OFFSET
1,1
COMMENTS
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.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Poulet Number
PROG
(PARI) 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Marius Coman, Aug 28 2012
STATUS
approved