A077717 -id:A077717 - cp's OEIS frontend

A235266 Primes whose base-2 representation is also the base-3 representation of a prime.

2, 7, 11, 13, 41, 47, 67, 73, 79, 109, 127, 151, 173, 181, 191, 193, 211, 223, 227, 229, 233, 251, 283, 331, 367, 421, 443, 487, 541, 557, 563, 587, 601, 607, 631, 641, 661, 677, 719, 733, 877, 941, 947, 967, 971, 1033, 1187, 1193, 1201, 1301, 1321, 1373, 1447, 1451, 1471, 1531, 1567, 1571, 1657, 1667, 1669, 1697, 1709, 1759

Offset: 1

M. F. Hasler, Jan 05 2014

Robert Israel, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

Cf. A077717, A005836.

f:= proc(n) local L,i;
  L:= convert(n,base,2);
  isprime(add(L[i]*3^(i-1),i=1..nops(L)))
end proc:
select(f, [seq(ithprime(i),i=1..1000)]); # Robert Israel, Jun 03 2019

Select[Prime@ Range@ 250, PrimeQ@ FromDigits[IntegerDigits[#, 2], 3] &] (* Michael De Vlieger, Jun 03 2019 *)

is(p,b=3,c=2)=isprime(vector(#d=digits(p,c),i,b^(#d-i))*d~)&&isprime(p) \\ This code can be used for other bases b,c when b>c. See A235265 for code valid for b

forprime(p=2, 1e3, if(isprime(fromdigits(binary(p), 3)), print1(p", "))) \\ Charles R Greathouse IV, Mar 28 2022

from sympy import isprime, nextprime
def agen(): # generator of terms
    p = 2
    while True:
        p3 = sum(3**i for i, bi in enumerate(bin(p)[2:][::-1]) if bi=='1')
        if isprime(p3):
            yield p
        p = nextprime(p)
g = agen()
print([next(g) for n in range(1, 65)]) # Michael S. Branicky, Jan 16 2022

a(n) is the number whose base-3 representation is the base-2 representation of A235265(n).

A235265 Primes whose base-3 representation also is the base-2 representation of a prime.

Original entry on oeis.org

3, 13, 31, 37, 271, 283, 733, 757, 769, 1009, 1093, 2281, 2467, 2521, 2551, 2917, 3001, 3037, 3163, 3169, 3187, 3271, 6673, 7321, 7573, 9001, 9103, 9733, 19801, 19963, 20011, 20443, 20521, 20533, 20749, 21871, 21961, 22123, 22639, 22717, 27253, 28711, 28759, 29173, 29191, 59077, 61483, 61507, 61561, 65701, 65881

Offset: 1

M. F. Hasler, Jan 05 2014

This sequence and A235383 and A229037 are winners in the contest held at the 2014 AMS/MAA Joint Mathematics Meetings. - T. D. Noe, Jan 20 2014
This sequence was motivated by work initiated by V.J. Pohjola's post to the SeqFan list, which led to a clarification of the definition and correction of some errors, in sequences A089971, A089981 and A090707 through A090721. These sequences use "rebasing" (terminology of A065361) from some base b to base 10. Sequences A065720 - A065727 follow the same idea but use rebasing in the other sense, from base 10 to base b. The observation that only (10,b) and (b,10) had been considered so far led to the definition of this and related sequences: In a systematic approach, it seems natural to start with the smallest possible pairs of different bases, (2,3) and (3,2), then (2 <-> 4), (3 <-> 4), (2 <-> 5), etc.
Among the two possibilities using the smallest possible bases, 2 and 3, the present one seems a little bit more interesting, among others because not every base-3 representation is a valid base-2 representation (in contrast to the opposite case). This is also a reason why the present sequence grows much faster than the partner sequence A235266.

			3 = 10_3 and 10_2 = 2 is prime. 13 = 111_3 and 111_2 = 7 is prime.

Robert Israel, Table of n, a(n) for n = 1..10000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime
Veikko Pohjola, A090709 Decimal primes whose decimal representation in base 6 is also prime, SeqFan list, Jan 03 2014.

Subset of A077717.

Cf. A235266, A065720 and A036952, A065721 - A065727, A235394, A235395, A089971 and A020449, A089981, A090707 - A091924, A235461 - A235482. See M. F. Hasler's OEIS wiki page for further cross-references.

N:= 1000: # to get the first N terms
count:= 0:
for i from 1 while count < N do
   p2:= ithprime(i);
   L:= convert(p2,base,2);
   p3:= add(3^(j-1)*L[j],j=1..nops(L));
   if isprime(p3) then
      count:= count+1;
      A235265[count]:= p3;
   fi
od:
[seq(A235265[i], i=1..N)]; # Robert Israel, May 04 2014

b32pQ[n_]:=Module[{idn3=IntegerDigits[n,3]},Max[idn3]<2&&PrimeQ[ FromDigits[ idn3,2]]]; Select[Prime[Range[7000]],b32pQ] (* Harvey P. Dale, Apr 24 2015 *)

is(p,b=2,c=3)=vecmax(d=digits(p,c))

from sympy import isprime, nextprime
def agen(): # generator of terms
    p = 2
    while True:
        p3 = sum(3**i for i, bi in enumerate(bin(p)[2:][::-1]) if bi=='1')
        if isprime(p3):
            yield p3
        p = nextprime(p)
g = agen()
print([next(g) for n in range(1, 52)]) # Michael S. Branicky, Jan 16 2022

A077718 Primes which can be expressed as sum of distinct powers of 4.

Original entry on oeis.org

5, 17, 257, 277, 337, 1093, 1109, 1297, 1301, 1361, 4177, 4357, 4373, 4421, 5189, 5381, 5393, 5441, 16453, 16657, 16661, 17477, 17489, 17669, 17681, 17729, 17749, 20549, 20753, 21521, 21569, 21589, 21841, 65537, 65557, 65617, 65809, 66629

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 4 representation contains only zeros and 1's.

As a subsequence of primes in A000695, these could be called Moser-de Bruijn primes. See also A235461 for those terms whose base 4 representation also represents a prime in base 2. - M. F. Hasler, Jan 11 2014

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

Cf. A020449, A000695, A077717, A077719, A077720, A077721, A077722.

f:= proc(n) local L,x;
  L:= convert(n,base,2);
  x:= 1+add(L[i]*4^i,i=1..nops(L));
  if isprime(x) then x fi
end proc:
map(f, [$1..1000]); # Robert Israel, Sep 06 2018

Select[Prime[Range[6650]],Max[IntegerDigits[#,4]]<=1&] (* Jayanta Basu, May 22 2013 *)

for(i=1,999,isprime(b=vector(#b=binary(i),j,4^(#b-j))*b~)&&print1(b",")) \\ - M. F. Hasler, Jan 12 2014

More terms from Sascha Kurz, Jan 03 2003

A077722 Primes which can be expressed as sums of distinct powers of 8.

Original entry on oeis.org

73, 521, 577, 4673, 32833, 33289, 33353, 36929, 37441, 262153, 262217, 262657, 295433, 299017, 299521, 2097673, 2101249, 2101313, 2134529, 2359369, 2359873, 2363393, 2363401, 2392073, 16777289, 16777729, 16810049, 16810561, 16814089

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 8 representations contain only 0's and 1's.

Intersection of A000040 and A033045. - Michel Marcus, Sep 14 2013

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

Cf. A020449, A000695, A033045, A077717, A077718, A077719, A077720, A077721, A077723.

isok(n) = {digs = digits(n, 8); for (i = 1, #digs, if (digs[i] > 1, return (0));); return (1);}
lista(nn) = {forprime (p=1, nn, if (isok(p), print1(p, ", ");););} \\ Michel Marcus, Sep 14 2013

forstep(n=7,999,2,t=fromdigits(binary(n),8); if(isprime(t), print1(t", "))) \\ Charles R Greathouse IV, Jun 08 2015

More terms from Francois Jooste (phukraut(AT)hotmail.com), Dec 23 2002

A077720 Primes which can be expressed as sum of distinct powers of 6.

Original entry on oeis.org

7, 37, 43, 223, 1297, 1303, 1549, 7993, 9109, 46663, 54469, 55987, 281233, 326593, 327889, 335917, 1679653, 1679659, 1679833, 1680919, 1681129, 1687393, 1726273, 1726489, 1727569, 1727827, 1734049, 1960891, 1961107, 1967587, 2006461

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 6 representation contains only zeros and 1's.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A020449, A000695, A033043, A077717, A077718, A077719, A077721, A077722.

Select[FromDigits[#,6]&/@Tuples[{0,1},9],PrimeQ] (* Harvey P. Dale, May 01 2018 *)

More terms from Sascha Kurz, Jan 03 2003

A077721 Primes which can be expressed as sum of distinct powers of 7.

Original entry on oeis.org

7, 2801, 17207, 19559, 120401, 134513, 134807, 137201, 840743, 842759, 842801, 941249, 943601, 958007, 958049, 958343, 960793, 5782001, 5784409, 5899307, 5899601, 5899657, 5901659, 6591089, 6607903, 6706393, 6708787, 6722801, 6722857, 6723193

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 7 representation contains only zeros and 1's.

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

Cf. A020449, A000695, A033044, A077717, A077718, A077719, A077720, A077722, A077723.

pos := 0:for i from 1 to 4000 do b := convert(i,base,2); s := sum(b[j]*7^(j-1),j=1..nops(b)): if(isprime(s)) then pos := pos+1:a[pos] := s:fi: od:seq(a[j],j=1..pos);

Select[Prime[Range[10^6]], Max[IntegerDigits[#, 7]]<=1 &] (* Vincenzo Librandi, Sep 07 2018 *)

More terms from Sascha Kurz, Jan 03 2003

A077719 Primes which can be expressed as sum of distinct powers of 5.

Original entry on oeis.org

5, 31, 131, 151, 631, 751, 3251, 3881, 16381, 19381, 19501, 19531, 78781, 78901, 81281, 81401, 81901, 82031, 93901, 94531, 97001, 97501, 97651, 390751, 390781, 393901, 394501, 406381, 468781, 469501, 471901, 472631, 484531, 485131, 487651, 1953151, 1953901

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 5 representation contains only zeros and 1's.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A020449, A000695, A033042, A077717, A077718, A077720, A077721, A077722.

from sympy import isprime
def aupton(terms):
  k, alst = 0, []
  while len(alst) < terms:
    k += 1
    t = sum(5**i*int(di) for i, di in enumerate((bin(k)[2:])[::-1]))
    if isprime(t): alst.append(t)
  return alst
print(aupton(37)) # Michael S. Branicky, May 31 2021

More terms from Sascha Kurz, Jan 03 2003

a(36) and beyond from Michael S. Branicky, May 31 2021

A077723 Primes which can be expressed as sum of distinct powers of 9.

Original entry on oeis.org

739, 811, 6571, 59779, 65701, 532261, 538093, 591301, 597133, 597781, 4783699, 4789621, 4842109, 4849399, 5314411, 5314501, 5373469, 5374279, 5380831, 43047541, 43112341, 43113061, 43643773, 43643863, 47837071, 47888821

Offset: 1

Amarnath Murthy, Nov 19 2002

nonn

Primes whose base 9 representation contains only zeros and 1's.

Zak Seidov, Table of n, a(n) for n = 1..3000

Cf. A020449, A000695, A033046, A077717, A077718, A077719, A077720, A077721, A077722.

Select[Prime[Range[3000000]],Union[Most[Rest[DigitCount[#,9]]]]=={0}&] (* Harvey P. Dale, Jul 31 2013 *)

lista(nn) = {forprime(p=2, nn, if (vecmax(digits(p, 9)) <= 1, print1(p, ", ")););} \\ Michel Marcus, Oct 10 2014

More terms from Sascha Kurz, Jan 03 2003

A082555 Primes whose base-3 representation does not contain a 0.

Original entry on oeis.org

2, 5, 7, 13, 17, 23, 41, 43, 53, 67, 71, 79, 131, 149, 151, 157, 211, 229, 233, 239, 241, 367, 373, 401, 449, 457, 607, 617, 619, 643, 647, 691, 701, 719, 727, 1093, 1097, 1103, 1123, 1129, 1187, 1201, 1213, 1367, 1373, 1427, 1429, 1447, 1453, 1823, 1831, 1861

Offset: 1

Randy L. Ekl, May 03 2003

Primes in A032924. - Robert Israel, Dec 28 2018

The analog "primes without digit 2 in ternary" is A077717. There is no prime > 2 not having the digit 1 in ternary, since then the number is divisible by 2. - M. F. Hasler, Feb 15 2023

			41 = 1112_3, which contains no 0.

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

Cf. A032924 (numbers without digit 0 in base 3), A073779, A077267.

Cf. A077717 (primes that are the sum of distinct powers of 3 <=> base-3 representation does not contain a digit 2).

select(t -> isprime(t) and not(has(convert(t,base,3),0)), [2,seq(i,i=5..10000,2)]); # Robert Israel, Dec 28 2018

dec3(s)=while(s>0,if(s%3==0,return(0),s=floor(s/3))); return(1)
forprime(i=1,20000,if(dec3(i)==1,print1(i,", "),))

def is_A082555(n): return is_A032924(n) and A010051(n)
[p for p in range(1888) if is_A082555(p)] # M. F. Hasler, Feb 15 2023

A077724 a(n) = smallest prime which can be expressed as a sum of distinct powers of n.

Original entry on oeis.org

2, 3, 5, 5, 7, 7, 73, 739, 11, 11, 13, 13, 197, 241, 17, 17, 19, 19, 401, 463, 23, 23, 577, 10171901, 677, 757, 29, 29, 31, 31, 32801, 1123, 1336337, 44101, 37, 37, 1483, 59359, 41, 41, 43, 43, 85229, 93151, 47, 47, 110641, 13847169701, 2551, 345157903, 53, 53

Offset: 2

Amarnath Murthy, Nov 19 2002

nonn

a(n) = smallest prime whose base n representation contains only zeros and 1's.

Values of n at which a(n) reach record values are: 2, 3, 4, 6, 8, 9, 25, 49, 91, 121, 187, 201, 301, 721, 799, 841... Notably, many of them are squares of primes. - Ivan Neretin, Sep 20 2017

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

Cf. A020449, A000695, A033046, A077717, A077718, A077719, A077720, A077721, A077722, A077723.

Table[i = p = 1; While[! PrimeQ[p], p = FromDigits[IntegerDigits[i++, 2], n]]; p, {n, 2, 53}] (* Ivan Neretin, Sep 20 2017 *)

from itertools import count
from sympy import isprime
def A077724(n): return next(filter(isprime,(sum(n**i for i, j in enumerate(bin(m)[-1:1:-1]) if j=='1') for m in count(1)))) # Chai Wah Wu, Apr 04 2025

More terms from Sascha Kurz, Jan 03 2003

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A235266 Primes whose base-2 representation is also the base-3 representation of a prime.

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A235265 Primes whose base-3 representation also is the base-2 representation of a prime.

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A077718 Primes which can be expressed as sum of distinct powers of 4.

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A077722 Primes which can be expressed as sums of distinct powers of 8.

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A077720 Primes which can be expressed as sum of distinct powers of 6.

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A077721 Primes which can be expressed as sum of distinct powers of 7.

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A077719 Primes which can be expressed as sum of distinct powers of 5.

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