Search: a232565 -id:a232565
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A156695
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Odd numbers that are not of the form p + 2^a + 2^b, a, b > 0, p prime.
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+10
51
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1, 3, 5, 6495105, 848629545, 1117175145, 2544265305, 3147056235, 3366991695, 3472109835, 3621922845, 3861518805, 4447794915, 4848148485, 5415281745, 5693877405, 6804302445, 7525056375, 7602256605, 9055691835, 9217432215
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OFFSET
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1,2
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COMMENTS
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Crocker shows that this sequence is infinite.
All members above 5 found so far (up to 2.5 * 10^11) are divisible by 255 = 3 * 5 * 17, and many are divisible by 257. I conjecture that all members of this sequence greater than 5 are divisible by 255. This implies that all odd numbers (greater than 7) are the sum of a prime and at most three positive powers of two.
Pan shows that, for every c > 1, a(n) << x^c. More specifically, there are constants C,D > 0 such that there are at least Dx/exp(C log x log log log log x/log log log x) members of this sequence up to x. - Charles R Greathouse IV, Apr 11 2016
All terms > 5 are numbers k > 3 such that k - 2^n is a de Polignac number (A006285) for every n > 0 with 2^n < k. Are there numbers K such that |K - 2^n| is a Riesel number (A101036) for every n > 0? If so, ||K - 2^n| - 2^m| is composite for every pair m,n > 0, by the dual Riesel conjecture. - Thomas Ordowski, Jan 06 2024
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LINKS
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EXAMPLE
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Prime factorization of terms:
F_0 = 3, F_1 = 5, F_2 = 17, F_3 = 257 are Fermat numbers (cf. A000215)
6495105 = 3 * 5 * 17 * 25471
848629545 = 3 * 5 * 17 * 461 * 7219
1117175145 = 3 * 5 * 17 * 257 * 17047
2544265305 = 3^2 * 5 * 17 * 257 * 12941
3147056235 = 3^2 * 5 * 17 * 257 * 16007
3366991695 = 3 * 5 * 17 * 83 * 257 * 619
3472109835 = 3 * 5 * 17 * 257 * 52981
3621922845 = 3 * 5 * 17^2 * 257 * 3251
3861518805 = 3^3 * 5 * 17 * 257 * 6547
4447794915 = 3^3 * 5 * 17 * 257 * 7541
4848148485 = 3^4 * 5 * 17 * 704161
5415281745 = 3 * 5 * 17 * 21236399
5693877405 = 3^2 * 5 * 17 * 257 * 28961
6804302445 = 3^2 * 5 * 17 * 53 * 257 * 653
7525056375 = 3^2 * 5^3 * 17 * 257 * 1531
7602256605 = 3 * 5 * 17 * 257 * 311 * 373
9055691835 = 3 * 5 * 17 * 257 * 138181
9217432215 = 3^2 * 5 * 17 * 173 * 257 * 271
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PROG
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(PARI) is(n)=if(n%2==0, return(0)); for(a=1, log(n)\log(2), for(b=1, a, if(isprime(n-2^a-2^b), return(0)))); 1 \\ Charles R Greathouse IV, Nov 27 2013
(Python)
from itertools import count, islice
from sympy import isprime
def A156695_gen(startvalue=1): # generator of terms >= startvalue
for n in count(max(startvalue+(startvalue&1^1), 1), 2):
l = n.bit_length()-1
for a in range(l, 0, -1):
c = n-(1<<a)
for b in range(min(a, l-1), 0, -1):
if isprime(c-(1<<b)):
break
else:
continue
break
else:
yield n
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CROSSREFS
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KEYWORD
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nonn,hard,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A268693
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Odd numbers that are not of the form p + 2^a + 2^b with b > a > 0, and p prime.
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+10
3
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1, 3, 5, 7, 255, 15045, 20655, 25755, 39525, 44115, 46665, 65535, 70125, 97155, 100215, 132855, 144465, 162945, 196605, 200175, 277695, 280755, 327675, 441915, 469965, 548505, 688245, 720375, 770355, 773925, 789225, 1017705, 1027395, 1110015, 1114095, 1127355
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OFFSET
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1,2
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COMMENTS
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This sequence is infinite; in particular, 2^(2^n) - 1 is in this sequence for each n > 2.
Not every member of this sequence greater than 7 is divisible by 255, see A268694.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A369375
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Numbers m such that the Mersenne number 2^m - 1 is a de Polignac number (A006285).
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+10
2
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1, 7, 15, 23, 27, 31, 37, 39, 43, 55, 58, 63, 71, 79, 82, 91, 95, 111, 123, 127, 133, 135, 139, 143, 148, 151, 159, 167, 169, 172, 173, 175, 179, 183, 191, 195, 199, 207, 211, 223, 239, 255, 286, 295, 313, 316, 319, 335, 337, 351, 367, 373, 383, 406, 415, 417, 433, 435, 447, 455, 461, 463, 479
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Integers m > 0 such that 2^m-1 - 2^n is not prime for every natural n < m.
For m > 2, a number m is a term of this sequence if and only if A208083(m) = 0.
All Mersenne number m = 2^k-1 for k > 2 are in this sequence. The proof is below.
Cf. A138290 (see Chai Wah Wu's conjecture in the third comment). By Crocker's (1971) theorem: if m > 2 and a <> b, then 2^(2^m)-1 - 2^a - 2^b is not prime.
If a = 2^m-1, then b < a, so for m > 2, 2^(2^m-1)-1 is a de Polignac number, QED.
Note that 2^(2^m-1)-1 - 2^n is divisible by some prime factor of 2^(2^m)-1.
Prime numbers of this sequence are Mersenne primes > 3, and many other primes.
Conjecture: if n > 5, then |2^(2^n-1)-1 - 2^m| is not prime for every m > 0.
If so, then by the dual Riesel conjecture, 2^(2^n-1)-1 is a (dual) Riesel number, i.e., if n > 5, then (2^(2^n-1)-1)2^m-1 is composite for every integer m > 0.
For example, the double Mersenne prime 2^(2^7-1)-1 may be a dual Riesel number.
It seems that the natural density of these numbers is about twice as high as the density of de Polignac numbers.
For many terms m, 2m+1 is also in this sequence. By iteration (x -> 2x+1), the subsequence b(n) = (m+1)2^n-1, for n >= 0, is infinite if m = 7 (which has already been proven) and probably if m = 27 (which is hard to prove).
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LINKS
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FORMULA
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For n > 1, a(n) = A138290(n-1) + 1.
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EXAMPLE
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7 is a term since {2^7-1-2, 2^7-1-2^2, 2^7-1-2^3, 2^7-1-2^4, 2^7-1-2^5, 2^7-1-2^6} = {125, 123, 119, 111, 95, 63} and all six of these numbers are composite.
Note that both 2^148-1 and 2^148+1 are de Polignac numbers.
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MATHEMATICA
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fQ[n_] := Block[{k = n -1}, While[k > 1 && !PrimeQ[2^n -1 -2^k], k--]; k == 1]; Select[ Range[3, 450], fQ] (* Robert G. Wilson v, Jan 22 2024 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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