A294912 Numbers n such that 2^(n-1), (2*n-1)*(2^((n-1)/2)), (4*ceiling((3/4)*n)-2), and (2^((n+1)/2) + floor((1/4)*n)*2^(((n+1)/2)+1)) are all congruent to 1 (mod n).

3, 11, 19, 43, 59, 67, 83, 107, 131, 139, 163, 179, 211, 227, 251, 283, 307, 331, 347, 379, 419, 443, 467, 491, 499, 523, 547, 563, 571, 587, 619, 643, 659, 683, 691, 739, 787, 811, 827, 859, 883, 907, 947, 971, 1019, 1051, 1091, 1123, 1163, 1171, 1187, 1259

Offset: 1

Author

Jonas Kaiser, Nov 10 2017

Keywords

• nonn

Comments

It appears that A007520 is a subsequence.

The first composite term is a(9969) = 476971 = 11*131*331. - Alois P. Heinz, Nov 10 2017

From Hilko Koning, Dec 03 2019: (Start)

The next composite terms < 1999979 are

a(17428) = 877099 = 307*2857

a(25090) = 1302451 = 571*2281

a(25518) = 1325843 = 499*2657

a(26785) = 1397419 = 67*20857

a(27549) = 1441091 = 347*4153

a(28715) = 1507963 = 971*1553

a(29117) = 1530787 = 619*2473

a(35635) = 1907851 = 11*251*691

(End)

From Hilko Koning, Dec 05 2019: (Start)

The next composite terms < 24999971 are

a(37344) = 2004403 = 307*6529

a(55773) = 3090091 = 1163*2657

a(56189) = 3116107 = 883*3529

a(91332) = 5256091 = 811*6481

a(102027) = 5919187 = 1777*3331

a(133230) = 7883731 = 811*9721

a(156407) = 9371251 = 1531*6121

a(182911) = 11081459 = 227*48817

a(189922) = 11541307 = 1699*6793

a(201043) = 12263131 = 811*15121

a(213203) = 13057787 = 467*27961

a(217484) = 13338371 = 3163*4217

a(257526) = 15976747 = 3739*4273

a(274961) = 17134043 = 1097*15619

a(299096) = 18740971 = 1531*12241

a(308928) = 19404139 = 2011*9649

a(321676) = 20261251 = 2251*9001

a(341902) = 21623659 = 1163*18593

a(348622) = 22075579 = 163*135433

a(380162) = 24214051 = 281*86171

The composite terms < 25*10^6 match the terms of A244628.

(End)

It appears that composites of the form 2k+1 such that 3*(2k+1) divides 2^k+1 are the composite terms of this sequence. - Hilko Koning, Dec 09 2019

Links

• Jonas Kaiser, On the relationship between the Collatz conjecture and Mersenne prime numbers, arXiv:1608.00862 [math.GM], 2016.

Crossrefs

Cf. A244626, A294717, A293394, A070179, A007520.

Programs

Mathematica

okQ[n_] := AllTrue[{2^(n-1), (2*n-1)*(2^((n-1)/2)), (4*Ceiling@((3/4)*n) - 2), (2^((n+1)/2) + Floor@(n/4)*2^(((n+1)/2)+1))}, Mod[#, n] == 1&];
Select[Range[1300], okQ] (* Jean-François Alcover, Feb 18 2019 *)

PARI

isok(n) = (n%2) && lift((Mod(2, n)^(n-1))==1)&&lift((Mod((2*n-1), n)*Mod(2, n)^((n-1)/2)) == 1)&&lift((Mod(((4*ceil((3/4)*n)-2)), n) )== 1)&&lift((Mod(2, n)^((n+1)/2) +Mod(floor((1/4)*n),n)*Mod(2, n)^(((n+1)/2)+1 ))== 1)

Extensions

More terms from Alois P. Heinz, Nov 10 2017