A073779 -id:A073779 - cp's OEIS frontend

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

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

A073780 Number of 1's in base 3 representation of n-th prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 3, 3, 3, 1, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 3, 1, 3, 3, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 3, 3, 3, 5, 3, 3, 3, 3, 1, 3, 5, 3, 3, 3, 3, 5, 5, 3, 3, 3, 3, 3, 3, 3, 3, 1, 3, 3, 3, 3, 3

Offset: 1

Zak Seidov Aug 11 2002

			a1(11)=3 as 11th prime, 31, is 1011 in base 3 representation.

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

Cf. A073779, A073781.

a1[n_] := Length[Cases[IntegerDigits[Prime[n], 3], 1]]
Table[DigitCount[n,3,1],{n,Prime[Range[100]]}] (* Harvey P. Dale, Feb 23 2015 *)

a(n) = A062756(A000040(n)). - Michel Marcus, Oct 02 2013

A073781 Number of 2's in base-3 representation of n-th prime.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 0, 0, 1, 1, 2, 3, 2, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 0, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 3, 3, 2, 4, 4, 4, 2, 1, 2, 3, 0, 1, 1, 0, 2, 1, 2, 2, 3, 1, 0, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 3

Offset: 1

Zak Seidov Aug 11 2002

			a(7)=2 as 7th prime 17 is 122 in base-3 representation.

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

Cf. A073779, A073780.

a2[n_] := Length[Cases[IntegerDigits[Prime[n], 3], 2]]
DigitCount[Prime[Range[90]],3,2] (* Harvey P. Dale, May 11 2016 *)

a(n) = A081603(A000040(n)). - Michel Marcus, Oct 02 2013

A181172 Primes whose base 4 representation does not contain a 0.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 89, 101, 103, 107, 109, 127, 149, 151, 157, 167, 173, 181, 191, 223, 229, 233, 239, 251, 347, 349, 359, 367, 373, 379, 383, 409, 421, 431, 439, 443, 479, 487, 491, 503, 509, 599, 601, 607, 613, 617, 619

Offset: 1

Jonathan D. B. Hodgson, Oct 08 2010

This sequence contains all Mersenne primes (i.e. this is a supersequence of A000668). - Iain Fox, Dec 25 2017

			53 = 311 (base 4), which contains no 0.

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

Cf. A082555, A000668 (subsequence).

Cf. A073779 (number of 0's in base-3 representation of n-th prime), A181173 (primes whose base 5 representation does not contain a 0). - Klaus Brockhaus, Oct 10 2010

[ p: p in PrimesUpTo(620) | not exists(t){d: d in Intseq(p, 4) | d eq 0 } ]; // Klaus Brockhaus, Oct 10 2010

The following code will store the first 200 terms into a sequence K. for i from 1 to 200 do if i=i then x[i]:=convert(ithprime(i),base,4) else x[i]:=0 end if: end do: S:={}: for i from 1 to 200 do if evalb(`in`(0, x[i]))=false then S:=S union {i} fi od; for i from 1 to nops(S)do z[i]:=ithprime(S[i]) od: K:=[seq((z[i]),i=1..nops(S))];
# Alternative:
select(t -> isprime(t) and not has(convert(t,base,4),0), [2,seq(i,i=3..10^4,2)]); # Robert Israel, Dec 24 2017

Select[Prime@ Range@ 120, DigitCount[#, 4, 0] == 0 &] (* Michael De Vlieger, Dec 24 2017 *)

lista(nn) = forprime(p=2, nn, if(!setsearch(Set(digits(p, 4)), 0), print1(p, ", "))) \\ Iain Fox, Dec 25 2017

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

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A073780 Number of 1's in base 3 representation of n-th prime.

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A073781 Number of 2's in base-3 representation of n-th prime.

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A181172 Primes whose base 4 representation does not contain a 0.

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