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A295196 Numbers n > 1 such that 2^(n-1) and (2*n-m)*2^(((n-1)/2) - floor(log_2(n))) are congruent to 1 (mod n) for at least one of m = 3, m = 7 and m = 15. 0
7, 23, 31, 47, 71, 79, 263, 271, 1031, 1039, 2063, 4111, 32783, 65543, 65551, 262151, 1048583, 4194319, 8388623, 67108879, 268435463, 1073741831, 1073741839, 4294967311 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This definition arises from the conjecture that pseudoprime numbers (A001567) occur only at certain distances m from the next smaller number of the form 2^n. So, if we know that a certain distance does not appear with pseudoprime numbers, we are able to calculate these numbers using Fermat's little theorem and we know that it has to be prime. To "plot" the distance of pseudoprime numbers to 2^n use m = A001567(n) - 2^floor(log_2(A001567(n))). So, the first values of m which do not have a "safe prime number distance" (values with "safe prime number distance" are those values for m which pseudoprime numbers never have) should be m = 1, 49, 81, 85, 129, 133, 273, 275, 289, 321, ....

Conjecture 1: There are no composite numbers in this sequence and perhaps infinitely many primes.

Conjecture 2: For m = 7 this definition generates A104066 and for m = 15 this definition generates A144487 (A057197).

Conjecture 3: There are (infinitely many?) m for which this definition generates nothing but (infinitely many?) primes of the form p = 2^k + m.

It appears that this sequence is a subsequence of A139035.

LINKS

Table of n, a(n) for n=1..24.

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

MAPLE

P:=proc(q) local m, n, ok; for n from 1 to q do

if 2^(n-1) mod n=1 then ok:=0; for m in [1, 3, 7] do

  if (((2*n-(2*m+1))*(2^(((n-1)/2)-floor(log(n)/log(2)))))) mod n=1

then ok:=1; break; fi; od; if ok=1 then print(n); fi; fi; od; end: P(10^5); # Paolo P. Lava, Nov 17 2017

MATHEMATICA

twoDistableQ[n_] := MemberQ[Mod[(2n - {3, 7, 15}) PowerMod[2, (n - 1)/2 - Floor@ Log2@ n, n], n], 1]; p = 3; twoDistablesList = {}; While[p < 1000000000, If[twoDistableQ@ p, AppendTo[ twoDistablesList, p]]; p = NextPrime@ p]; twoDistablesList (* Robert G. Wilson v, Nov 17 2017 *)

PROG

(PARI) a(n) = (n%2) && lift((Mod(2, n)^(n-1))==1) && (lift((Mod((2*n-3), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1)||lift((Mod((2*n-7), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1)||lift((Mod((2*n-15), n)*Mod(2, n)^(((n-1)/2)-floor(log(n)/log(2)))) == 1))

(PARI) is(n)=if(Mod(2, n)^(n-1)!=1, return(0)); my(m=Mod(2, n)^(n\2-logint(n, 2))); ((2*n-3)*m==1 || (2*n-7)*m==1 || (2*n-15)*m==1) && n>1 \\ Charles R Greathouse IV, Nov 17 2017

CROSSREFS

Cf. A057197, A104066, A139035, A144487, A244626, A294717, A293394.

Sequence in context: A014663 A007522 A141175 * A287309 A275777 A329931

Adjacent sequences:  A295193 A295194 A295195 * A295197 A295198 A295199

KEYWORD

more,nonn

AUTHOR

Jonas Kaiser, Nov 16 2017

EXTENSIONS

a(17)-a(24) from Charles R Greathouse IV, Nov 17 2017

STATUS

approved

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Last modified August 3 21:01 EDT 2021. Contains 346441 sequences. (Running on oeis4.)