A057733 -id:A057733 - cp's OEIS frontend

A197918 Pythagorean primes p such that for all primes q < p, p XOR q is not equal to p - q.

2, 5, 17, 41, 73, 97, 137, 193, 257, 521, 577, 641, 1033, 1153, 2081, 2113, 4129, 7681, 8353, 8737, 9281, 10369, 10753, 12289, 16417, 17921, 18433, 21569, 25601, 32801, 32833, 36353, 37889, 38921, 39041, 40961, 50177, 53377, 65537, 131617, 133121, 136193, 139273, 139297, 139393, 147457, 163841

Offset: 1

Brad Clardy, Oct 24 2011

It is conjectured that with the exception of the first three terms (2,5,17) all of the terms are a subset of all primes p such that p XOR 22 = p + 22.

If the inequality in the definition is replaced with equality the result are the Mersenne primes A000668, which is equivalent to for all primes q

This sequence is apparently a subset of A081091 Primes of the form 2^i + 2^j + 1, i>j>0, with the added conditions that j <> 1 or 2, and if j can be written as 2n then i cannot be 2n+1. This removes A123250 Primes of form 2^n + 5 (or 2^n + 2^2 +1) for n>0, primes from A140660 3*4^n + 1 (or 2^(2n+1) + 2^(2n) + 1) for n>0, and A057733 Primes of form 2^n + 3 (2^n + 2^1 + 1) for n>1.

			5 is a Pythagorean prime (1^2 + 2^2) and a member since ((5 XOR 2) <> (5 - 2)) and ((5 XOR 3) <> (5 - 3)).
13 is a Pythagorean prime (2^2 + 3^2) however it is not a member because 5, a prime less than 13, (13 XOR 5) = (13 - 5).

Charles R Greathouse IV, Table of n, a(n) for n = 1..303

Cf. A000668, A057733, A081091, A123250, A140660, A214643.

XOR := func;
i:=0; k:=0; pn:=0;
for n:= 5 to 10000 by 4 do
       if IsPrime(n)  then  pn:=n;  end if;
       if (pn eq n) then k:=0;
           for j in [2 .. n-2] do
                if IsPrime(j)  then pj:=j;
                     if (XOR(pn,pj) ne pn-pj) then i+:=1;
                         else k+:=1;
                     end if;
                end if;
           end for;
       end if;
       if ((i ne 0) and (k eq 0))  then pn; end if;
       i:=0; k:=0;
end for;

forprime(p=2,1e6,if(p%4-3==0,next);forprime(q=2,p-1,if(bitxor(p,q)==p-q, next(2)));print1(p", ")) \\ Charles R Greathouse IV, Jul 31 2012

A228028 Primes of the form 5^n + 4.

Original entry on oeis.org

5, 29, 15629, 9765629

Offset: 1

Vincenzo Librandi, Aug 11 2013

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..8

Cf. A124621 (associated n).

Cf. Primes of the form k^n + h: A092506 (k=2, h=1), A057733 (k=2, h=3), A123250 (k=2, h=5), A104066 (k=2, h=7), A104070 (k=2, h=9), A057735 (k=3, h=2), A102903 (k=3, h=4), A102870 (k=3, h=8), A102907 (k=3, h=10), A290200 (k=4, h=1), A228027 (k=4, h=9), A182330 (k=5, h=2), this sequence (k=5, h=4), A228029 (k=5, h=6), A102910 (k=5, h=8), A182331 (k=6, h=1), A104118 (k=6, h=5), A104115 (k=6, h=7), A104065 (k=7, h=4), A228030 (k=7, h=6), A228031 (k=7, h=10), A228032 (k=8, h=3), A228033 (k=8, h=5), A144360 (k=8, h=7), A145440 (k=8, h=9), A228034 (k=9, h=2), A159352 (k=10, h=3), A159031 (k=10, h=7).

[a: n in [0..200] | IsPrime(a) where a is  5^n+4];

Select[Table[5^n + 4, {n, 0, 200}], PrimeQ]

Corrected cross-references - Robert Price, Aug 01 2017

A286843 Even k such that k - A001065(k) = 2^m (for some m > 0).

Original entry on oeis.org

10, 14, 22, 38, 44, 92, 110, 130, 134, 136, 152, 170, 184, 248, 250, 262, 284, 376, 410, 442, 632, 730, 752, 884, 988, 1012, 1052, 1276, 1292, 1370, 1628, 2144, 2168, 2272, 2332, 2528, 3068, 4064, 4124, 5210, 6112, 6364, 6556, 7372, 8198, 8312, 8384, 8648

Offset: 1

XU Pingya, Aug 01 2017

nonn

Tianxin Cai conjectured that the sequence is infinite.
When p is prime, A001065(2p) = 1 + 2 + p = 3 + p. 2p - A001065(2p) = 2^m iff 2^m + 3 = p. Therefore if A057733 is infinite, Cai's conjecture is correct.
In general, for j = 2, 3, ..., if the number of primes of the form 2^m + 2^j - 1 is infinite, then Cai's conjecture is correct.
When 2^p - 1 is prime, let k = 2^p*(2^p - 1). A001065(k) = 1 + 2 + 2^2 + ... + 2^p + 2^p - 1 + 2(2^p - 1) + 2^2*(2^p - 1) + ... + 2^(p - 1)*(2^p - 1) = 2^(p + 1) - 1 + (2^p - 1)^2 = 2^(2p). k - A001065(k) = -2^p. Therefore if the number of Mersenne primes (A000668) is infinite, then there are infinitely many even k such that k - A001065(k) = -2^p.

			10 is a term as 10 - A001065(10) = 10 - 8 = 2.
22 is a term as 22 - A001065(22) = 22 - 14 = 2^3.

Amiram Eldar, Table of n, a(n) for n = 1..893 (terms below 10^11)

Cf. A000079, A001065, A005843, A057733.

Select[Table[2n, {n, 1, 5000}], DivisorSigma[1, 2# - DivisorSigma[1, #]] + 1 == 2(2# - DivisorSigma[1, #]) > 2 &] (* or *)
Select[2 Range[5000], IntegerQ@ Log2[2 # - DivisorSigma[1,#]] && !IntegerQ@ Log2@ # &] (* Giovanni Resta, Aug 07 2017 *)

ispower2(n) = n > 1 && n >> valuation(n, 2) == 1;
is(n) = !(n%2) && ispower2(2*n - sigma(n)); \\ Amiram Eldar, Mar 22 2024

A156974 Primes of the form 2^k + 29.

Original entry on oeis.org

31, 37, 61, 157, 541, 8221, 32797, 131101, 8388637, 134217757, 8589934621, 137438953501, 8796093022237, 9223372036854775837, 590295810358705651741, 9444732965739290427421, 604462909807314587353117, 618970019642690137449562141, 166153499473114484112975882535043101, 170141183460469231731687303715884105757, 883423532389192164791648750371459257913741948437809479060803100646309917

Offset: 1

Edwin Dyke (ed.dyke(AT)btinternet.com), Feb 19 2009

nonn

Cf. A000040, A156982.

Cf. A057733 (2^k+3), A123250 (2^k+5), A104066 (2^k+7), A156940 (2^k+11).

[a: n in [0..300] | IsPrime(a) where a is 2^n+29 ]; // Vincenzo Librandi, Nov 27 2010

a := proc (n) if isprime(2^n+29) = true then 2^n+29 else end if end proc: seq(a(n), n = 1 .. 110); # Emeric Deutsch, Mar 14 2009

Delete[Union[Table[If[PrimeQ[2^n + 29], 2^n + 29, 0], {n, 1, 500}]],1]

a(n) = 2^A156982(n) + 29. - Elmo R. Oliveira, Nov 08 2023

More terms from Emeric Deutsch, Mar 14 2009

More terms from Vincenzo Librandi, Nov 27 2010

A193109 Least k such that 2^x + k produces primes for x=1..n and composite for x=n+1.

Original entry on oeis.org

0, 1, 9, 3, 225, 15, 65835, 1605, 19425, 2397347205, 153535525935

Offset: 1

Arkadiusz Wesolowski, Jul 21 2011

All terms except the first four are congruent to 15 mod 30.
a(10) was found in 2005 by T. D. Noe and a(11) was found in the same year by Don Reble.
Other known values: a(13) = 29503289812425.
a(12) > 10^13. - Tyler Busby, Feb 19 2023

Another version of A110096.

Cf. A008597, A057733, A092506, A104070, A144487.

Table[k = 0; While[i = 1; While[i <= n && PrimeQ[2^i + k], i++]; i <= n || PrimeQ[2^i + k], k++]; k, {n, 9}] (* T. D. Noe, Jul 21 2011 *)

is(k, n) = for(x=1, n, if(!isprime(k+2^x), return(0))); 1;
a(n) = {my(s=2); forprime(p=3, n, if(znorder(Mod(2, p))==(p-1), s*=p)); forstep(k=s*(n>1)/2, oo, s, if(is(k, n) && !isprime(k+2^(n+1)), return(k))); } \\ Jinyuan Wang, Jul 30 2020

A229222 Smallest prime p such that p contains a digit larger than 1 and the sum of the n-th powers of the decimal digits of p is a prime number.

Original entry on oeis.org

2, 23, 113, 23, 191, 223, 191, 41, 223, 113, 157, 191, 137, 113, 113, 43, 137, 191, 179, 337, 577, 223, 227, 113, 263, 113, 199, 229, 263, 199, 467, 89, 223, 179, 223, 113, 443, 683, 1279, 337, 661, 463, 827, 2281, 577, 223, 223, 661, 137, 229, 11399, 461, 577

Offset: 1

Michel Lagneau, Sep 16 2013

We impose the condition that p is not in A020449 in order to avoid trivial sequences with infinite repetitions with the numbers 11 if p>1, or 101 if p>11, or 101111 if p > 101, ... for example if p > 1 the sequence is {2, 11, 11, 11, ...}, if p > 11 the sequence is {23, 23, 101, 23, 101, 101, 41, 101, 101, 101, 101, 101, ...}.
a(n) is an unification of a family of sequences mentioned hereafter:
A082101: primes of the form 2^n+3^n => 23 is in the sequence;
A057735: primes of the form 3^n+2 => 113 is in the sequence;
A153133: primes of the form 2^n+3^(n-1) => 223 is in the sequence;
A228034: primes of the form 9^n+2 => 191 is in the sequence;
A057733: primes of the form 2^n+3 => 2111 is in the sequence;
A228026: primes of the form 4^n+3 => 4111 is in the sequence;
A228034: primes of the form 9^n+2 => 191 is in the sequence;
A182330: primes of the form 5^n+2 => 151 is in the sequence;
A111974: primes of the form 2*3^n+1 => 313 is in the sequence;
A102903: primes of the form 3^n+4 => 11113 is in the sequence.
In this sequence, we observe repetitions of numbers such that 23, 113, 223, 191, 199, 223,... and this problem is very difficult, because it is probable that there exists both finite and infinite repetitions according to the numbers: for example, if we consider the number 23 of this sequence, it is probable that the number of element "23" is finite (see the comment in A082101 for the primes of form 2^k + 3^k). But, if we consider the number 113 of this sequence, is the number of the elements "113" infinite ? (see A057735 with the primes of the form 2+3^n). We observe that a(n) = 113 for n = 3, 14, 15, 24, 26,..., 123, 126, 139,..., 386, 391, 494, ....

			a(3) = 113 because 1^3+1^3+3^3 = 29 is prime.

Michel Lagneau, Table of n, a(n) for n = 1..500

Cf. A020449.

with(numtheory) :lst:={11, 101, 101111, 10011101, 10101101, 10110011, 10111001, 11000111, 11100101, 11110111, 11111101 }:for n from 1 to 300 do :ii:=0:for k from 1 to 10^8 while(ii=0) do:x:=convert(k,base,10):n1:=nops(x):it:=0:jj:=0:s:= sum('x[i]^n', 'i'=1..n1):lst1:={k} intersect lst:if type(k,prime)=true and type(s,prime)=true and (lst1<>{k}) then ii:=1: printf(`%d, `,k):else fi:od:od:

Table[p = 2; While[d = IntegerDigits[p]; Union[d][[-1]] < 2 || ! PrimeQ[Total[d^n]],  p = NextPrime[p]]; p, {n, 60}]

a(n)=forprime(p=2,,my(d=digits(p)); if(vecmax(d)>1 && isprime(sum(i=1,#d,d[i]^n)), return(p))) \\ Charles R Greathouse IV, Sep 19 2013

A267413 Dropping any binary digit gives a prime number.

Original entry on oeis.org

6, 7, 11, 15, 35, 39, 63, 135, 255, 999, 2175, 8223, 16383, 57735, 131075, 131079, 262143, 524295, 1048575, 536870919, 1073735679, 2147483655, 4294967295, 17179770879, 4260641103903, 4611686018427387903, 4720069647059686260735, 1237940039285380274899124223

Offset: 1

Stanislav Sykora, Jan 14 2016

This is the binary analog of A034895. The sequence contains mostly numbers with very few binary digit runs (BDR, A005811). Those with one BDR are of the type 2^k-1, such that 2^(k-1)-1 is a Mersenne prime (A000668). Vice versa, if M is any Mersenne prime, then 2*M+1 is a term. The number 6 is the only term with an even number of BDRs. There are many terms with 3 BDRs. The first term with 5 BDRs is 57735. The next terms with at least 5 BDRs (if they exist at all) are larger than 10^10. So far, I could test that a(24) > 10^10.
From Robert Israel, Jan 14 2016: (Start)
For n >= 2, a(n) == 3 (mod 4).
2^k+3 is in the sequence if 2^(k-1)+1 and 2^(k-1)+3 are primes, i.e., 2^(k-1)+1 is in the intersection of A019434 and A001359.  The only known terms of the sequence in this class are 7, 11, 35, 131075.
2^k+7 is in the sequence if 2^(k-1)+3 and 2^(k-1)+7 are primes: thus 2^(k-1)+3 is in A057733 and 2^(k-1)+7 is in A104066.  Terms of the sequence in this class include 15, 39, 135, 131079, 524295, 536870919, 2147483655 (but no more for k <= 2000).
(End)
a(25) > 2^38. - Giovanni Resta, Apr 10 2016
For n > 1, a(n) = 2p+1 for some prime p. - Chai Wah Wu, Aug 27 2021
From Bert Dobbelaere, Aug 07 2023: (Start)
There are no more terms with an odd number of binary digits: from any number having an odd number of binary digits, one can always drop a digit and obtain a multiple of 3. Numbers of the form 2^k+3 (k even and k > 2) cannot be terms because 2^(k-1)+1 is a multiple of 3.
(End)

			Decimal and binary forms of the known terms:
   1           6                                110
   2           7                                111
   3          11                               1011
   4          15                               1111
   5          35                             100011
   6          39                             100111
   7          63                             111111
   8         135                           10000111
   9         255                           11111111
  10         999                         1111100111
  11        2175                       100001111111
  12        8223                     10000000011111
  13       16383                     11111111111111
  14       57735                   1110000110000111 <--- (a binary palindrome
  15      131075                 100000000000000011       with 5 digit runs)
  16      131079                 100000000000000111
  17      262143                 111111111111111111
  18      524295               10000000000000000111
  19     1048575               11111111111111111111
  20   536870919     100000000000000000000000000111
  21  1073735679     111111111111111110011111111111
  22  2147483655   10000000000000000000000000000111
  23  4294967295   11111111111111111111111111111111
  24 17179770879 1111111111111111100111111111111111

Cf. A000668, A001359, A005811, A019434, A034895 (base 10), A051362, A057733, A104066.

filter:= proc(n) local B,k,y;
   if not isprime(floor(n/2)) then return false fi;
   B:= convert(n,base,2);
   for k from 2 to nops(B) do
     if B[k] <> B[k-1] then
       y:= n mod 2^(k-1);
       if not isprime((y+n-B[k]*2^(k-1))/2) then return false fi
     fi
   od;
   true
end proc:
select(filter, [6, seq(i,i=7..10^6,4)]); # Robert Israel, Jan 14 2016

Select[Range[2^20], AllTrue[Function[w, Map[FromDigits[#, 2] &@ Drop[w, {#}] &, Range@ Length@ w]]@ IntegerDigits[#, 2], PrimeQ] &] (* Michael De Vlieger, Jan 16 2016, Version 10 *)

DroppingAnyDigitGivesAPrime(N,b) = {
\\ Property-testing function; returns 1 if true for N, 0 otherwise
\\ Works with any base b. Here used with b=2.
  my(k=b,m); if(N=(k\b), m=(N\k)*(k\b)+(N%(k\b));
    if ((m<2)||(!isprime(m)),return(0)); k*=b);
  return(1);
}

from sympy import isprime
def ok(n):
    if n < 7 or n%4 != 3: return n == 6
    b = bin(n)[2:]
    return all(isprime(int(b[:i]+b[i+1:], 2)) for i in range(len(b)))
print(list(filter(ok, range(2, 2**20)))) # Michael S. Branicky, Jun 07 2021

a(24) from Giovanni Resta, Apr 10 2016

a(25)-a(28) from Bert Dobbelaere, Aug 07 2023

A172183 a(n) is the smallest prime of the form p^q+n, where p and q are prime, or zero if no such prime exists.

Original entry on oeis.org

5, 11, 7, 13, 13, 31, 11, 17, 13, 19, 19, 37, 17, 23, 19, 41, 8209, 43, 23, 29, 29, 31, 31, 73, 29, 53, 31, 37, 37, 79, 0, 41, 37, 43, 43, 61, 41, 47, 43, 67, 73, 67, 47, 53, 53, 71, 79, 73, 53, 59, 59, 61, 61, 79, 59, 83, 61, 67, 67, 109, 0, 71, 67, 73, 73, 191, 71, 193, 73, 79

Offset: 1

Cheng Zhang (cz1(AT)rice.edu), Jan 28 2010, Mar 02 2010

nonn

If n mod 6 = 1, both p and q must be 2, and a(n)=0 if n + 4 is not a prime. The values of a(n) for n=257,297,353,383,557 are either greater than 176 000 or 0. Several large entries: a(87) = 2^25633 + 87, a(717) = 2^3217 + 717, a(773) = 2^2539 + 773, a(927) = 2^1117 + 927.

			a(1)=5 because 5=2^2+1 is the smallest prime of the form p^q+1. a(2)=11 because 11=3^2+2. a(3)=7, because 7=2^2+3. a(17)=8209, because 8209=2^13+17. a(31)=0, because p^q+31 cannot be a prime.

Cf. A123318, A056208, A056206, A057733, A123250, A104066, A156940, A104067, A156973

For[l = {}; n = 1, n <= 70, n++, found = False; If[Mod[n, 2] == 0, For[rm = Infinity; i = 1, i < 100, i++, For[j = 1, j < 100, j++, p = Prime[i]; q = Prime[j]; r = p^q + n; If[r >= rm, Break[], If[PrimeQ[r], rm = r; found = True]]; ]; ], (* if n is odd, r=2^q+n *) If[Mod[n, 6] == 1, r = 4 + n; If[PrimeQ[r], found = True], For[j = 1, j < 1000, j++, q = Prime[j]; r = 2^q + n; If[PrimeQ[r], found = True; rm = r; Break[]]; ]; ]; ]; If[ ! found, rm = 0]; l = Append[l, rm]; ]; l

A238739 Numbers n such that 2^n + 3 and 3*2^n + 1 are both prime.

Original entry on oeis.org

1, 2, 6, 12, 18, 30

Offset: 1

Juri-Stepan Gerasimov, Mar 04 2014

Intersection of A057732 and A002253. - Joerg Arndt, Mar 04 2014
By checking primality of 2^n+3 for values n in A002253, it follows a(7) > 7033641. - Giovanni Resta, Mar 08 2014
Exponents of second Fermat prime pairs. - Juri-Stepan Gerasimov, Mar 08 2014
From Juri-Stepan Gerasimov, Mar 04 2014: (Start)
If prime pair {2^n + (2k+1), (2k+1)*2^n + 1} is called a Fermat prime pair, then numbers n such that 2^n + (2k + 1) and (2k + 1)*2^n + 1 are both prime:
for k = 0: 0, 1, 2, 4, 8, 16, ...   the exponents first Fermat prime pairs;
for k = 1: 1, 2, 6, 12, 18, 30, ... the exponents second Fermat prime pairs;
for k = 2: 1, 3, ...                the exponents third Fermat prime pairs;
for k = 3: 2, 4, 6, 20, 174, ...    the exponents fourth Fermat prime pairs;
for k = 4: 1, 2, 3, 6, 7, ...       the exponents fifth Fermat prime pairs;
for k = 5: 1, 3, 5, 7, ...          the exponents sixth Fermat prime pairs;
for k = 6: 2, 8, 20, ...            the exponents seventh Fermat prime pairs;
for k = 7: 1, 2, 4, 10, 12, ...     the exponents eighth Fermat prime pairs;
for k = 8:
for k = 9: 6, ...                   the exponents tenth Fermat prime pairs;
for k = 10: 1, 4, 5, 7, 16, ...     the exponents eleventh Fermat prime pairs;
for k = 11:
for k = 12: 2, 4, 6, 10, 20, 22, ...the exponents thirteenth Fermat prime pairs;
for k = 13: 2, 4, 16, 40, 44, ...   the exponents fourteenth Fermat prime pairs;
for k = 14: 1, 3, 5, 27, 43, ...    the exponents fifteenth Fermat prime pairs.
Semiprimes of the form (2^m+2k+1)*((2k+1)*2^m+1): 4, 9, 25, 35, 77, 91, 209, 289, 319, 481, 527, 533, 901, 989, ...
(End)

			a(1) = 1 because 2^1 + 3 = 5 and 3*2^1 + 1 = 7 are both prime,
a(2) = 2 because 2^2 + 3 = 7 and 3^2^2 + 1 = 13 are both prime,
a(3) = 6 because 2^6 + 3 = 67 and 3*2^6 + 1 = 193 are both prime.

Cf. A002253, A019434, A039687, A057732, A057733, A238554, A239038, A267984.

[n: n in [0..30] | IsPrime(2^n+3) and IsPrime(3*2^n+1)]; // Arkadiusz Wesolowski, Jan 23 2016

Select[Range[30],AllTrue[{2^#+3,3*2^#+1},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 08 2015 *)

isok(n) = isprime(2^n + 3) && isprime(3*2^n + 1); \\ Michel Marcus, Mar 04 2014

Previous Showing 21-30 of 33 results. Next

cp's OEIS Frontend

A156983 Primes of the form 2^k + 21.

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A197918 Pythagorean primes p such that for all primes q < p, p XOR q is not equal to p - q.

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A228028 Primes of the form 5^n + 4.

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A286843 Even k such that k - A001065(k) = 2^m (for some m > 0).

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A156974 Primes of the form 2^k + 29.

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A193109 Least k such that 2^x + k produces primes for x=1..n and composite for x=n+1.

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A229222 Smallest prime p such that p contains a digit larger than 1 and the sum of the n-th powers of the decimal digits of p is a prime number.

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A267413 Dropping any binary digit gives a prime number.

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