A332075 -id:A332075 - cp's OEIS frontend

A332079 Number of primes between 2^n and the least prime p > 2^n in A332075, i.e., such that k + 2^m is prime, where k and m are the odd part and 2-valuation, respectively, of p-1.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 8, 1, 9, 7, 0, 0, 7, 5, 1, 2, 4, 9, 1, 7, 8, 6, 11, 0, 4, 0, 1, 1, 0, 0, 10, 17, 3, 0, 8, 0, 10, 20, 3, 23, 15, 3, 20, 13, 7, 36, 17, 15, 4, 4, 0, 9, 15, 10, 21, 8, 22, 36, 6, 13, 2, 7, 36, 14, 10, 9, 4, 0, 44, 10, 8, 27, 5, 1, 0, 2, 22, 3, 2, 33, 20, 21, 19, 12, 12, 5

Offset: 1

M. F. Hasler, Aug 13 2020

nonn

It appears that the sequence of odd numbers k*2^m+1 such that k + 2^m is prime (A332075) mainly consists of primes, and many primes are in this sequence. This sequence attempts to measure in how far this remains true for large numbers.

T. Ordowski, Problem, post to the SeqFan list, Aug 11 2020.

Cf. A332075, A000040 (primes), A000265 (odd part), A007814 (2-valuation).

a[n_] := Module[{count = 0, p = NextPrime[2^n]}, While[!PrimeQ[(m = 2^IntegerExponent[p - 1, 2]) + (p - 1)/m], count++; p = NextPrime[p]]; count]; s = Array[a, 100] (* Amiram Eldar, Aug 14 2020 *)

apply( {A332079(n,c=0)=forprime(p=2^n,,is_A332075(p)&&return(c);c++)}, [1..99])

A332078 Primes p = k*2^m + 1 such that k + 2^m is not prime, where k and m are the odd part and 2-valuation, respectively, of p-1.

Original entry on oeis.org

47, 67, 97, 107, 127, 137, 151, 167, 179, 181, 227, 239, 263, 283, 293, 307, 347, 349, 367, 431, 439, 457, 461, 467, 487, 491, 503, 547, 557, 571, 587, 599, 607, 617, 641, 643, 647, 661, 683, 719, 727, 733, 739, 751, 769, 787, 797, 811, 821, 823, 827, 853, 857, 887, 907

Offset: 1

M. F. Hasler, Aug 13 2020

nonn

It appears that the sequence of odd numbers k*2^m+1 such that k + 2^m is prime (A332075) mainly consists of the primes. This sequence lists the "exceptions": the complement of A332075 within the primes. (The exceptions become more frequent as the numbers grow, the asymptotic density of this subset within the primes might well approach one. See also A332079.)

These are primes of the form p = (w-2^m)*2^m + 1, where w is an odd composite number and 1 < 2^m < w. There are infinitely many primes of this form, because all primes p > 7 such that p == 7 (mod 20) are in this sequence. - Thomas Ordowski, Aug 13 2020

Robert Israel, Table of n, a(n) for n = 1..10000
Thomas Ordowski, Problem, post to the SeqFan list, August 2020.

Cf. A000040 (primes), A000265 (odd part), A007814 (2-valuation), A332075.

The terms A141882 > 7 are an infinite subsequence. - Thomas Ordowski, Aug 13 2020

filter:= proc(p) local k,m;
   if not isprime(p) then return false fi;
   m:= padic:-ordp(p-1,2);
   k:= (p-1)/2^m;
   not isprime(k+2^m);
end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Sep 14 2020

Select[Range[1000], PrimeQ[#] && !PrimeQ[(m = 2^IntegerExponent[# - 1, 2]) + (# - 1)/m] &] (* Amiram Eldar, Aug 14 2020 *)

(A332078_upto(N)=[p|p<-primes([1,N]),!is_A332075(p)])(1000)

A332076 Indices n of odd numbers 2n+1 such that k + 2^m is prime, where k and m are the odd part and 2-valuation, respectively, of 2n.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 24, 26, 27, 29, 30, 35, 36, 38, 39, 41, 44, 45, 50, 51, 54, 56, 57, 59, 60, 65, 66, 69, 71, 74, 77, 78, 80, 81, 84, 86, 87, 92, 95, 96, 98, 99, 101, 104, 105, 107, 110, 111, 114, 116, 120, 125, 126, 128, 129, 132, 134

Offset: 1

M. F. Hasler, Aug 13 2020

nonn

It appears that about 1/log_10(N) of the odd numbers below 2N have this property: for n < 10^k with k = (1, 2, 3, 4, 5, 6), there are (7, 51, 364, 2675, 20668, 167185) numbers as defined in NAME.

See the sequence A332075 of the corresponding odd numbers for more information.

T. Ordowski, Problem, post to the SeqFan list, Aug. 11, 2020

Cf. A000040 (primes), A000265 (odd part), A007814 (2-valuation), A332075 (the corresponding odd numbers).

Select[Range[134], PrimeQ[(m = 2^IntegerExponent[2*#, 2]) + 2*#/m] &] (* Amiram Eldar, Aug 16 2020 *)

select( {is_A332076(n)=ispseudoprime((n>>n=valuation(n,2))+2<

cp's OEIS Frontend

A332079 Number of primes between 2^n and the least prime p > 2^n in A332075, i.e., such that k + 2^m is prime, where k and m are the odd part and 2-valuation, respectively, of p-1.

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A332078 Primes p = k*2^m + 1 such that k + 2^m is not prime, where k and m are the odd part and 2-valuation, respectively, of p-1.

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A332076 Indices n of odd numbers 2n+1 such that k + 2^m is prime, where k and m are the odd part and 2-valuation, respectively, of 2n.

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Author

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Comments

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Crossrefs

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Mathematica

PARI