A156695 -id:A156695 - cp's OEIS frontend

A157237 Number of ways to write the n-th positive odd integer in the form p+2^x+11*2^y with p a prime congruent to 1 mod 6 and x,y positive integers.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 2, 2, 0, 1, 1, 1, 2, 2, 2, 4, 1, 2, 5, 2, 1, 3, 1, 1, 2, 1, 3, 3, 1, 3, 5, 2, 2, 5, 4, 0, 5, 4, 2, 4, 3, 3, 4, 3, 3

Offset: 1

Zhi-Wei Sun, Feb 25 2009

nonn

On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if n<16 or n=18, 21, 24, 51, 84, 1011, 59586; in other words, except for 35, 41, 47, 101, 167, 2021, 119171, any odd integer greater than 30 can be written as the sum of a prime congruent to 1 mod 6, a positive power of 2 and eleven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7, and Qing-Hu Hou continued the verification for odd integers below 1.5*10^8 (on Sun's request). Compare the conjecture with Crocker's result that there are infinitely many positive odd integers not of the form p+2^x+2^y with p an odd prime and x,y positive integers.

			For n=19 the a(19)=2 solutions are 2*19-1=7+2^3+2*11=13+2+2*11.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.

Zhi-Wei Sun, Table of n, a(n) for n=1..200000
Zhi-Wei Sun, A webpage: Mixed Sums of Primes and Other Terms, 2009.
Zhi-Wei Sun, A project for the form p+2^x+k*2^y with k=3,5,...,61
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075

A000040, A000079, A157218, A157225, A155860, A155904, A156695, A154257, A154285, A155114, A154536

PQ[x_]:=x>1&&Mod[x,6]==1&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-11*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/11]},{y,1,Log[2,Max[2,2n-1-11*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]

a(n)=|{: p+2^x+11*2^y=2n-1 with p a prime congruent to 1 mod 6 and x,y positive integers}|

A188903 a(n) is the least power of 2 such that 2n+1 - a(n) is prime, or 0 if no such prime exists.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 16, 2, 2, 4, 8, 2, 4, 2, 2, 4, 2, 4, 16, 2, 4, 16, 2, 2, 4, 8, 2, 4, 2, 2, 4, 8, 2, 4, 2, 4, 16, 2, 4, 16, 8, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 16, 8, 16, 16, 0, 2, 4, 2, 4, 64, 2, 2, 4, 8, 8, 0, 2, 2, 4, 8, 2, 4, 32, 2, 4, 2, 4, 16, 2, 4, 16, 2, 2, 4, 8, 8, 64, 2, 2, 4, 2, 2, 4, 8, 8, 16, 32, 2, 4, 128, 8, 64, 32, 2, 4, 2, 2, 4, 2, 4, 16, 2, 2, 4, 8, 8, 0, 2

Offset: 0

Michel Lagneau, Apr 13 2011

nonn

The second Polignac's Conjecture states that every odd positive integer is the sum of a prime and a power of two. This conjecture was proved false, and the smallest counterexample is 127 because subtracting powers of 2 from 127 produces the composite numbers 126, 123, 119, 111, 95, and 63.

The sequence A006285 gives the odd numbers for which the conjecture fails. Hence, a(n) = 0 for n = (A006285(k)-1)/2 = {0, 63, 74, 125, 165, 168, 186, ...}.

			a(1) = 1 because 2*1 + 1 = 3 = 1 + 2 ;
a(2) = 2 because 2*2 + 1 = 5 = 2 + 3 ;
a(3) = 2 because 2*3 + 1 = 7 = 2 + 5 ;
a(63) = 0 ; a(74) = 0 ; a(125) = 0, ....

David Wells, Prime Numbers: The Most Mysterious Figures In Math, John Wiley & Sons, 2005, p. 175-176.

Carlos Rivera, Puzzle 219, Polignac numbers, The Prime Puzzles and Problems Connection.

Cf. A065381 (primes not of the form p + 2^k, p prime and k >= 0), A156695.

with(numtheory):for n from 1 to 126 do:x:=2*n+1:id:=0:for k from 0 to 50 while(id=0)
  do: for q from 1 to 100 while(id=0) do: p:=ithprime(q): y:=2^k+p:if y=x then
  id:=1:printf(`%d, `,2^k):else fi:od:od:if id=0 then printf(`%d, `,0):else fi:od:

Table[d = 2*n + 1; k = 1; While[k < d && ! PrimeQ[d - k], k = 2*k]; If[k < d, k, 0], {n, 0, 126}]

def A188903(n):
    return next((2**k for k in (0..floor(log(2*n+1,2))) if is_prime(2*n+1-2**k)), 0)
# D. S. McNeil, Apr 14 2011

A232565 a(n) is the smallest k such that 2^(2^n) - 2^k - 1 is prime, or -1 if no such k exists.

Original entry on oeis.org

0, 1, 2, 4, 2, 8, 18, 76, 32, 151, 692, 592, 154, 580, 27365, 11267

Offset: 1

Charles R Greathouse IV, Nov 26 2013

Crocker showed that 2^(2^n) - 1 - 2^a - 2^b is not prime (with n > 2) if a and b are distinct. This sequence demonstrates that the theorem is sharp in the sense that distinctness is required.

If n > 2, then the (largest) prime P(n) = 2^(2^n)-2^a(n)-1 is a de Polignac number (A065381); i.e., P(n)-2^m is not prime. It seems that if n > 6, then |P(n)-2^m| is composite for every natural m and P(n)*2^m-1 is composite (by the dual Riesel conjecture). So if n > 6, then the prime P(n) may be a Riesel number (A182296). For example, the prime P(7) = 2^(2^7)-(2^18+1) is the first candidate (note that 2^18+1 is the smallest de Polignac number of form 2^k+1). Also, by Crocker's theorem, the smallest number of form 2^(2^n)-2^m-1, namely 2^(2^n-1)-1 is a de Polignac number (A006285) and for n > 6 may be a dual Riesel number (A101036). For example, the double Mersenne prime 2^(2^7-1)-1 probably is a dual Riesel number. It is not known whether these are Riesel numbers with a covering set. - Thomas Ordowski, Jan 24 2024

Roger Crocker, "On the sum of a prime and of two powers of two", Pacific Journal of Mathematics 36:1 (1971), pp. 103-107.

Cf. A156695.

a(n)=my(N=2^2^n-1);for(a=1,2^n-1,if(ispseudoprime(N-2^a), return(a)));0

a(15) from Charles R Greathouse IV, Dec 02 2013
a(16) from Daniel Suteu, Oct 11 2020
Name edited by Thomas Ordowski, Jan 24 2024

A268693 Odd numbers that are not of the form p + 2^a + 2^b with b > a > 0, and p prime.

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Feb 11 2016

nonn

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.

Kevin P. Thompson, Table of n, a(n) for n = 1..392

Cf. A232565, A268694. Supersequence of A156695.

A304122 Squarefree numbers of the form 2^k + 5^m, where k is a positive integer and m is a nonnegative integer.

Original entry on oeis.org

3, 5, 7, 13, 17, 21, 29, 33, 37, 41, 57, 65, 69, 89, 127, 129, 133, 141, 157, 253, 257, 281, 381, 517, 537, 627, 629, 633, 641, 689, 753, 881, 1049, 1137, 1149, 1649, 2049, 2053, 2073, 2173, 3127, 3129, 3133, 3157, 3189, 3253, 3637, 4097, 4101, 4121

Offset: 1

Zhi-Wei Sun, May 07 2018

nonn

The conjecture in A304081 has the following equivalent version: Any even number greater than 4 can be written as the sum of a prime and a term of the current sequence, and also any odd number greater than 8 can be written as the sum of a prime and twice a term of the current sequence.

			a(1) = 3 since 3 = 2^1 + 5^0 is squarefree.
a(6) = 21 since 21 = 2^4 + 5^1 = 3*7 is squarefree.

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
Zhi-Wei Sun, Mixed sums of primes and other terms, in: Additive Number Theory (edited by D. Chudnovsky and G. Chudnovsky), pp. 341-353, Springer, New York, 2010.
Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)

Cf. A000040, A000079, A000351, A005117, A118955, A156695, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601, A303637, A303639, A303656, A303660, A303702, A303821, A303932, A303934, A304034, A304081.

V={}; Do[If[SquareFreeQ[2^k+5^m],V=Append[V,2^k+5^m]],{k,1,12},{m,0,5}];
LL:=LL=Sort[DeleteDuplicates[V]];
a[n_]:=a[n]=LL[[n]];
Table[a[n],{n,1,50}]

A337487 De Polignac numbers k > 1 such that k - 2^m is a de Polignac number for every 1 < 2^m < k.

Original entry on oeis.org

1117175145, 2544265305, 3147056235, 3366991695, 3472109835, 3621922845, 3861518805, 4447794915, 4848148485, 5415281745, 5693877405, 7525056375, 7602256605, 9055691835, 9217432215, 13431856995, 16819230075, 19373391165, 21468020835, 24358769685, 27002844795, 30252463305, 33359739795

Offset: 1

Thomas Ordowski, Aug 29 2020

nonn

Odd integers k > 3 that are not of the form p + 2^m + 2^n with m,n >= 0, where p is a prime.
These are de Polignac numbers k > 1 in A156695. Numbers k in A156695 such that k - 2 is composite.
Problem: are there infinitely many such numbers?

Amiram Eldar, Table of n, a(n) for n = 1..204 (terms below 10^12, calculated using the b-file at A156695)

An intersection of A006285 > 1 and A156695 (with m,n >= 1).

A156695 = Cases[Import["https://oeis.org/A156695/b156695.txt", "Table"], {, }][[;; , 2]]; dePolQ[n_] := n > 3 && AllTrue[n - 2^Range[Floor[Log[2, n]]], !PrimeQ[#] &]; Select[A156695, dePolQ] (* Amiram Eldar, Aug 29 2020 *)

A152073 a(n) = largest prime < prime(n) such that prime(n) - a(n) is a power of 2, where prime(n) is the n-th prime; a(n) = 0 if no such prime exists.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 13, 29, 29, 37, 41, 43, 37, 43, 59, 59, 67, 71, 71, 79, 73, 89, 97, 101, 103, 107, 109, 0, 127, 73, 137, 0, 149, 149, 131, 163, 157, 163, 179, 127, 191, 193, 197, 179, 191, 223, 227, 229, 223, 239, 0, 241, 199, 13, 269, 269, 277, 281, 277, 179

Offset: 2

Leroy Quet, Nov 23 2008

a(n) = 0 for odd primes prime(n) appearing in A065381.

Primes p(n) for which there is no such prime a(n) (in which case a(n)=0) are listed in A065381 = (2,127,149,251,331,337,373,...). - M. F. Hasler, Nov 23 2008

			Looking at the primes less than the 10th prime = 29: 29 - 23 = 6, not a power of 2. 29-19 = 10, not a power of 2. 29-17 = 12, not a power of 2. But 29-13 = 16, a power of 2. Since p = 13 is the largest prime p such that 29 - p = a power of 2, then a(10) = 13.

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A065381, A139758, A152075, A152076, A156695.

Table[Max[0, Select[# - 2^Range[0, Log2@#] &@Prime[n], PrimeQ]], {n, 2, 63}] (* Ivan Neretin, Jun 10 2018 *)

A152073(n)=local( q=n=prime(n)); while( q=precprime(q-1), n-q==1<M. F. Hasler, Nov 23 2008

Edited and extended by M. F. Hasler and Ray Chandler, Nov 23 2008

A154421 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and an even Lucas number.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 2, 5, 2, 5, 2, 4, 3, 4, 4, 5, 2, 6, 2, 7, 5, 7, 3, 9, 3, 9, 4, 7, 3, 6, 4, 9, 3, 10, 3, 8, 4, 6, 5, 8, 6, 8, 3, 9, 4, 8, 6, 8

Offset: 1

Zhi-Wei Sun, Jan 09 2009

nonn

On Jan 09 2009, Zhi-Wei Sun conjectured that a(n)>0 for all n=6,7,.... ; in other words, any integer n>5 can be written in the form p+F_s+L_{3t} with p an odd prime, s positive and t nonnegative. [Compare this with the conjecture related to the sequence A154290.] Sun verified the above conjecture up to 5*10^6 and Qing-Hu Hou continued the verification up to 2*10^8. If we set v_0=2, v_1=4 and v_{n+1}=4v_n+v_{n-1} for n=1,2,3,..., then L_{3t}=v_t is at least 4^t for every t=0,1,2,.... On Jan 17 2009, D. S. McNeil found that 36930553345551 cannot be written as the sum of a prime, a Fibonacci number and an even Lucas number.

			For n=8 the a(8)=3 solutions are 3 + F_4 + L_0, 3 + F_2 + L_3, 5 + F_2 + L_0.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.

Zhi-Wei SUN, Table of n, a(n), n=1..50000.
D. S. McNeil, Sun's strong conjecture
Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t
Zhi-Wei Sun, A summary concerning my conjecture n=p+F_s+F_t (II)
Terence Tao, A remark on primality testing and decimal expansions, Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x^m-2^n and x^2-F_{3n}/2, Math. Comp. 78 (2009) 1853, [DOI], arXiv:math.NT/0702382

Cf. A000040, A000045, A000032, A154257, A154285, A154290, A154364, A154417, A156695.

PQ[m_]:=m>2&&PrimeQ[m] RN[n_]:=Sum[If[PQ[n-2*Fibonacci[3x+1]+Fibonacci[3x]-Fibonacci[y]],1,0], {x,0,Log[2,n]},{y,2,2*Log[2,Max[2,n-2*Fibonacci[3x+1]+Fibonacci[3x]]]}] Do[Print[n," ",RN[n]];Continue,{n,1,50000}]

a(n) = |{: p+F_s+L_{3t}=n with p an odd prime, s>1 and t nonnegative}|.

McNeil's counterexample added by Zhi-Wei Sun, Jan 20 2009

A157225 Number of ways to write the n-th positive odd integer in the form p+2^x+7*2^y with p a prime congruent to 5 mod 6 and x,y positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 2, 3, 1, 2, 4, 1, 2, 4, 2, 2, 3, 2, 2, 4, 2, 4, 4, 1, 5, 5, 2, 5, 7, 1, 3, 7, 2, 4, 8, 2, 4, 3, 2, 4

Offset: 1

Zhi-Wei Sun, Feb 25 2009

On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if n<11 or n=13,16,992; in other words, except for 25, 31, 1983, any odd integer greater than 20 can be written as the sum of a prime congruent to 5 mod 6, a positive power of 2 and seven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7, and Qing-Hu Hou continued the verification for odd integers below 1.5*10^8 (on Sun's request). Compare the conjecture with Crocker's result that there are infinitely many positive odd integers not of the form p+2^x+2^y with p an odd prime and x,y positive integers.

			For n=18 the a(18)=3 solutions are 2*18-1=5+2+7*2^2=5+2^4+7*2=17+2^2+7*2.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.

Zhi-Wei Sun, Table of n, a(n) for n=1..200000
Zhi-Wei Sun, A webpage: Mixed Sums of Primes and Other Terms, 2009.
Zhi-Wei Sun, A project for the form p+2^x+k*2^y with k=3,5,...,61
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075

Cf. A000040, A000079, A157218, A155860, A155904, A156695, A154257, A154285, A155114, A154536, A154404, A154940.

PQ[x_]:=x>1&&Mod[x,6]==5&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-7*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/7]},{y,1,Log[2,Max[2,2n-1-7*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]

a(n)=|{: p+2^x+7*2^y=2n-1 with p a prime congruent to 5 mod 6 and x,y positive integers}|

A157242 Number of ways to write the n-th positive odd integer in the form p+2^x+11*2^y with p a prime congruent to 5 mod 6 and x,y positive integers.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Feb 25 2009

On Feb. 24, 2009, Zhi-Wei Sun conjectured that a(n)=0 if and only if n<15 or n=17, 20, 23, 86, 124; in other words, except for 33, 39, 45, 171 and 247, any odd integer greater than 28 can be written as the sum of a prime p=5 (mod 6), a positive power of 2 and eleven times a positive power of 2. Sun verified the conjecture for odd integers below 5*10^7. Knowing the conjecture from Sun, Qing-Hu Hou and D. S. McNeil have continued the verification for odd integers below 1.5*10^8 and 10^12 respectively, and they have found no counterexample. Compare the conjecture with Crocker's result that there are infinitely many positive odd integers not of the form p+2^x+2^y with p an odd prime and x,y positive integers.

			For n=18 the a(18)=2 solutions are 2*18-1=5+2^3+2*11=11+2+2*11.

R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36(1971), 103-107.
Z. W. Sun and M. H. Le, Integers not of the form c(2^a+2^b)+p^{alpha}, Acta Arith. 99(2001), 183-190.

Zhi-Wei Sun, Table of n, a(n) for n=1..200000
Zhi-Wei Sun, A webpage: Mixed Sums of Primes and Other Terms, 2009.
Zhi-Wei Sun, A project for the form p+2^x+k*2^y with k=3,5,...,61
Zhi-Wei Sun, A promising conjecture: n=p+F_s+F_t
Z. W. Sun, Mixed sums of primes and other terms, preprint, 2009. arXiv:0901.3075

A000040, A000079, A157237, A155860, A155904, A156695, A154257, A154285, A155114, A154536

PQ[x_]:=x>1&&Mod[x,6]==5&&PrimeQ[x] RN[n_]:=Sum[If[PQ[2n-1-11*2^x-2^y],1,0], {x,1,Log[2,(2n-1)/11]},{y,1,Log[2,Max[2,2n-1-11*2^x]]}] Do[Print[n," ",RN[n]],{n,1,200000}]

a(n)=|{: p+2^x+11*2^y=2n-1 with p a prime congruent to 5 mod 6 and x,y positive integers}|

cp's OEIS Frontend

A157237 Number of ways to write the n-th positive odd integer in the form p+2^x+11*2^y with p a prime congruent to 1 mod 6 and x,y positive integers.

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A188903 a(n) is the least power of 2 such that 2n+1 - a(n) is prime, or 0 if no such prime exists.

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A232565 a(n) is the smallest k such that 2^(2^n) - 2^k - 1 is prime, or -1 if no such k exists.

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A268693 Odd numbers that are not of the form p + 2^a + 2^b with b > a > 0, and p prime.

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A304122 Squarefree numbers of the form 2^k + 5^m, where k is a positive integer and m is a nonnegative integer.

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A337487 De Polignac numbers k > 1 such that k - 2^m is a de Polignac number for every 1 < 2^m < k.

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A152073 a(n) = largest prime < prime(n) such that prime(n) - a(n) is a power of 2, where prime(n) is the n-th prime; a(n) = 0 if no such prime exists.

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A154421 Number of ways to express n as the sum of an odd prime, a positive Fibonacci number and an even Lucas number.

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