A067526 -id:A067526 - cp's OEIS frontend

A039669 Numbers n > 2 such that n - 2^k is a prime for all k > 0 with 2^k < n.

4, 7, 15, 21, 45, 75, 105

Offset: 1

Erdős conjectures that these are the only values of n with this property.
No other terms below 2^120. - Max Alekseyev, Dec 08 2011
Curiously, Mientka and Weitzenkamp say there are 9 such numbers below 20000. - Michel Marcus, May 12 2013
Presumably, Mientka and Weitzenkamp are including 1 and 2. - Robert Israel, Dec 23 2015
Observation: The prime numbers of the form (n-2) associated with each element of the series are (2,5,13,19,43,73,103). These prime numbers are exactly the first elements of A068374 (primes n such that positive values of n - A002110(k) are all primes for k>0). - David Morales Marciel, Dec 14 2015

			45 is here because 43, 41, 37, 29 and 13 are primes.

R. K. Guy, Unsolved Problems in Number Theory, A19.
F. Le Lionnais, Les Nombres Remarquables, Paris, Hermann, 1983, p. 96, 1983.
D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, p. 306.
D. Wells, Curious and interesting numbers, Penguin Books, p. 118.

P. Erdős, On integers of the form 2^k + p and some related questions, Summa Bras. Math., 2 (1950), 113-123.
Walter E. Mientka and Roger C. Weitzenkamp, On f-plentiful numbers, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, pages 374-377.

Cf. A067526 (n such that n-2^k is prime or 1), A067527 (n such that n-3^k is prime), A067528 (n such that n-4^k is prime or 1), A067529 (n such that n-5^k is prime), A100348 (n such that n-4^k is prime), A100349 (n such that n-2^k is prime or semiprime), A100350 (primes p such that p-2^k is prime or semiprime), A100351 (n such that n-2^k is semiprime).

Cf. A022005.

N = 10^8; % to get terms < N
p = primes(N);
A = [3:N];
for k = 1:floor(log2(N))
  A = intersect(A, [1:(2^k), (p+2^k)]);
end
A % Robert Israel, Dec 23 2015

lst={}; Do[k=1; While[p=n-2^k; p>0 && PrimeQ[p], k++ ]; If[p<=0, AppendTo[lst, n]], {n, 3, 1000}]; lst (* T. D. Noe, Sep 15 2002 *)

isok(n) = {my(k = 1); while (2^k < n, if (! isprime(n-2^k), return (0)); k++;); return (1);} \\ Michel Marcus, Dec 14 2015

Additional comments from T. D. Noe, Sep 15 2002

Definition edited by Robert Israel, Dec 23 2015

A067528 Numbers n such that n - 4^k is a prime or 1 for all k > 0 and n > 4^k.

Original entry on oeis.org

5, 6, 7, 9, 11, 15, 17, 21, 23, 27, 33, 35, 45, 47, 57, 63, 75, 77, 83, 87, 105, 117, 143, 153, 167, 195, 215, 227, 243, 245, 255, 287, 297, 413, 437, 447, 483, 495, 507, 525, 573, 635, 657, 677, 755, 825, 1113, 1133, 1295, 1487, 1515, 1547, 1617, 1623, 2015

Offset: 1

Amarnath Murthy, Feb 17 2002

nonn

Is the sequence finite?
The last term appears to be 5833497. - T. D. Noe, Nov 23 2004
A less strict version of A039669, n - 2^k is prime for 0 < k < log_2 k. If a number is in that sequence, then obviously it is also in this sequence. As of this writing, 105 is believed to be the last term of that sequence. - Alonso del Arte, May 24 2017

			167 is a term as 167 - 4 = 163, 167 - 16 = 151, 167 - 64 = 103 are primes.

T. D. Noe, Table of n, a(n) for n=1..102 (no others < 2*10^9)

Cf. A067526.

filter:= proc(n) local k, t;
  for k from 1 do
    if 4^k >= n-1 then return true
    elif not isprime(n-4^k) then return false
    fi
  od
end proc:
select(filter, [$5..3000]); # Robert Israel, May 24 2017

A067528 = {}; Do[k = 1; While[p = n - 4^k; p > 0 && (p == 1 || PrimeQ[p]), k++]; If[p <= 0, AppendTo[A067528, n]], {n, 5, 10^7}]; A067528 (* T. D. Noe *)

More terms from Sascha Kurz, Mar 19 2002

A067527 n - 3^k is a prime for all k > 0 such that 3^k < n.

Original entry on oeis.org

5, 6, 8, 14, 16, 20, 22, 26, 32, 40, 46, 50, 56, 70, 110, 140, 260, 470, 1190, 1330

Offset: 1

Amarnath Murthy, Feb 17 2002

Is the sequence finite?

No other terms below 3^36. - Max Alekseyev, Dec 12 2011

			40 is a member as 40 - 3 = 37, 40 - 9 = 31, 40 - 27 = 13 are all primes.

Cf. A067526.

Cf. A039669 (n - 2^k is prime).

A067527 = {}; Do[k = 1; While[p = n - 3^k; p > 0 && PrimeQ[p], k++]; If[p <= 0, AppendTo[A067527, n]], {n, 4, 10000}]; A067527 (* T. D. Noe, Feb 20 2005 *)

More terms from Sascha Kurz, Mar 18 2002

Edited by Max Alekseyev, Dec 12 2011

A067529 n - 5^k is a prime for all k > 0 and n > 5^k.

Original entry on oeis.org

7, 8, 10, 12, 16, 18, 22, 24, 28, 36, 42, 48, 66, 72, 78, 84, 108, 114, 132, 156, 162, 198, 204, 276, 282, 288, 318, 336, 492, 504, 546, 582, 612, 624, 666, 864, 882, 1044, 1134, 1218, 1242, 1326, 1452, 1998, 2136, 2472, 2922, 3234, 3948, 4032, 4572, 4914, 6342

Offset: 1

Amarnath Murthy, Feb 17 2002

Is the sequence finite?

The last term appears to be 7726572. - T. D. Noe, Nov 23 2004

			624 is a term as 624-5, 624-25,624-125 or 619,599 and 499 are primes.

Michel Marcus, Table of n, a(n) for n = 1..66
Walter E. Mientka and Roger C. Weitzenkamp, On f-plentiful numbers, Journal of Combinatorial Theory, Volume 7, Issue 4, December 1969, Pages 374-377

Cf. A067526, A067527, A067528.

Cf. A039669 (n-2^k is prime).

lst={}; Do[k=1; While[p=n-5^k; p>0 && PrimeQ[p], k++ ]; If[p<=0, AppendTo[lst, n]], {n, 6, 10^7}]; lst (T. D. Noe)

More terms from Sascha Kurz, Mar 19 2002

A282459 Number of composite numbers of the form 2n - 2^k + 1 (k > 0, 2^k < 2n + 1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 2, 2, 1, 3, 2, 1, 2, 3, 1, 4, 3, 0, 3, 2, 2, 4, 2, 3, 4, 2, 1, 4, 4, 1, 4, 4, 0, 3, 4, 3, 3, 4, 2, 5, 3, 3, 4, 5, 3, 4, 4, 0, 4, 4, 1, 4, 3, 2, 5, 4, 4, 4, 6, 3, 4, 4, 2, 6, 3, 3, 4, 4, 3, 7, 5, 3, 5, 5, 3, 5, 6, 2, 4, 4, 2, 5, 4, 5, 6, 3, 3, 6, 5, 3, 6, 6, 1, 5, 3, 2, 5, 5, 4, 6, 5, 3, 4, 6

Offset: 0

Altug Alkan, Feb 15 2017

nonn

It is conjectured that a(n) > 0 for all n > 52. See related conjecture and findings in A039669. Also see the graph of this sequence.

			a(7) = 0 because 2*7 + 1 - 2^1 = 13, 2*7 + 1 - 2^2 = 11, 2*7 + 1 - 2^3 = 7 are prime numbers.

Altug Alkan, Table of n, a(n) for n = 0..10000

Cf. A002808, A039669, A067526, A109925.

isA002808(n) = n>1 && !isprime(n);
a(n) = sum(k=1, log(2*n+1)\log(2), isA002808(2*n+1-2^k))

A067530 Numbers k such that k - m! is a prime or 1 for all m > 1 and k > m!.

Original entry on oeis.org

3, 4, 5, 7, 9, 13, 19, 25, 43, 85, 103, 133, 403, 763, 943, 1573, 1603, 2713, 5233, 26023, 37363, 177133, 186043, 276043, 277603, 305863, 968833, 1449313, 1540033, 5854363, 6013873, 26114323, 35088793, 291865753, 724927333

Offset: 1

Amarnath Murthy, Feb 17 2002

nonn

Is the sequence finite?

Next term, if it exists, exceeds 6*10^9. - Sean A. Irvine, Dec 18 2023

			85 is a term as 85 - 2!, 85 -3!, 85-4! or 83,79 and 61 are primes.

Cf. A067526, A067527, A067528.

More terms from Sascha Kurz, Mar 19 2002

Offset corrected and a(33)-a(34) from Sean A. Irvine, Dec 18 2023

cp's OEIS Frontend

A039669 Numbers n > 2 such that n - 2^k is a prime for all k > 0 with 2^k < n.

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A067528 Numbers n such that n - 4^k is a prime or 1 for all k > 0 and n > 4^k.

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A067527 n - 3^k is a prime for all k > 0 such that 3^k < n.

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A067529 n - 5^k is a prime for all k > 0 and n > 5^k.

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A282459 Number of composite numbers of the form 2*n - 2^k + 1 (k > 0, 2^k < 2*n + 1).

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A067530 Numbers k such that k - m! is a prime or 1 for all m > 1 and k > m!.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A282459 Number of composite numbers of the form 2n - 2^k + 1 (k > 0, 2^k < 2n + 1).