A039669 -id:A039669 - cp's OEIS frontend

A100348 Numbers n such that n-4^k is a prime for all k > 0 with 4^k < n.

6, 7, 9, 11, 15, 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, 2043, 2397

Offset: 1

T. D. Noe, Nov 18 2004

nonn

The largest term appears to be 5833497. No others < 10^9; conjectured to be finite. Similar to A067528, which also contains 5 and 17, but a more direct generalization of A039669, a problem due to Erdos.

			27 is here because 27-4 and 27-16 are primes.

Michel Marcus, Table of n, a(n) for n = 1..100
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. A039669 (n such that n-2^k is prime), A067528 (n such that n-4^k is prime or 1).

lst={}; Do[k=1; While[p=n-4^k; p>0 && PrimeQ[p], k++ ]; If[p<=0, AppendTo[lst, n]], {n, 5, 10^7}]; lst

A108957 Values of n such that n - 2^k is deficient for all 1 <= 2^k < n.

Original entry on oeis.org

2, 3, 4, 5, 6, 9, 11, 12, 15, 17, 18, 23, 27, 33, 35, 39, 45, 47, 51, 53, 54, 59, 63, 65, 66, 69, 75, 77, 83, 87, 93, 95, 99, 107, 111, 117, 119, 123, 125, 126, 129, 131, 135, 137, 138, 143, 147, 149, 150, 153, 155, 159, 165, 167, 171, 173, 174, 179, 183, 185, 186

Offset: 1

Jason Earls, Jul 22 2005

Conjectures: a. Sequence is infinite. b. There are infinitely many consecutive pairs, such as (5:6), (11:12), (17:18), (53:54), ... (204005:204006).

			53 is a term because 52, 51, 49, 45, 37 and 21 are all deficient numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A005100, A039669.

aQ[n_] := AllTrue[n - 2^Range[0, Floor[Log2[n]]], # == 0 || DivisorSigma[1, #] < 2 # &]; Select[Range[2, 186], aQ] (* Amiram Eldar, Sep 21 2019 *)

A057652 Numbers n such that n-2^k is a lucky number for all k such that 1 < 2^k < n.

Original entry on oeis.org

1, 2, 3, 5, 11, 17, 647

Offset: 1

Naohiro Nomoto, Oct 14 2000

Perhaps there are no more terms?

Lucky numbers have asymptotic properties very similar to prime numbers, so one can conjecture finiteness of this sequence in the same way as Erdős did for A039669, and this should generalize to any sequence created using a similar sieve. - M. F. Hasler, Oct 15 2010

			647 is in this sequence since 647-2, 647-4, 647-8, 647-16, 647-32, 647-64, 647-128, 647-256, 647-512 are all members of the sequence A000959 of lucky numbers. - _M. F. Hasler_, Oct 15 2010

Cf. A000959, A039669.

A057652(Nmax) = { my(v=vector(Nmax\2,i,2*i-1)); for(i=2,#v,v[i]>#v && break; v=vecextract(v,2^#v-1-sum(k=1,#v\v[i],2^(v[i]*k))>>1)); v=Set(v); for(n=1,Nmax, for(k=1,Nmax, 2^kM. F. Hasler, Oct 15 2010 */

Added initial terms {1, 2}, reworded definition following a suggestion from D. Forgues. - M. F. Hasler, Oct 15 2010

A091155 Numbers m such that m - 2^k is squarefree for all 1 <= 2^k < m.

Original entry on oeis.org

2, 3, 4, 7, 15, 23, 39, 63, 75, 87, 111, 135, 147, 159, 195, 219, 231, 255, 267, 315, 387, 399, 411, 423, 435, 447, 459, 495, 519, 567, 615, 663, 675, 699, 711, 735, 747, 759, 771, 819, 867, 915, 999, 1011, 1023, 1035, 1047, 1071, 1095, 1119, 1155, 1167, 1263

Offset: 1

T. D. Noe, Dec 23 2003

nonn

Erdős conjectures that this sequence is infinite. It appears that m == 3 (mod 12) except for m = 2, 4, 7 and 23.

			39 is on the list because 38, 37, 35, 31, 23 and 7 are all squarefree.

R. K. Guy, Unsolved Problems in Number Theory, A19.

T. D. Noe, Table of n, a(n) for n = 1..10000
P. Erdős, On integers of the form 2^k + p and some related problems, Summa Brasil. Math. 2 (1950), pp. 113-123.

Cf. A039669 (m such that m-2^k are all primes).

a={}; Do[k=1; While[sf=SquareFreeQ[n-k]; sf&&2k

is(n)=for(k=1,log(n+.5)\log(2),if(!issquarefree(n-2^k),return(0))); 1 \\ Charles R Greathouse IV, Apr 13 2014

A240842 Numbers n such that n - 2*k^2 is a prime for all k > 0 with k^2 < n/2.

Original entry on oeis.org

1, 2, 4, 5, 7, 13, 15, 21, 25, 31, 49, 55, 61, 91, 181, 199

Offset: 1

Ilya Lopatin and Juri-Stepan Gerasimov, Apr 13 2014

nonn

No other terms found for n < 2000000. - Colin Barker, Apr 13 2014
No other terms with n < 10^17. - Charles R Greathouse IV, Apr 14 2014
All terms > 4 must be odd, since otherwise n - 2*1^2 is composite. The initial terms 1 and 2 satisfy the condition voidly (no k^2 < n/2 exists). They could be excluded explicitly, but including them can only improve search results. - M. F. Hasler, Apr 16 2014

			91 is in this sequence because 91-2*1^2 = 89, 91-2*2^2 = 83, 91-2*3^2 = 73, 91-2*4^2 = 59, 91-2*5^2 = 41, 91-2*6^2 = 19 where 89, 83, 73, 59, 41, 19 are all primes.

Cf. A039669.

isOK(n) = k=1; until(k^2>=n/2, if(!isprime(n-2*k^2), return(0)); k++); 1;
for(n=1, 20000, if(isOK(n), print1(n, ", "))) \\ Colin Barker, Apr 14 2014

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))

A370523 Numbers k > 2 such that all positive values of k - 2^(2^m) are prime, with integer m >= 0.

Original entry on oeis.org

4, 7, 9, 15, 21, 33, 45, 63, 75, 105, 153, 183, 195, 243, 273, 285, 435, 525, 573, 603, 813, 825, 1065, 1233, 1305, 1623, 2145, 2595, 2715, 2805, 3375, 3465, 3933, 4023, 4245, 4275, 4653, 4803, 4935, 5655, 6303, 6705, 7563, 8865, 10095, 10503, 10863, 12165, 12243, 12825, 13713, 13725, 14013

Offset: 1

Thomas Ordowski, Feb 22 2024

nonn

If k > 4 is a term of this sequence, then (k-2, k-4) is a twin prime pair.
So all terms k > 7 are divisible by 3, and k = 7 is the only prime here.
It seems that there are infinitely many such numbers.
Note that A039669 is finite and probably complete.

			The number 15 is a term, since 15-2^(2^0) and 15-2^(2^1) are primes 13 and 11.

Cf. A039669, A129613.

q[k_] := Module[{m = 0}, While[2^(2^m) < k && PrimeQ[k - 2^(2^m)], m++]; 2^(2^m) >= k]; Select[Range[4, 15000], q] (* Amiram Eldar, Feb 22 2024 *)

More terms from Amiram Eldar, Feb 22 2024

A371303 Numbers k > 4 such that both k - 2^(2^m) and k + 2^(2^m) are prime for every natural m > 0 with 2^(2^m) < k.

Original entry on oeis.org

7, 9, 15, 27, 57, 63, 195, 267, 363, 405, 483, 603, 1197, 1233, 1443, 1737, 2715, 4257, 5403, 6117, 21855, 22287, 26817, 40755, 63777, 260007, 617253, 986733, 1151655, 1167837, 1174503, 1199373, 1331595, 3233307, 4128873, 4138707, 4609527, 5938107, 7203945, 7605213, 8379405, 8587545, 9596223

Offset: 1

Thomas Ordowski, Mar 18 2024

nonn

It seems that there are infinitely many such numbers.
If k > 7 is such a number, then it is odd and divisible by 3.
Conjecture: numbers k > 2 such that both k - 2^(2^m) and k + 2^(2^m) are prime for every integer m >= 0 with 2^(2^m) < k are only 9, 15, and 195 (Amiram Eldar checked that there are no more terms k < 10^8).

Cf. A039669, A129613, A370523.

q[k_] := Module[{m = 1}, While[2^(2^m) < k && PrimeQ[k - 2^(2^m)] && PrimeQ[k + 2^(2^m)], m++]; 2^(2^m) > k]; Select[Range[5, 10^6, 2], q] (* Amiram Eldar, Mar 18 2024 *)

More terms from Amiram Eldar, Mar 18 2024

cp's OEIS Frontend

A100348 Numbers n such that n-4^k is a prime for all k > 0 with 4^k < n.

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A108957 Values of n such that n - 2^k is deficient for all 1 <= 2^k < n.

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A057652 Numbers n such that n-2^k is a lucky number for all k such that 1 < 2^k < n.

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A091155 Numbers m such that m - 2^k is squarefree for all 1 <= 2^k < m.

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A240842 Numbers n such that n - 2*k^2 is a prime for all k > 0 with k^2 < n/2.

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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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A370523 Numbers k > 2 such that all positive values of k - 2^(2^m) are prime, with integer m >= 0.

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A371303 Numbers k > 4 such that both k - 2^(2^m) and k + 2^(2^m) are prime for every natural m > 0 with 2^(2^m) < k.

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