A077720 -id:A077720 - cp's OEIS frontend

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

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

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

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

A235463 Primes whose base-6 representation also is the base-2 representation of a prime.

Original entry on oeis.org

7, 37, 43, 223, 1297, 1303, 1549, 7993, 9109, 46663, 54469, 55987, 326593, 1679659, 1681129, 1727569, 1734049, 1967587, 2006461, 2007763, 2014027, 2015287, 10077919, 10125649, 10125691, 10133467, 10412107, 10413397, 11757349, 11766421, 11766427, 11766637

Offset: 1

M. F. Hasler, Jan 11 2014

This sequence is part of the two-dimensional array of sequences based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.
For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.
When the smaller base is b=2 such that only digits 0 and 1 are allowed, these are primes that are the sum of distinct powers of the larger base, here c=6, thus a subsequence of A077720.

			7 = 11_6 and 11_2 = 3 are both prime, so 7 is a term.
37 = 101_6 and 101_2 = 5 are both prime, so 37 is a term.

Alois P. Heinz, 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. A065720 ⊂ A036952, A065721 - A065727, A235394, A235395, A089971 ⊂ A020449, A089981, A090707 - A091924, A235461 - A235482. See the LINK for further cross-references.

  b62Q[n_]:=Module[{idn6=IntegerDigits[n,6]},Max[idn6]<2&&AllTrue[ {FromDigits[ idn6,6],FromDigits[idn6,2]},PrimeQ]]; Select[Prime[ Range[ 4,780000]],b62Q] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 29 2020 *)

is(p,b=2,c=6)=vecmax(d=digits(p,c))

forprime(p=1,1e3,is(p,6,2)&&print1(vector(#d=digits(p,2),i,6^(#d-i))*d~,",")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(.,2,6)

A235476 Primes whose base-2 representation also is the base-6 representation of a prime.

Original entry on oeis.org

3, 5, 7, 11, 17, 19, 29, 41, 53, 67, 101, 127, 193, 263, 281, 337, 353, 431, 461, 479, 487, 499, 523, 593, 599, 631, 743, 757, 773, 821, 823, 829, 857, 883, 887, 941, 1013, 1021, 1093, 1117, 1259, 1279, 1303, 1367, 1373, 1429, 1439, 1459, 1471, 1483, 1493, 1511, 1583, 1619, 1699, 1759, 1831, 1847, 1879, 1931, 1951, 1987

Offset: 1

M. F. Hasler, Jan 12 2014

This sequence is part of a two-dimensional array of sequences, given in the LINK, based on this same idea for any two different bases b, c > 1. Sequence A235265 and A235266 are the most elementary ones in this list. Sequences A089971, A089981 and A090707 through A090721, and sequences A065720 - A065727, follow the same idea with one base equal to 10.

For further motivation and cross-references, see sequence A235265 which is the main entry for this whole family of sequences.

			5 = 101_2 and 101_6 = 37 are both prime, so 5 is a term.
7 = 111_2 and 111_6 = 43 are both prime, so 7 is a term.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

Cf. A235463 ⊂ A077720, A235475, A152079, A235266, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

Select[Prime[Range[300]],PrimeQ[FromDigits[IntegerDigits[#,2],6]]&] (* Harvey P. Dale, Jan 03 2022 *)

is(p,b=6)=isprime(vector(#d=binary(p),i,b^(#d-i))*d~)&&isprime(p)

cp's OEIS Frontend

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

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

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

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