A077722 -id:A077722

A079107

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..83.

EXAMPLE

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

MAPLE

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);

CROSSREFS

Cf. A077722.

KEYWORD

nonn,base

AUTHOR

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

EXTENSIONS

More terms from Sascha Kurz, Feb 10 2003

STATUS

approved

A079108

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..105.

EXAMPLE

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

MAPLE

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);

CROSSREFS

Cf. A077722.

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

More terms from Sascha Kurz, Feb 10 2003

STATUS

approved

A079109

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

+20
0

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,5

LINKS

Table of n, a(n) for n=1..105.

EXAMPLE

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

CROSSREFS

Cf. A077722.

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

More terms from Sascha Kurz, Jan 02 2003

STATUS

approved

A077718

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

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Nov 19 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

STATUS

approved

A077717

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

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

(PARI) is_A077717(n)=vecmax(digits(n, 3))<2 && isprime(n)

select(is_A077717, [1..9111]) \\ M. F. Hasler, Feb 15 2023

(Python)

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Amarnath Murthy, Nov 19 2002

EXTENSIONS

More terms from John W. Layman, Nov 22 2002

STATUS

approved

A077720

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Nov 19 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

STATUS

approved

A077721

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

MAPLE

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);

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Nov 19 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

STATUS

approved

A077719

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

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

PROG

(Python)

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Nov 19 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

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

STATUS

approved

A077723

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Nov 19 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

STATUS

approved

A077724

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

+10
3

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

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

LINKS

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Nov 19 2002

EXTENSIONS

More terms from Sascha Kurz, Jan 03 2003

STATUS

approved