A077722 -id:A077722 - cp's OEIS frontend

A079107 Number of digits of A077722(n) written in base 8.

3, 4, 4, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

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

			a(5)=6 since A077722(5)=32833, which has an octal form of 100101 (6 digits).

Cf. A077722.

s := 0:for n from 1 to 7000 do b := convert(n,base,2): q := sum(b[i]*8^(i-1),i=1..nops(b)): if(isprime(q)) then s := s+1:a[s] := nops(b):fi: od:seq(a[k],k=1..s);

More terms from Sascha Kurz, Feb 10 2003

A079108 Number of 1's in the base 8 form of A077722(n).

Original entry on oeis.org

3, 3, 3, 4, 3, 4, 5, 4, 5, 3, 4, 3, 5, 5, 5, 4, 3, 4, 5, 5, 5, 4, 5, 5, 4, 3, 4, 5, 5, 6, 6, 4, 5, 6, 8, 4, 5, 6, 6, 5, 3, 4, 4, 5, 6, 5, 3, 4, 6, 5, 6, 8, 6, 5, 4, 6, 8, 5, 5, 5, 6, 6, 8, 3, 4, 5, 4, 6, 6, 5, 6, 8, 4, 5, 5, 6, 5, 8, 4, 6, 8, 6, 8, 6, 8, 8, 8, 5, 6, 6, 5, 5, 5, 8, 4, 6, 6, 8, 8, 9, 6, 8, 5, 5, 4

Offset: 1

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

nonn

			a(5)=3 since A077722(5)=32833, which has an octal form of 100101, which has 3 ones.

Cf. A077722.

s := 0:for n from 1 to 7000 do b := convert(n,base,2):q := sum(b[i]*8^(i-1),i=1..nops(b)): if(isprime(q)) then s := s+1:a[s] := sum(b[i],i=1..nops(b)):fi:od:seq(a[k],k=1..s);

More terms from Sascha Kurz, Feb 10 2003

A079109 Number of zeros in the base 8 form of A077722(n).

Original entry on oeis.org

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

Offset: 1

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

nonn

			a(5)=3 since A077722(5)=32833 which has octal form 100101, which has 3 zeros.

Cf. A077722.

More terms from Sascha Kurz, Jan 02 2003

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

A077717 Primes which can be expressed as a sum of distinct powers of 3.

Original entry on oeis.org

3, 13, 31, 37, 109, 271, 283, 337, 733, 739, 757, 769, 811, 823, 1009, 1063, 1093, 2269, 2281, 2467, 2521, 2539, 2551, 2917, 2953, 3001, 3037, 3163, 3169, 3187, 3253, 3271, 6571, 6673, 6679, 6841, 7321, 7411, 7537, 7561, 7573, 8761, 8779, 8839, 9001

Offset: 1

Amarnath Murthy, Nov 19 2002

Primes whose base 3 representation contains only 0's and 1's.

			31 = 3^3 + 3 + 1 belongs to this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A020449, A077718, A077719, A077720, A077721, A077722, A077724.

Select[FromDigits[#,3]&/@Tuples[{0,1},10],PrimeQ] (* Harvey P. Dale, Mar 30 2015 *)

print1(3); forstep(n=3,1e3,2, if(isprime(t=fromdigits(binary(n),3)), print1(", "t))) \\ Charles R Greathouse IV, Mar 28 2022

is_A077717(n)=vecmax(digits(n,3))<2 && isprime(n)
select(is_A077717, [1..9111]) \\ M. F. Hasler, Feb 15 2023

def is_A077717(n): return A039966(n) and A010051(n) # M. F. Hasler, Feb 15 2023

More terms from John W. Layman, Nov 22 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

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

cp's OEIS Frontend

A079107 Number of digits of A077722(n) written in base 8.

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A079108 Number of 1's in the base 8 form of A077722(n).

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A079109 Number of zeros in the base 8 form of A077722(n).

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

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A077717 Primes which can be expressed as a sum of distinct powers of 3.

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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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