A077375 -id:A077375 - cp's OEIS frontend

A078583 Numbers k such that 2^k - prime(k) is prime.

3, 11, 19, 24, 26, 61, 96, 175, 189, 312, 483, 741, 4741, 7082, 8421, 10695, 14802, 18824, 18892, 20655, 21653, 39937, 84160

Offset: 1

Robert G. Wilson v, Nov 30 2002

a(23) is greater than 50000 - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Dec 05 2004

a(24) is greater than 200000. - Michael S. Branicky, Jan 03 2025

Corresponding prime(k) are in A078686.

Cf. A077375.

Do[ If[ PrimeQ[2^n - Prime[n]], Print[n]], {n, 1, 10^5}]

a(14)-a(22) from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Dec 05 2004

a(23) from Michael S. Branicky, Dec 31 2024

A242944 Primes prime(k) such that 2^k + prime(k) is also prime.

Original entry on oeis.org

3, 5, 7, 11, 37, 41, 43, 83, 269, 577, 1429, 1433, 2063, 2549, 8353, 10639, 15299, 16927, 18637, 20201, 24007, 30097, 34039, 41777, 146609, 394249, 839203, 2955319

Offset: 1

Robert G. Wilson v, Jun 20 2014

If instead we ask for odd primes, and therefore the index is one less than that for all primes, the sequence would begin: 3, 29, 89, 251, 659, 937, 1307, 1453, 8179, 9391, 12097, 28499, 83969, 101209, 120739, ..., .

If we count 1 amongst the primes (A008578), then the sequence would begin: 1, 3, 31, 71, 97, 107, 277, 307, 641, 907, 967, 1009, 1447, 3463, 3527, 7757, 8167, ..., .

Corresponding k are in A077375.

Cf. A078583, A078686, A056206, A243428.

p = 2; lst = {}; While[p < 760001, If[ PrimeQ[p + 2^PrimePi@ p], AppendTo[ lst, p]; Print@ p]; p = NextPrime@ p; c++]; lst
Select[Table[{n,Prime[n]},{n,3000}],PrimeQ[#[[2]]+2^#[[1]]]&][[;;,2]] (* The program generates the first 21 terms of the sequence. *) (* Harvey P. Dale, Mar 04 2024 *)

a(27) from Michael S. Branicky, May 29 2025 using A077375.

a(28) from Michael S. Branicky, Jun 01 2025

A077371 Fibonacci numbers whose internal digits form a Fibonacci number. Equivalently, Fibonacci numbers from which deleting the MSD and LSD leaves a Fibonacci number.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 233, 610, 987

Offset: 1

Amarnath Murthy, Nov 06 2002

Conjecture: The sequence is finite.
No more terms < 10^6. - Lars Blomberg, May 20 2015
From Manfred Scheucher, Jun 02 2015 (Start)
No more terms < 10^10000.
When considering binary representations, the sequence would be 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 144, and no further terms < 2^150000 (about 10^44095).
When considering k-ary representations with k=2..100, each of the sequences has some small terms in the beginning (as in the 10-ary case) and no further terms <10^1000.
The sequence seems to be finite for any base, not just for base 10.
Another observation: When considering k-ary representations with k=55,144,377,... (Fibonacci numbers with even index, A001906), the number of "initial terms" (terms <10^1000) increases very fast.
(End)

Manfred Scheucher, Sage Script

Cf. A077372, A077373, A077374, A077375.

A228021 Prime(k) such that 2^(k - 1) + prime(k) is also prime.

Original entry on oeis.org

2, 3, 29, 89, 251, 659, 937, 1307, 1453, 8179, 9391, 12097, 28499, 83969, 101209, 120739, 730993

Offset: 1

Juri-Stepan Gerasimov, Aug 03 2013

The primes indices k are 1, 2, 10, 24, 54, 120, 159, 214, 231, 1027, 1161, 1447, 3100, 8188, 9695, 11363 ...
The corresponding primes 2^(k - 1) + prime(k) are 3, 5, 541, 8388697,...
a(18) > 2*10^6. - Michael S. Branicky, Apr 16 2025

			29 is in the sequence because 29 = prime(10) and 2^(10 - 1) + 29 = 512 + 29 = 541 is prime.

Cf. A077375, A227126, A242944, A244913, A244916.

for i from 1 do
    p := ithprime(i) ;
    if isprime(p+2^(i-1)) then
       printf("%d,\n",p) ;
    end if;
end do: # R. J. Mathar, Jul 12 2014

p = 2; lst = {}; While[p < 730001, If[ PrimeQ[ 2^(PrimePi@ p-1) + p], AppendTo[lst, p]; Print@ p]; p = NextPrime@ p]; lst (* Robert G. Wilson v, Jul 09 2014 *)

lista(nn) = {ip = 1; forprime(p=2, nn, if (isprime(2^(ip-1)+p), print1(p, ", ")); ip++;);} \\ Michel Marcus, Jul 12 2014

a(3) - a(9) from _Olivier Gérard_, Aug 01 2013
a(10) - a(15) from Robert G. Wilson v, Aug 01 2013
a(16) from Robert G. Wilson v, Jul 09 2014
a(17) from Michael S. Branicky, Apr 14 2025

A236344 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2^m + prime(m) prime}|, where phi(.) is Euler's totient function.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 3, 4, 2, 4, 3, 5, 2, 3, 4, 3, 3, 4, 6, 5, 6, 6, 7, 7, 5, 4, 6, 6, 5, 7, 5, 3, 3, 3, 7, 4, 5, 5, 8, 4, 6, 5, 5, 5, 6, 4, 5, 4, 5, 4, 3, 4, 5, 6, 3, 6, 9, 6, 9, 8, 13, 5, 11, 5, 6, 7, 11, 4, 9, 9, 5, 6, 6, 11, 7, 8, 9, 9, 4

Offset: 1

Zhi-Wei Sun, Jan 22 2014

nonn

a(n) = 0 for n = 1, ..., 15, 481, 564, 66641, 70965, 72631, .... If a(n) > 0 infinitely often, then there are infinitely many positive integers m with 2^m + prime(m) prime.

			a(26) = 1 since phi(5)/2 + phi(21)/12 = 2 + 1 = 3 with 2^3 + prime(3) = 8 + 5 = 13 prime.
a(5907) = 1 since phi(3944)/2 + phi(5907-3944)/12 = 896 + 150 = 1046 with 2^(1046) + prime(1046) = 2^(1046) + 8353 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000040, A000079, A077375, A236241, A236256.

p[n_]:=IntegerQ[n]&&PrimeQ[2^n+Prime[n]]
f[n_,k_]:=EulerPhi[k]/2+EulerPhi[n-k]/12
a[n_]:=Sum[If[p[f[n,k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,100}]

A130640 Numbers k such that either 2^k + prime(k) or 2^k - prime(k) is prime.

Original entry on oeis.org

2, 3, 4, 5, 11, 12, 13, 14, 19, 23, 24, 26, 57, 61, 96, 106, 175, 189, 226, 227, 311, 312, 373, 483, 741, 1046, 1298, 1787, 1952, 2130, 2285, 2670, 3254, 3642, 4369, 4741, 7082, 8421, 10695, 13559, 14802, 18824, 18892, 20655, 21653, 33418, 39937, 66830, 84160

Offset: 1

J. M. Bergot, Jun 19 2007

nonn

			2^5 + p(5) = 32 + 11 = 43; 43 is prime, hence 5 is in the sequence.
2^11 - p(11) = 2048 - 31 = 2017; 2017 is prime, therefore 11 is in the sequence.

Union of A077375 and A078583.

Select[Range[2000], PrimeQ[2^# - Prime[ # ]] || PrimeQ[2^# + Prime[ # ]] &]

Edited and extended by Stefan Steinerberger, Jun 24 2007

a(45)-a(49) from Michael S. Branicky, May 30 2025

cp's OEIS Frontend

A078583 Numbers k such that 2^k - prime(k) is prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A242944 Primes prime(k) such that 2^k + prime(k) is also prime.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A077371 Fibonacci numbers whose internal digits form a Fibonacci number. Equivalently, Fibonacci numbers from which deleting the MSD and LSD leaves a Fibonacci number.

Views

Author

Keywords

Comments

Links

Crossrefs

A228021 Prime(k) such that 2^(k - 1) + prime(k) is also prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A236344 a(n) = |{0 < k < n: m = phi(k)/2 + phi(n-k)/12 is an integer with 2^m + prime(m) prime}|, where phi(.) is Euler's totient function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A130640 Numbers k such that either 2^k + prime(k) or 2^k - prime(k) is prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Extensions