A076481 -id:A076481 - cp's OEIS frontend

A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

831167, 1154567, 2502767, 3019787, 3675197, 5056577, 6352487, 14519177, 26724377, 43003577, 47378927, 47695607, 56406197, 86332457, 86611757, 99568757, 121967987, 126435527, 127990997, 128149127, 128975057, 145281557, 155715407

Offset: 1

David W. Wilson

nonn

			First chain is {831167, 6649337, 53194697, 425557577, 3404460617, 27235684937};
If p is congruent to {1,3,7,9} mod 10, then consecutive iterates are congruent to {9,5,7,3}, {3,1,7,5}, {5,9,7,1} respectively; so only 10k+7 may remain prime through five iterations, as sequence demonstrates nicely. - _Labos Elemer_, Jul 23 2003

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A085956, A086361, A086362, A023330, A059766, A023287, A000668, A076481, A086127.

Subsequence of A005123, A023228, A023260, A023291, and A023319.

k=0; m=8; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
it5Q[n_]:=AllTrue[Rest[NestList[8#+1&,n,5]],PrimeQ]; Select[Prime[Range[ 9*10^6]],it5Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2014 *)

{p, 8p+1, 64p+9, 512p+73, 4096p+585, 32768p+4681} are all primes, where the initial p is prime.

a(n) == 197 (mod 210). - John Cerkan, Nov 04 2016

A173933 The number of numbers m < k/2 such that m/k is a reduced fraction in the Cantor set, where k= A173931(n).

Original entry on oeis.org

1, 2, 3, 3, 4, 8, 6, 15, 6, 6, 8, 15, 8, 12, 8, 8, 10, 24, 27, 16, 12, 9, 63, 10, 16, 12, 63, 20, 12, 11, 10, 36, 12, 56, 12, 12, 44, 12, 15, 36, 12, 16, 120, 60, 110, 24, 16, 18, 24, 225

Offset: 1

T. D. Noe, Mar 03 2010

nonn

When k is a prime of the form (3^r-1)/2, then the m are 2^r-1 numbers (greater than 0) whose base-3 representation consists of only 0's and 1's. Hence, for r=3,7, and 13, the primes k are 13, 1093, and 797161, and the number of m < k/2 is 3, 63, and 4095.

			When k=40, then 1/k, 3/k, 9/k, and 13/k have base-3 representations containing only the digits 0 and 2.

T. D. Noe, Table of n, a(n) for n = 1..185

Cf. A005823, A005836, A007734, A076481, A173931, A173934.

Length /@ Last[Transpose[cantor]] (* see A173931 *)

Name qualified by Peter Munn, Jul 14 2019

A272106 Absolute primes in base 3: every permutation of digits in base 3 is a prime (only the smallest representatives of the permutation classes are shown).

Original entry on oeis.org

2, 5, 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013

Offset: 1

Chai Wah Wu, Apr 20 2016

For n <= 7, only a(2) = 5 is not a repunit in base 3. Supersequence of A076481. Base 3 analog of A258706.

Cf. A003459, A076481, A258706, A268812, A272107.

A268812 Absolute primes in base 16: every permutation of digits in base 16 is a prime (only the smallest representatives of the permutation classes are shown).

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 31, 53, 59, 61, 89, 191, 277, 283, 887, 1373, 1979, 3037

Offset: 1

Chai Wah Wu, Apr 20 2016

Base 16 analog of A258706.

Cf. A003459, A076481, A258706, A272106, A272107.

A272107 Absolute primes in base 8: every permutation of digits in base 8 is a prime (only the smallest representatives of the permutation classes are shown).

Original entry on oeis.org

2, 3, 5, 7, 13, 29, 31, 47, 73, 1759

Offset: 1

Chai Wah Wu, Apr 20 2016

Base 8 analog of A258706.

Cf. A003459, A076481, A258706, A268812, A272106.

A381974 Primes of the form Sum_{k >= 0} floor(m/3^k) for some number m.

Original entry on oeis.org

2, 5, 13, 17, 19, 23, 31, 41, 53, 59, 61, 67, 71, 89, 97, 101, 103, 107, 127, 131, 139, 149, 151, 157, 163, 167, 179, 191, 193, 197, 211, 223, 227, 229, 233, 251, 257, 263, 269, 277, 283, 313, 317, 331, 337, 349, 353, 373, 379, 383, 409, 419, 421, 431, 439

Offset: 1

Clark Kimberling, Apr 01 2025

nonn

			[9/1] + [9/3] + [9/9] = 13, where [ ] = floor, so 13 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A381973. Includes A076481.

f:= proc(n) local k; add(floor(n/3^k),k=0..ilog[3](n)) end proc:
select(isprime, map(f, [$2..100])); # Robert Israel, Apr 21 2025

f[n_] := Sum[Floor[n/3^k], {k, 0, Floor[Log[3, n]]}]  (* A004128 *)
u = Select[Range[400], PrimeQ[f[#]] &]  (* A381973 *)
Map[f, u]   (* A381974 *)

A086124 Primes generated by linear recursion: f(n) = prime(n) * f(n-1) + 2, f(1) = 1.

Original entry on oeis.org

5, 191, 8831183, 559832762721564181, 3655022053493602810873312808337814473758207442937

Offset: 1

Labos Elemer, Jul 23 2003

nonn

f(n) = 1, 5, 27, 191, 2103, 27341, 464799, 8831183, 203117211, ... .

a(6) has 298 decimal digits.

Cf. A000668, A076481, A086122, A086123, A086125.

f[x_] := Prime[x]*f[x-1]+2; f[1]=1; Do[s=f[n]; If[PrimeQ[s], Print[n]], {n, 1, 1000}]

a(n) = f(A086125(n)).

A086125 Values of k such that f(k) is a prime, where f(1) = 1, f(i) = prime(i)*f(i-1) + 2.

Original entry on oeis.org

2, 4, 8, 15, 31, 128, 163, 12284

Offset: 1

Labos Elemer, Jul 23 2003

No additional terms up to k = 1000. - Harvey P. Dale, Feb 02 2019

No additional terms up to k = 20000. - Michael S. Branicky, May 31 2025

The primes are in A086124.

Cf. A000668, A076481, A086122, A086123.

f[1]=1; f[x_] := f[x] = Prime[x]*f[x - 1] + 2; Do[ If[ PrimeQ[ f[n]], Print[n]], {n, 1, 1900}]
nxt[{n_,a_}]:={n+1,a*Prime[n+1]+2}; Select[NestList[nxt,{1,1},200], PrimeQ[ #[[2]]]&][[All,1]] (* Harvey P. Dale, Feb 02 2019 *)

Edited by Robert G. Wilson v, Jul 25 2003

a(8) from Michael S. Branicky, May 29 2025

A086127 Numbers k such that k remains prime after five iteration of function f(j) = 14*f(j)+1, starting at f(1) = prime.

Original entry on oeis.org

4889, 18059, 62639, 225527, 557093, 604973, 700703, 804077, 806903, 837077, 1341203, 1363403, 1932197, 2004269, 2062703, 2284637, 2797463, 3157379, 3493103, 3746399, 3995687, 4155413, 4227893, 4493297, 5534939, 5708603

Offset: 1

Labos Elemer, Jul 23 2003

nonn

{p, 14p+1, 196p+15, 2744p+211, 38416p+2955, 537824p+41371} are all primes, where p is prime.

			First chain is: {4889,68447,958259,13415627,187818779,2629462907}.
10th chain is {837077,11719079,164067107,2296939499,32157152987,450200141819}.

Cf. A085956, A086361, A086362, A023330, A059766, A023287, A000668, A076481.

k=0; m=14; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
Select[Range[6000000],And@@PrimeQ[NestList[14#+1&,#,5]]&] (* Harvey P. Dale, Sep 17 2012 *)

A129734 List of primitive prime divisors of the numbers 3^n-2^n (A001047) in their order of occurrence.

Original entry on oeis.org

5, 19, 13, 211, 7, 29, 71, 97, 1009, 11, 23, 331, 61, 53, 29927, 463, 3571, 17, 401, 129009091, 577, 1559, 745181, 4621, 43, 6217, 35839, 47, 2002867877, 5521, 101, 39756701, 79, 4057, 397760329, 369181, 68629840493971, 31, 241, 617671248800299, 3041, 14177

Offset: 1

N. J. A. Sloane, May 13 2007

nonn

Read A001047 term-by-term, factorize each term, write down any primes not seen before.

Amiram Eldar, Table of n, a(n) for n = 1..1893
Graham Everest, Shaun Stevens, Duncan Tamsett and Tom Ward, Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265-284.

Cf. A001047, A058765, A057468; A003462, A076481, A028491, A129733; A000225, A004668, A000043, A108974.

a(41) and a(42) switched by Amiram Eldar, Jun 30 2023

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A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

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A173933 The number of numbers m < k/2 such that m/k is a reduced fraction in the Cantor set, where k= A173931(n).

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A272106 Absolute primes in base 3: every permutation of digits in base 3 is a prime (only the smallest representatives of the permutation classes are shown).

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A268812 Absolute primes in base 16: every permutation of digits in base 16 is a prime (only the smallest representatives of the permutation classes are shown).

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A272107 Absolute primes in base 8: every permutation of digits in base 8 is a prime (only the smallest representatives of the permutation classes are shown).

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A381974 Primes of the form Sum_{k >= 0} floor(m/3^k) for some number m.

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A086124 Primes generated by linear recursion: f(n) = prime(n) * f(n-1) + 2, f(1) = 1.

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A086125 Values of k such that f(k) is a prime, where f(1) = 1, f(i) = prime(i)*f(i-1) + 2.

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A086127 Numbers k such that k remains prime after five iteration of function f(j) = 14*f(j)+1, starting at f(1) = prime.

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Mathematica

A129734 List of primitive prime divisors of the numbers 3^n-2^n (A001047) in their order of occurrence.

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