A076336 -id:A076336 - cp's OEIS frontend

A271080 Integers k such that s(k) = 7523267 + 11184810*k and s(k) + 14 are consecutive primes.

8, 16, 82, 101, 132, 187, 201, 253, 265, 300, 318, 351, 393, 408, 429, 449, 474, 489, 508, 660, 662, 673, 687, 772, 869, 877, 880, 924, 945, 958, 963, 984, 1028, 1042, 1070, 1083, 1124, 1134, 1226, 1249, 1257, 1265, 1319, 1340, 1345, 1352, 1365, 1389, 1463, 1664, 1816, 1834, 1878, 1969

Offset: 1

Altug Alkan, Mar 30 2016

nonn

s(k) and s(k) + 14 are always Sierpiński numbers for k >= 0.
Motivated by the question: What are the consecutive Sierpiński numbers with difference 14 that are also consecutive primes?
See A270971 and A270993 for the reason for the definition's focus on 14.
How does the graph of this sequence look for larger values of n?

			8 is a term because 7523267 + 11184810*8 = 97001747 and 97001761 are consecutive (provable) Sierpiński numbers and they are also consecutive primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A076336, A270971, A270993.

Select[Range@ 2000, And[PrimeQ@ #, NextPrime@ # == # + 14] &@(7523267 + 11184810 #) &] (* Michael De Vlieger, Mar 30 2016 *)
cpQ[n_]:=Module[{c=7523267+11184810n},PrimeQ[c]&&NextPrime[c]==c+14]; Select[Range[ 2000],cpQ] (* Harvey P. Dale, Oct 07 2023 *)

lista(nn) = for(n=0, nn, if(ispseudoprime(s=7523267 + 11184810*n) && nextprime(s+1) == (s+14), print1(n, ", ")));

is(n)=my(s=11184810*n+7523267); isprime(s) && isprime(s+14) && !isprime(s+6) && !isprime(s+12) \\ Charles R Greathouse IV, Mar 31 2016

A288477 a(n) = (2^49 - 2)*n/3 + 247371098957.

Original entry on oeis.org

247371098957, 187897355572727, 375547340046497, 563197324520267, 750847308994037, 938497293467807, 1126147277941577, 1313797262415347, 1501447246889117, 1689097231362887, 1876747215836657, 2064397200310427, 2252047184784197, 2439697169257967

Offset: 0

Arkadiusz Wesolowski, Jun 09 2017

For all n, the numbers a(n) and a(n) + 2 form a pair of consecutive Sierpiński numbers.

Conjecture: a(0) + 1 = 247371098958 is the smallest nonnegative even number m such that for all k >= 1 the numbers m + 2^k + 1 and m + 2^k - 1 are composite.

Muniru A Asiru, Table of n, a(n) for n = 0..5000
Carlos Rivera, Collection 20th - 019
Wikipedia, Sierpinski number
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A076336.

List([0..15],n->(2^49-2)*n/3+247371098957); # Muniru A Asiru, Oct 01 2018

[(2^49-2)*n/3+247371098957: n in [0..13]];

seq(coeff(series((247371098957+187402613374813*x)/(1-x)^2,x,n+1), x, n), n = 0 .. 15); # Muniru A Asiru, Oct 01 2018

Table[(2^49 - 2) n/3 + 247371098957, {n, 0, 13}] (* or *)
CoefficientList[Series[(247371098957 + 187402613374813 x)/(1 - x)^2, {x, 0, 13}], x] (* Michael De Vlieger, Jun 09 2017 *)
LinearRecurrence[{2,-1},{247371098957,187897355572727},20] (* Harvey P. Dale, Feb 22 2025 *)

PARI
```
a(n)=(2^49-2)*n/3+247371098957
```

G.f.: (247371098957 + 187402613374813*x)/(1 - x)^2.

A335653 Smallest prime whose binary expansion has Hamming distance 1 from 2n+1, or 0 if no such prime exists.

Original entry on oeis.org

3, 2, 7, 3, 11, 3, 5, 7, 19, 3, 5, 7, 17, 11, 13, 23, 37, 3, 5, 7, 43, 11, 13, 43, 17, 19, 37, 23, 41, 43, 29, 31, 67, 3, 5, 7, 89, 11, 13, 71, 17, 19, 1109, 23, 73, 83, 29, 31, 101, 67, 37, 71, 41, 43, 101, 47, 97, 83, 53, 103, 89, 59, 61, 383, 131, 3, 5, 7, 139, 11, 13, 139, 17, 19, 151, 23

Offset: 0

Ross Dempsey, Jun 15 2020

If 2n+1 is a dual Sierpiński number, and if 2n+1 cannot be made prime by flipping any of the ones in its binary representation to zero, then a(n) = 0.
2131099 is Sierpiński, and it is conjectured that the Sierpiński numbers are the same as the dual Sierpiński numbers. Furthermore 2131099 is the smallest Sierpiński number whose binary representation has the property stated above. If the dual Sierpiński conjecture holds, and if A076336 is complete up to 2131099, then a(1065549) = 0 and this is likely the first 0 in the sequence.
From Robert Israel, Jul 08 2020: (Start)
Every prime is in the sequence.
  Proof: Since 2 = a(1), we may assume prime p is odd. Take k so 2^k > p, and consider 2n+1 = p + 2^k. Then p has Hamming distance 1 from 2n+1. On the other hand, if q < p is prime, then q + 2^j < p + 2^k if j <= k while q + 2^j >= q + 2*2^k > p + 2^k if j > k, so q can't be at Hamming distance 1 from 2n+1. Thus p = a(n). (End)

			For n = 4, 2n+1 = 1001_2, and the smallest prime with Hamming distance 1 is 1011_2 = 11.

Robert Israel, Table of n, a(n) for n = 0..2234
MathOverflow, Hamming Distance to Primes

Cf. A067760, A076336.

f:= proc(n) local L, nL, k, v;
    L:= convert(n, base, 2);
    nL:= nops(L);
    for k from nL to 1 by -1 do
      if L[k] = 1 then
        v:= n - 2^(k-1);
        if isprime(v) then return v fi;
      fi
    od;
    for k from 1 to nL do
      if L[k] = 0 then
        v:= n + 2^(k-1);
        if isprime(v) then return v fi;
      fi
    od;
    for k from nL+1 do
      v:= n+2^(k-1);
      if isprime(v) then return v fi;
    od
end proc:
map(f, [seq(i, i=1..200, 2)]); # Robert Israel, Jun 15 2020

a[n_Integer] := a[IntegerDigits[2 n + 1, 2]];
a[bin_List] := Module[{flips, primes},
   flips =
    Sort[FromDigits[bin,
       2] + (1 - 2 bin) Power[2, Length[bin] - Range[Length[bin]]]];
   primes = Select[flips, PrimeQ];
   If[Length[primes] >= 1, First[primes],
    a[FromDigits[bin, 2], Length[bin]]]
   ];
a[n_Integer, k_Integer] :=
  Module[{test = n + Power[2, k]}, test /; PrimeQ[test]];
a[n_Integer, k_Integer] := a[n, k + 1];
Table[a[n],{n,0,50}]

a(n) = my(p=2); while(norml2(binary(bitxor(p, 2*n+1))) != 1, p = nextprime(p+1)); p; \\ Michel Marcus, Jun 16 2020

a((p+2^k-1)/2) = p if p is an odd prime and 2^k > p-3. - Robert Israel, Jun 16 2020

A350118 Primes p for which the smallest m such that p*2^m + 1 is prime increases. Sequence terminates with the smallest prime Sierpiński number.

Original entry on oeis.org

2, 3, 7, 17, 19, 31, 47, 383, 2897, 3061, 5297, 7013, 10223

Offset: 1

Jianing Song, Dec 14 2021

The smallest prime Sierpiński number is likely to be 271129.
Related to A058887: this sequence is A058887 with repeated values removed. The following list shows that relation between these two sequences:
a(2) = 3, A350119(2) = 1 => A058887(0..0) = 3;
a(3) = 7, A350119(3) = 2 => A058887(1..1) = 7;
a(4) = 17, A350119(4) = 3 => A058887(2..2) = 17;
a(5) = 19, A350119(5) = 6 => A058887(3..5) = 19;
a(6) = 31, A350119(6) = 8 => A058887(6..7) = 31;
a(7) = 47, A350119(7) = 583 => A058887(8..582) = 47;
a(8) = 383, A350119(8) = 6393 => A058887(583..6392) = 383;
...
a(N) is the smallest prime Sierpiński number, A350119(N) = -1 => A058887(k) = a(N) for all k >= A350119(N-1).

			Let b(p) be the smallest m such that p*2^m + 1 is prime. We have a(1) = 2 with b(2) = 0.
The least prime p such that b(p) > 0 is p = 3 with b(3) = 1, so a(2) = 3.
The least prime p such that b(p) > 1 is p = 7 with b(7) = 2, so a(3) = 7.
The least prime p such that b(p) > 2 is p = 17 with b(17) = 3, so a(4) = 17.
The least prime p such that b(p) > 3 is p = 19 with b(19) = 6, so a(5) = 19.
The least prime p such that b(p) > 6 is p = 31 with b(31) = 8, so a(6) = 31.
The least prime p such that b(p) > 8 is p = 47 with b(47) = 583, so a(7) = 47.

Cf. A058887, A057192, A350119, A064699, A076336 (Sierpiński numbers).

b(p) = for(k=0, oo, if(isprime(p*2^k+1), return(k)))
list(lim) = if(lim>=2, my(v=[2],r=0); forprime(p=2, lim, if(b(p)>r, r=b(p); v=concat(v,p))); v)

A350119 Record values in A057192, until A350118(n) reaches the smallest prime Sierpiński number and then sequence terminates with a(n) = -1.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 583, 6393, 9715, 33288, 50011, 126113, 31172165

Offset: 1

Jianing Song, Dec 14 2021

			Let b(p) be the smallest m such that p*2^m + 1 is prime. We have a(1) = b(2) = 0.
The least prime p such that b(p) > 0 is p = 3 with b(3) = 1, so a(2) = 1.
The least prime p such that b(p) > 1 is p = 7 with b(7) = 2, so a(3) = 2.
The least prime p such that b(p) > 2 is p = 17 with b(17) = 3, so a(4) = 3.
The least prime p such that b(p) > 3 is p = 19 with b(19) = 6, so a(5) = 6.
The least prime p such that b(p) > 6 is p = 31 with b(31) = 8, so a(6) = 8.
The least prime p such that b(p) > 8 is p = 47 with b(47) = 583, so a(7) = 583.

Cf. A057192, A350118, A103964, A076336 (Sierpiński numbers).

b(p) = for(k=0, oo, if(isprime(p*2^k+1), return(k)))
list(lim) = if(lim>=2, my(v=[0],r=0); forprime(p=2, lim, if(b(p)>r, r=b(p); v=concat(v,r))); v) \\ gives a(n) for A350118(n) <= lim

a(n) = A057192(primepi(A350118(n))).

A361546 a(n) is the least odd number k such that k*2^prime(n) + 1 is prime, or -1 if no such number k exists.

Original entry on oeis.org

1, 5, 3, 5, 9, 5, 9, 11, 45, 23, 35, 15, 3, 9, 27, 51, 27, 53, 9, 39, 23, 249, 51, 51, 131, 221, 29, 105, 321, 179, 5, 221, 111, 411, 191, 65, 83, 75, 95, 101, 147, 83, 149, 111, 203, 131, 9, 245, 281, 15, 83, 65, 299, 39, 51, 51, 225, 65, 81, 125, 611, 143, 65, 107, 21

Offset: 1

Jean-Marc Rebert, Mar 15 2023

nonn

			prime(1) = 2 and 1*2^2 + 1 = 5 is prime, and no lesser odd k satisfies this, so a(1) = 1.

Jean-Marc Rebert, Table of n, a(n) for n = 1..1000

Cf. A076336. Essentially the same as A253398.

a[n_] := Module[{m = 2^Prime[n], k = 1}, While[!PrimeQ[k*m + 1], k += 2]; k]; Array[a, 65] (* Amiram Eldar, Mar 15 2023 *)

a(n)=my(m=2^prime(n),k=1);while(!isprime(k*m+1),k+=2);k

A369378 a(n) is the smallest k such that 2^(2^n+1)-1 + 2^k is prime, or -1 if no such k exists.

Original entry on oeis.org

1, 2, 4, 16, 1968, 64, 64

Offset: 0

Thomas Ordowski, Jan 22 2024

Conjecture: if n > 0, then 2^(2^n+1)-1 + 2^k is not prime for every k < 2^n.
This conjecture seems provable, because it is not too strong for large n > 6.
If, for n > 6, a(n) does not exist, then 2^(2^n+1)-1 + 2^k is composite for every natural k. Thus, by the dual Sierpinski conjecture, for n > 6, 2^(2^n+1)-1 is a Sierpinski number, i.e., if n > 6, then (2^(2^n+1)-1)2^k+1 is composite for every natural k. For example, the Mersenne number 2^(2^8+1)-1 may be a dual Sierpinski number.
Similarly, if for n > 5, |2^(2^n-1)-1 - 2^m| is not prime for every m > 0, then by the dual Riesel conjecture, 2^(2^n-1)-1 is a Riesel number, i.e., if n > 5, then (2^(2^n-1)-1)2^m-1 is composite for every integer m > 0. For example, the double Mersenne prime 2^(2^7-1)-1 may be a dual Riesel number. By Crocker's theorem; if n > 2, then positive 2^(2^n-1)-1 - 2^m are composite. Let b(n) be the smallest k such that 2^k - (2^(2^n-1)-1) is prime, for n >= 0: {1, 2, 39, 47, 447, 191, ?}.
a(7) > 65000, a(8) thru a(12) > 25000, if they exist. - Robert G. Wilson v, Jan 22 2024

			a(1) = 2 since for 2^(2^1+1)-1 = 7 we have 7 + 2^1 = 9 and 7 + 2^2 = 11 is prime.

Cf. A076336, A101036, A156695, A369375.

a[n_] := Module[{m = 2^(2^n + 1) - 1, k = 1}, While[! PrimeQ[m + 2^k], k++]; k]; Array[a, 7, 0] (* Amiram Eldar, Jan 22 2024 *)

A373210 Least k such that the smallest prime >= 2^k is 2^k + 2*n + 1, or -1 if no such k exists.

Original entry on oeis.org

0, 3, 5, 10, 9, 29, 64, 22, 13, 162, 19, 39, 34, 14, 17, 36, 60, 25, 74, 87, 121, 24, 151, 209, 170, 111, 35, 50, 188, 45, 96, 247, 193, 124, 49, 115, 258, 83, 173, 254, 56, 167, 136, 138, 279, 148, 314, 153, 158, 106, 199, 434, 93, 161, 6954, 104, 719, 240, 164

Offset: 0

Jianing Song, May 28 2024

sign

a(n) = -1 if 2*n + 1 is a Sierpiński number (for example when 2*n + 1 = 78557); cf. A076336. See also A067760.
Conjecture: a(n) != -1 if 2*n + 1 is not a Sierpiński number. In other words, if 2*n + 1 is not a Sierpiński number, then there exists some k >= 1 such that 2^k + 1, 2^k + 3, ..., 2^k + 2*n - 1 are all composite while 2^k + 2*n + 1 is prime.
a(54), a(75), a(83), a(128), a(159), a(176), ... > 5000 (if not equal to -1), which means that 109, 151, 167, 257, 319, 353, ... do not present among the first 5000 terms of A092131.
a(75) = 5880, a(83) = 5513. - Michael S. Branicky, May 28 2024
a(128) > 7000. - Michael S. Branicky, May 30 2024

			a(6) = 64, because the smallest prime >= 2^k is not 2^k + 13 for 0 <= k <= 63, while the smallest prime >= 2^64 is 2^64 + 13.

Michael S. Branicky, Table of n, a(n) for n = 0..127

Cf. A067760, A076336, A092131.

A373210_first_N_terms(N) = my(v = vector(N+1, i, -1), dist); v[1] = 0; for(i=2, oo, dist = nextprime(2^i) - 2^i; if(dist <= 2*N+1 && v[(dist+1)/2] == -1, v[(dist+1)/2] = i); if(vecmin(v) > -1, return(v))) \\ Warning: ignoring Sierpinski numbers

a(54) and beyond from Michael S. Branicky, May 29 2024

A258074 Table read by rows: each row represents the constant and exponent of a Colbert number.

Original entry on oeis.org

5359, 5054502, 33661, 7031232, 28433, 7830457, 27653, 9167433, 19249, 13018586, 10223, 31172165

Offset: 1

Tom Edgar, May 18 2015

Each row has two entries [k,n]. With this notation, the corresponding Colbert number is k*2^n+1.
A Colbert number is an integer with more than 1,000,000 digits that is prime and has contributed to the in-progress computational proof that 78557 is the smallest Sierpiński number (A076336).
a(11)-a(12) number, corresponding to [10223,31172165], is the second largest known prime that is not a Mersenne prime as of August 2023. - Hermann Stamm-Wilbrandt, Aug 13 2023
This table can only have (and is expected to have) five more rows corresponding to constants k equal to 21181, 22699, 24737, 55459, and 67607.

			The table is as follows:
  5359, 5054502;
  33661, 7031232;
  28433, 7830457;
  27653, 9167433;
  19249, 13018586
  10223, 31172165

C. K. Caldwell, The Prime Glossary, Colbert Number
C. K. Caldwell, The Prime Database, 10223*2^31172165+1
W. Sierpiński, Sur un problème concernant les nombres k * 2^n + 1, Elem. Math., 15 (1960), pp. 73-74.
Hermann Stamm-Wilbrandt, Colbert numbers, contains entries [k,n,s,x,y] for the 6 Colbert numbers, with p=k*2^n+1, s^2%p==p-1 and p==x^2+y^2.
Wikipedia, Seventeen or Bust

Cf. A076336, A064699, A064721.

a(11)-a(12) from Richard N. Smith, Jul 15 2019, following by the prime 10223*2^31172165+1 found by PrimeGrid.

A283619 a(n) = (conjectured) smallest positive integer k which is neither of the form p + n^x nor of the form p - n^x with x >= 0 and p prime, where gcd(k, n) = 1 and gcd(k^2-1, n-1) = 1.

Original entry on oeis.org

30666137, 3902132276156, 2473929, 1015214, 464437, 40743218950116, 47, 2344, 61863, 32660, 4367, 7974, 11, 2021170066180678, 92343, 784, 571, 2364594, 13, 20450, 136113, 2596, 176011, 262638, 3223, 512, 59217, 26, 18973, 6360528, 23, 11848, 99, 292226, 832573

Offset: 2

Arkadiusz Wesolowski, Mar 12 2017

nonn

The definition is similar to that for A123159, but considering "p + n^x" and "p - n^x".
What does "conjectured" mean? A positive integer k is a candidate if:
1) gcd(k, n) = 1,
2) gcd(k^2-1, n-1) = 1,
3) every term in the sequence k + n^x is divisible by one of the prime numbers of a covering set,
4) all numbers of the form k - n^x are composite, k > n^x + 1, x >= 0.
The main problem is to prove that the given terms are indeed correct.
A quick search showed that a(8) = 47, a(14) = 11, a(20) = 13, a(27) = 512, a(29) = 26, a(32) = 23, a(34) = 99.
This is an interesting sequence: it leads to new classes of numbers. For example, the integer 30666137 is probably the smallest number that is simultaneously a Polignac number and a Sierpinski number.

Cf. A076336, A123159, A263644, A283622.

cp's OEIS Frontend

A271080 Integers k such that s(k) = 7523267 + 11184810*k and s(k) + 14 are consecutive primes.

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A288477 a(n) = (2^49 - 2)*n/3 + 247371098957.

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A335653 Smallest prime whose binary expansion has Hamming distance 1 from 2n+1, or 0 if no such prime exists.

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A350118 Primes p for which the smallest m such that p*2^m + 1 is prime increases. Sequence terminates with the smallest prime Sierpiński number.

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A350119 Record values in A057192, until A350118(n) reaches the smallest prime Sierpiński number and then sequence terminates with a(n) = -1.

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A361546 a(n) is the least odd number k such that k*2^prime(n) + 1 is prime, or -1 if no such number k exists.

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A369378 a(n) is the smallest k such that 2^(2^n+1)-1 + 2^k is prime, or -1 if no such k exists.

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A373210 Least k such that the smallest prime >= 2^k is 2^k + 2*n + 1, or -1 if no such k exists.

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