A196697 -id:A196697 - cp's OEIS frontend

A196788 a(n) is the first occurrence of n in A196697.

1, 1805, 133, 2, 3, 4, 5, 13, 6, 9, 8, 10, 19, 16, 32, 24, 74, 30, 18, 60, 168, 20, 42, 90, 180, 210, 186, 408, 144, 1020, 1050, 900, 2520, 3348, 2850, 5520, 3390, 774, 5760

Offset: 1

Lei Zhou, Oct 06 2011

a(39+k) > 9594, for any k >= 1. It is getting much harder for n>39. It took the Mathematica program weeks to get the first 39 items.

			A196697(1)=1, a(1)=1;
A196697(2)=4, a(4)=2;
...
A196697(1805)=2, a(2)=1805; (for any k<1805, A196697(k)<>2)

Cf. A196697.

b = 2; max = 39; Array[fa, max]; Do[
fa[k] = 0, {k, 1, max}]; filled = 0; i = 0; While[filled < max, i++;
c1 = b^i; cs = {};
Do[c2 = b^j; cp = c1 + c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 - c2 - 1;
  If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 0, i - 1}];
ct = Length[cs];
If[ct <= max, If[fa[ct] == 0, fa[ct] = i; filled++]]]; Table[
fa[k], {k, 1, max}]

A196698 Number of primes of the form 3^n +- 3^k +- 1 with 0 <= k < n.

Original entry on oeis.org

2, 4, 6, 8, 7, 11, 7, 10, 11, 11, 8, 10, 9, 11, 14, 11, 10, 14, 7, 16, 12, 12, 7, 17, 10, 7, 15, 13, 4, 11, 11, 11, 13, 6, 12, 18, 9, 12, 17, 14, 13, 11, 10, 11, 13, 6, 7, 17, 9, 14, 9, 10, 13, 20, 8, 11, 10, 9, 8, 16, 12, 12, 13, 8, 12, 14, 8, 8, 10, 13, 9

Offset: 1

Lei Zhou, Oct 05 2011

nonn

Conjecture: all elements of this sequence are greater than 0.
Conjecture verified up to n = 7399.
I conjecture the contrary: infinitely many elements of this sequence are equal to 0. Probably the first n with a(n) = 0 is less than a million. - Charles R Greathouse IV, Nov 21 2011
This is also number of primes in n-digit balanced ternary form with no more than three nonzero digits for n > 1. - Lei Zhou, Dec 04 2013

			n = 1, 3 = 3^1 + 3^0 - 1 = 3^1 - 3^0 + 1; 5 = 3^1 + 3^0 + 1, two primes found, so a(1) = 2;
n = 2, 5 = 3^2 - 3^1 - 1; 7 = 3^2 - 3^1 + 1 = 3^2 - 3^0 - 1; 11 = 3^2 + 3^1 - 1 = 3^2 + 3^0 + 1; 13 = 3^2 + 3^1 + 1, four primes found, so a(2) = 4;
...
n = 7, 1459 = 3^7 - 3^6 + 1; 2161 = 3^7 - 3^3 + 1; 2179 = 3^7 - 3^1 + 1; 2213 = 3^7 + 3^3 - 1; 2267 = 3^7 + 3^4 - 1; 2269 = 3^7 + 3^4 + 1; 2917 = 3^7 + 3^6 + 1, seven primes found, so a(7) = 7.

Lei Zhou, Table of n, a(n) for n = 1..6205
Lei Zhou, A 400,000 decimal digits balanced ternary prime with three non-zero digits, found on Jan 02 2015.

Cf. A196697.

Table[c1 = 3^i; cs = {};
Do[c2 = 3^j; cp = c1 + c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 - c2 - 1;
  If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 1, i - 1}];
Length[cs], {i, 2, 100}]
(* Alternative: *)
Table[s = 3^i; ct = 0; Do[t = 3^j; a1 = s + t; a2 = s - t; If[PrimeQ[a1 + 1], ct++]; If[PrimeQ[a1 - 1], ct++]; If[PrimeQ[a2 + 1], ct++]; If[PrimeQ[a2 - 1], ct++], {j, 1, i - 1}]; ct, {i, 2, 100}] (* Lei Zhou, Mar 19 2015 *)

a(n)=sum(k=0, n-1, isprime(3^n-3^k-1)+isprime(3^n-3^k+1)+isprime(3^n+3^k-1)+isprime(3^n+3^k+1)) \\ Charles R Greathouse IV, Oct 06 2011

A238900 Least k such that one of 2^n +- 2^k +- 1 is prime, where 0 < k < n, or 0 if there is no such prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 12, 2, 11, 1, 1, 1, 1, 2, 3, 9, 5, 2, 3, 3, 3, 4, 5, 4, 8, 3, 7, 4, 2, 6, 17, 14, 6, 12, 2, 5, 1, 2, 3, 6, 11, 5, 1, 16, 8, 8, 20, 2, 1, 5, 7, 19, 6, 4, 19, 8, 5, 4, 5, 3, 9, 6, 4, 3, 13, 1, 24

Offset: 2

T. D. Noe, Mar 17 2014

nonn

Does a(n) = 0 for some n?

Giovanni Resta, Table of n, a(n) for n = 2..10000 (first 1999 terms from T. D. Noe)

Cf. A196697.

Table[c1 = 2^n; k = 1; While[c2 = 2^k; k < n && ! (PrimeQ[c1 + c2 + 1] || PrimeQ[c1 + c2 - 1] || PrimeQ[c1 - c2 + 1] || PrimeQ[c1 - c2 - 1]), k++]; If[k == n, 0, k], {n, 2, 100}]

A196778 a(n) is the number of primes in the form of 4^n+/-4^k+/-1, while 0 <= k < n.

Original entry on oeis.org

1, 3, 5, 6, 7, 7, 9, 8, 9, 12, 7, 9, 4, 4, 8, 11, 6, 11, 7, 8, 14, 7, 8, 11, 6, 10, 9, 8, 8, 11, 6, 10, 13, 7, 6, 9, 10, 8, 8, 10, 5, 10, 15, 6, 11, 9, 14, 7, 8, 16, 12, 10, 5, 10, 9, 8, 10, 8, 7, 10, 11, 13, 12, 6, 12, 9, 4, 10, 12, 13, 8, 14, 7, 2, 13, 7

Offset: 1

Lei Zhou, Oct 06 2011

Conjecture: all elements of this sequence is greater than 0.
Conjecture tested hold up to n=2355.  Further test is still running
The Mathematica program gives the first 100 terms.
Terms for all n are tend to be small integers.
4^n+/-4^k+/-1=2^2n+/-2^2k+/-1

			n=1, 2=4^1-4^0-4^0, 1 prime found, so a(1)=1;
n=2, 11=4^2-4^1-1; 13=4^2-4^1+1; 19=4^2+4^1-1, 3 primes found, so a(2)=3;
...
n=13, 67043329=4^13-4^8+1; 67104769=4^13-4^6+1; 67108859=4^13-4^1-1; 67108879=4^13+4^2-1, 4 primes found, so a(13)=4;

Cf. A196697, A196698.

b = 4; Table[c1 = b^i; cs = {};
Do[c2 = b^j; cp = c1 + c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
  cp = c1 - c2 - 1;
  If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 0, i - 1}];
ct = Length[cs]; ct, {i, 1, 100}]

A232190 a(n) is the number of primes of the form 2^b + 2n +- 2^k +- 1 and 2^(b+2) - 2^b - 2n +- 2^k +- 1, where b is the length of the binary representation of 2n, and 0

Original entry on oeis.org

5, 9, 7, 10, 11, 10, 10, 13, 14, 14, 15, 12, 13, 11, 12, 15, 18, 15, 15, 15, 17, 17, 18, 12, 15, 14, 14, 12, 16, 14, 13, 14, 16, 23, 20, 16, 18, 16, 17, 16, 17, 16, 16, 13, 17, 15, 15, 15, 20, 18, 20, 19, 17, 18, 18, 14, 15, 18, 18, 13, 17, 14, 15, 17, 17, 16

Offset: 1

Lei Zhou, Nov 20 2013

nonn

Tested up to n = 1000000000, a(n)> 0.
If any zero terms exist, it is likely that the first one will appear in the interval [2*10^9, 2*10^10].
The terms of this sequence form a bell-shaped distribution with the commonest value of 21 when n is large enough. Up to the first 100 million terms, the range of a(n) is [3..55].

			When n=1, 2n=2, b=2, the set of numbers of the form 2^b + 2n + 2^k + 1 is {9, 11}; form 2^b + 2n + 2^k - 1: {7, 9}; form 2^b + 2n - 2^k - 1: {1, 3}; form 2^b + 2n - 2^k + 1: {3, 5}; form 2^(b+2) - 2^b - 2n - 2^k - 1: {7, 5}; form 2^(b+2) - 2^b - 2n - 2^k + 1: {9, 7}; form 2^(b+2) - 2^b - 2n + 2^k + 1: {15, 13}; form 2^(b+2) - 2^b - 2n + 2^k - 1: {13, 11}. The union of the above sets is {1, 3, 5, 7, 9, 11, 13, 15}. Among the 8 numbers, 5 are primes. So a(1)=5.
When n=11, using the same rule, the candidate number set is {21, 23, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 105, 107}. Among these 32 numbers, 15 are prime: {23, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 107}. So a(11)=15.

Lei Zhou, Table of n, a(n) for n = 1..10000

Cf. A196697, A196698.

Table[n2 = 2*n; b = Ceiling[Log[2, n2 + 1]]; sdm = 2^b + n2 - 1;
sdp = 2^b + n2 + 1; cset = {}; Do[cpmp = sdm + 2^k; cpmm = sdm - 2^k; cppp = sdp + 2^k; cppm = sdp - 2^k; upl = 2^(b + 2); cset = Join[
    cset, {cpmp, upl - cpmp, cpmm, upl - cpmm, cppp, upl - cppp, cppm,
      upl - cppm}], {k, 1, b}]; cset = Union[cset];
size = Length[cset]; ct = 0;
Do[If[PrimeQ[cset[[j]]], ct++], {j, 1, size}]; ct, {n, 1, 66}]

Edited by Jon E. Schoenfield, Mar 28 2015

A196779 a(n) is the smallest number m such that no prime takes the form of n^m+/-n^k+/-1, while 0 <= k < m and m > 1.

Original entry on oeis.org

1147, 113, 113, 400, 866, 131, 399, 32, 26, 29, 23, 58, 77, 21, 42, 3, 817, 4, 2, 37, 80, 29, 181, 39, 120, 382, 76, 5, 29, 20, 48, 19, 36, 7, 43, 7, 62, 22, 7, 43, 5, 17, 23, 44, 52, 137, 103, 2, 5, 49, 31, 10, 30, 5, 25, 25, 49, 10, 72, 50, 13, 4, 7, 6

Offset: 5

Lei Zhou, Oct 06 2011

nonn

Conjecture: a(n) has finite value when a>4
already tested: a(4)>2364; a(3)>7399; and a(2)>9594.
Hypothesis is that a(2), a(3), and a(4) are infinite.
Mathematica program ran about an hour and gave the first 96 items.
When n is larger, a(n) tends to be 2 for most of n.

			n=5, there is no prime number in the form of 5^1147+/-5^k+/-1 for 0 <= k < 1147

Cf. A196697, A196698, A196778.

Table[i = 1;  While[i++; c1 = b^i; cs = {};
  Do[c2 = b^j; cp = c1 + c2 + 1;
   If[PrimeQ[cp], cs = Union[cs, {cp}]];
   cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
   cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
   cp = c1 - c2 - 1;
   If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 0, i - 1}];
  ct = Length[cs]; ct > 0]; i, {b, 5, 100}]

cp's OEIS Frontend

A196788 a(n) is the first occurrence of n in A196697.

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A196698 Number of primes of the form 3^n +- 3^k +- 1 with 0 <= k < n.

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A238900 Least k such that one of 2^n +- 2^k +- 1 is prime, where 0 < k < n, or 0 if there is no such prime.

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A196778 a(n) is the number of primes in the form of 4^n+/-4^k+/-1, while 0 <= k < n.

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A232190 a(n) is the number of primes of the form 2^b + 2n +- 2^k +- 1 and 2^(b+2) - 2^b - 2n +- 2^k +- 1, where b is the length of the binary representation of 2n, and 0

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A196779 a(n) is the smallest number m such that no prime takes the form of n^m+/-n^k+/-1, while 0 <= k < m and m > 1.

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