);
(15) n*(4*n - 3)*(12*n - 11): the number 1104349 (3277);
(16) n*(4*n - 3)*(31*n - 30): the number 1398101 (15709);
(17) n*(5*n - 4)*(6*n - 5): the number 847261 (4681);
(18) n*(5*n - 4)*(8*n - 7): the number 1128121 (4681);
(19) n*(5*n - 4)*(11*n - 10): the number 1549411 (4681);
(20) n*(6*n - 5)*(11*n - 10): the number 1857241 (10261);
(21) n*(6*n - 5)*(16*n - 15): the number 423793 (4369);
(22) n*(7*n - 6)*(16*n - 15): the number 493697 (4369);
(23) n*(15*n - 14)*(16*n - 15): the number 1052929 (4369);
(24) n*(16*n - 15)*(21*n - 20): the number 1472353 (4369).
The only few numbers from the sequence above that can’t be written this way are multiples of the Poulet number 5461 and can be, instead, written as 5461*(42*k - 13): 158369 = 5461*29, 387731 = 5461*71, 617093 = 5461*113 and 1534541 = 5461*281.
Conjecture: The only Fermat pseudoprimes to base 2 divisible by a smaller Fermat pseudoprime to base 2 that can’t be written as p*((m + 1)*p - m)*((n + 1)*p - n), where m, n, p are natural numbers, are multiples of 5461 and can be written as 5461*(42*k - 13).
Conjecture is checked for the numbers from the sequence above and for the first 15 Poulet numbers with four prime factors.
Note: There are Fermat pseudoprimes to base 2 divisible by 5461 that can be written as p*((m + 1)*p - m)*((n + 1)*p - n); these ones can be written as 5461*(42*k + 43). Numbers from this category are: 1152271 = 5461*211, 1840357 = 5461*337, 2299081 = 5461*421.

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

Cf. A001567, A215150, A215672.

list(lim)=my(v=List(),t,psp); forprime(p=3,sqrtint(lim\3), forprime(q=p,lim\p\3, t=p*q; if(Mod(2,t)^t!=2, next); forprime(r=3, lim\t, psp=t*r; if(Mod(2,psp)^psp==2, listput(v,psp))))); Set(v) \\ Charles R Greathouse IV, Jun 30 2017

A216364 Fermat pseudoprimes to base 2 divisible by 15.

Original entry on oeis.org

645, 1905, 18705, 55245, 62745, 72885, 215265, 451905, 831405, 1246785, 1472505, 1489665, 1608465, 1815465, 2077545, 2113665, 2882265, 4535805, 6135585, 6242685, 8322945, 9063105, 9816465, 16263105, 18137505, 19523505, 53661945, 63560685, 81612105, 81722145

Offset: 1

Marius Coman, Sep 05 2012

nonn

Most of the numbers in the sequence above can be written in one of just two forms: 15*(42*n + 1) and 15*(42*n - 13):
(I) numbers of the first form and the corresponding n in the brackets: 645(1), 1905(3), 1246785(1979), 2113665(3355), 2882265(4575), 6135585(9739); 6242685(9909); 8322945(13211), 81612105(129543);
(II) numbers of the second form and the corresponding n in the brackets: 18705(30), 55245(88), 72885(116), 215265(342), 831405(1320), 1815465(2882), 2077545(3298), 4535805(7200), 9816465(15582), 18137505(28790), 19523505(30990), 53661945(85178), 81722145(129718).
But these pseudoprimes can be categorized in many ways taking, beside 42, p - 1, where p is a prime divisor common to many of them (e.g., numbers of the form 15*(46*n + 43) and the corresponding n in the brackets: 62745 (90); 451905 (654); 1489665(2158); 9063105(13134); 63560685(92116)) or p + 1 (e.g., numbers of the form 15*(90*n + 67) and the corresponding n in the brackets: 1472505(1090); 16263105(12046)).
What is also interesting about these numbers: the Fermat pseudoprimes to base 2 formed with their prime divisors, different from 3 and 5 (e.g., 645 = 15*43 and 1905 = 15*127) are Fermat pseudoprimes to base 2, but also 5461 = 43*127; 18705 = 15*29*43 and 55245 = 15*29*127 are Fermat pseudoprimes to base 2, and 158369 = 29*43*127.
Note: Fermat pseudoprimes to base 2 divisible by 5 are mostly of the form 3*k or 3*k + 1; of the first 100 numbers divisible by 5 checked, fewer than 10 are of the form 3*k + 2.

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

Intersection of A001567 and A008597.

Cf. A215672, A215944, A216023.

Select[15*Range[10^6], PowerMod[2, # - 1, #] == 1 &] (* Amiram Eldar, Mar 07 2020 *)

is_a216364(n) = {Mod(2, n)^n==2 & !isprime(n) & Mod(n, 15)==0} \\ Michael B. Porter, Jan 27 2013

A296117 Base-2 pseudoprimes of the form 2pq where p and q are primes.

Original entry on oeis.org

161038, 49699666, 760569694, 4338249646, 357647681422, 547551530002, 3299605275646, 22999986587854, 42820164121582, 55173914702146, 69345154539266, 353190859033982

Offset: 1

Max Alekseyev, Dec 05 2017

a(5) and a(10) are found by McDaniel (1989).

Terms in this sequence are of the form 2pq where p and q are distinct odd primes (A075819). - Charles R Greathouse IV, Dec 05 2017

W. L. McDaniel, Some pseudoprimes and related numbers having special forms, Math. Comp. 53:187 (1989), 407-409.

Subsequence of A006935 and hence of A015919.
The even terms of A215672.
Intersection of A006935 and A215672. - Felix Fröhlich, Dec 05 2017
Cf. A270973, A075819, A001220.

list(lim)=my(v=List(),pq); forprime(p=3,lim\6, forprime(q=3,min(lim\(2*p),p), pq=p*q; if(Mod(4,pq)^pq==2, listput(v,2*pq)))); Set(v) \\ Charles R Greathouse IV, Dec 05 2017

A306487 Poulet numbers which are not super-Poulet numbers.

Original entry on oeis.org

561, 645, 1105, 1729, 1905, 2465, 2821, 4371, 6601, 8481, 8911, 10585, 11305, 12801, 13741, 13981, 15841, 16705, 18705, 23001, 25761, 29341, 30121, 30889, 33153, 34945, 39865, 41041, 41665, 46657, 52633, 55245, 57421, 62745, 63973, 68101, 72885, 74665, 75361

Offset: 1

Amiram Eldar, Feb 18 2019

nonn

Subsequence of A080747 from which this differs for the first time at n=78, with A080747(78) = 294409, a term not present here.
Is this sequence infinite?
According to Sierpinski there are infinitely many Poulet numbers which are not super-Poulet numbers. But his definition of Poulet numbers includes the even pseudoprimes to base 2 (A006935), and the proof is based on the infinitude of this sequence and that super-Poulet numbers are never even.

			561 is in the sequence because 2^561 % 561 == 2 but 33|561 and 2^33 % 33 = 8 <> 2. - _David A. Corneth_, Feb 28 2019

W. Sierpinski, Elementary Theory of Numbers, ed. A. Schinzel, North-Holland Mathematical Library (2nd ed.), Amsterdam: North Holland, 1988, Chapter V, p. 234, Exercise 1.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Poulet Number
Eric Weisstein's World of Mathematics, Super-Poulet Numbers
Wikipedia, Super-Poulet number

Cf. A001567, A050217, A006935, A080747.

Cf. A215672 (differs from a(13) = 11305 on, which is not in A215672).

Select[Select[Range[3, 100000, 2], !PrimeQ[ # ] && PowerMod[2, (# - 1), # ] == 1 &], Union[PowerMod[2, Rest[Divisors[#]], #]] != {2}& ]

is_A001567(n) = {Mod(2, n)^(n-1)==1 && !isprime(n) && n>1}; \\ From A001567 by M. F. Hasler
is_A050217(n) = if(isprime(n), 0, fordiv(n, d, if(Mod(2, d)^d!=2, return(0))); (n>1)); \\ After Charles R Greathouse IV's Aug 27 2016 PARI-program in A050217.
is_A306487(n) = (is_A001567(n) && !is_A050217(n)); \\ (Probably could be reduced to a simpler program). - Antti Karttunen, Feb 28 2019

is(n) = {if(isprime(n) || n < 2 || n%2 == 0, return(0)); if(Mod(2, n)^n!=2, return(0) , d = divisors(n); for(i = 1, #d-1, if(Mod(2, d[i])^d[i]!=2, return(1) ) ) ); 0 } \\ David A. Corneth, Feb 28 2019

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A215944 Fermat pseudoprimes to base 2 with three prime factors divisible by a smaller Fermat pseudoprime to base 2.

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A216364 Fermat pseudoprimes to base 2 divisible by 15.

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A296117 Base-2 pseudoprimes of the form 2*p*q where p and q are primes.

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A306487 Poulet numbers which are not super-Poulet numbers.

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A296117 Base-2 pseudoprimes of the form 2pq where p and q are primes.