A036964 -id:A036964 - cp's OEIS frontend

A065727 Primes p such that the decimal expansion of its base-9 conversion is also prime.

2, 3, 5, 7, 37, 43, 61, 109, 127, 199, 271, 277, 379, 457, 487, 523, 541, 613, 619, 673, 727, 757, 883, 907, 919, 991, 997, 1033, 1117, 1249, 1447, 1483, 1531, 1549, 1567, 1627, 1693, 1699, 1747, 1753, 1987, 2053, 2161, 2221, 2287, 2341, 2347, 2437, 2473

Offset: 1

Patrick De Geest, Nov 15 2001

In general rebase notation (Marc LeBrun): p9 = (9) [p] (10).

			E.g., 997_10 = 1327_9 is prime, and so is 1327_10.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A065720, A065721, A065722, A065723, A065724, A065725, A065726, A065361.

Intersection of A000040 and A036964.

Select[ Range[2500], PrimeQ[ # ] && PrimeQ[ FromDigits[ IntegerDigits[ #, 9]]] & ]
NestList[NestWhile[NextPrime, #, ! PrimeQ[FromDigits[IntegerDigits[#2, 9]]] &, 2] &, 2, 48] (* Jan Mangaldan, Jul 01 2020 *)
Select[Prime[Range[400]],PrimeQ[FromDigits[IntegerDigits[#,9],10]]&] (* Harvey P. Dale, Sep 19 2021 *)

isok(p) = isprime(p) && isprime(fromdigits(digits(p, 9))); \\ Michel Marcus, Jul 02 2020

A036952 Numbers whose binary expansion is a decimal prime.

Original entry on oeis.org

3, 5, 23, 47, 89, 101, 149, 157, 163, 173, 179, 185, 199, 229, 247, 253, 295, 313, 329, 331, 355, 367, 379, 383, 405, 425, 443, 453, 457, 471, 523, 533, 539, 565, 583, 587, 595, 631, 643, 647, 653, 659, 671, 675, 689, 703, 709, 755, 781, 785, 815, 841, 855

Offset: 1

Patrick De Geest, Jan 04 1999

A100051(f(a(n))) = 1 with f(x) = if x<2 then x else 10*f(floor(x/2)) + x mod 2. - Reinhard Zumkeller, Mar 31 2010

Primes in A007088. - N. J. A. Sloane, Feb 17 2023

			1 = 1_2 is not a prime.
2 = 10_2 is not OK because 10 = 2*5 is not a prime.
3 = 11_2 is OK because 11 is a prime.
4 = 100_2 is not OK because 100 = 4*25 is not a prime.
5 = 101_2 is OK because 101 is a prime.
7 = 111_2 is not OK because 111 = 3*37.
11 = 1011_2 is not OK because 1011 = 3*337.
313 = 100111001_2 is OK because 100111001 is prime.

Cf. A007088, A020449, A036953-A036964.

Union of A156059 and A065720.

A007088 := proc(n)
dgs := convert(n,base,2) ;
add(op(i,dgs)*10^(i-1),i=1..nops(dgs)) ;
end proc:
isA036952 := proc(n)
isprime( A007088(n)) :
end proc:
A036952 := proc(n)
if n =1 then
3;
else
for a from procname(n-1)+1 do
if isA036952(a) then
return a ;
end if;
end do:
end if;
end proc:
seq(A036952(n),n=1..80) ;
# R. J. Mathar, Mar 12 2010
A036952 := proc() if isprime(convert(n,binary)) then RETURN (n); fi; end: seq(A036952(), n=1..1000);  # K. D. Bajpai, Jul 04 2014

f[n_,k_]:=FromDigits[IntegerDigits[n,k]];lst={};Do[If[PrimeQ[f[n,2]],AppendTo[lst,n]],{n,7!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 12 2010 *)
NestList[NestWhile[# + 2 &, #, ! PrimeQ[FromDigits[IntegerDigits[#2, 2]]] &, 2] &, 3, 52] (* Jan Mangaldan, Jul 02 2020 *)

is(n)=my(v=binary(n));isprime(sum(i=1,#v,v[i]*10^(#v-i))) \\ Charles R Greathouse IV, Jun 28 2013

Entry revised by R. J. Mathar and N. J. A. Sloane, Mar 12 2010

A036953 Primes having only {0, 1, 2} as digits.

Original entry on oeis.org

2, 11, 101, 211, 1021, 1201, 2011, 2111, 2221, 10111, 10211, 12011, 12101, 12211, 20011, 20021, 20101, 20201, 21001, 21011, 21101, 21121, 21211, 21221, 22111, 101021, 101111, 101221, 102001, 102101, 102121, 110221, 111121, 111211, 112111

Offset: 1

Patrick De Geest, Jan 04 1999

Number of n-digit terms d(n) = (1, 1, 2, 5, 16, 34, 76, 194, 543, 1469, 4094, 11017, ...); e.g., there are five 4-digit terms: 1021, 1201, 2011, 2111, 2221, hence d(4) = 5. - Zak Seidov, Jun 30 2013

Also, primes in A007089. - M. F. Hasler, Jul 25 2015

Zak Seidov, Table of n, a(n) for n = 1..10000
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Index to entries for primes with digits in a given set

Subsequence of A107715.
Cf. A036952-A036964.
Cf. A020450 - A020472, A260044, A260267 - A260271, A199325 - A199329, A061247, A199340 - A199349, A217039, A079651.

Select[FromDigits/@Tuples[{0,1,2},6],PrimeQ] (* Harvey P. Dale, Jul 11 2017 *)

lista(n) = {forprime(p=2, n, if (vecmax(digits(p)) <= 2, print1(p, ", ")))} \\ Michel Marcus, Aug 02 2014

A036953={(n,show=0)->for(d=1,1e9,my(u=vector(d,i,10^(d-i))~);forvec(v=vector(d,i,if(i>1,if(iM. F. Hasler, Jul 25 2015

from gmpy2 import digits
from sympy import isprime
[int(digits(n,3)) for n in range(1000) if isprime(int(digits(n,3)))] # Chai Wah Wu, Jul 31 2014

Edited by M. F. Hasler, Jul 25 2015

A036956 Primes containing only digits from the set (0,1,2,3,4).

Original entry on oeis.org

2, 3, 11, 13, 23, 31, 41, 43, 101, 103, 113, 131, 211, 223, 233, 241, 311, 313, 331, 401, 421, 431, 433, 443, 1013, 1021, 1031, 1033, 1103, 1123, 1201, 1213, 1223, 1231, 1301, 1303, 1321, 1423, 1433, 2003, 2011, 2111, 2113, 2131, 2141, 2143, 2203, 2213

Offset: 1

Patrick De Geest, Jan 04 1999

Jason Bard, Table of n, a(n) for n = 1..10000
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019).

Subsequence of A036958.

Cf. A036952-A036964.

Offet 1 from Michel Marcus, Oct 10 2019

A036954 Primes with digits in {0,1,2} taken as base 3 and converted to base 10.

Original entry on oeis.org

2, 4, 10, 22, 34, 46, 58, 67, 79, 94, 103, 139, 145, 157, 166, 169, 172, 181, 190, 193, 199, 205, 211, 214, 229, 277, 283, 295, 298, 307, 313, 349, 367, 373, 391, 394, 409, 421, 433, 439, 463, 466, 478, 505, 517, 523, 529, 535, 541, 547, 556, 559, 571, 577

Offset: 1

Patrick De Geest, Jan 04 1999

Equivalently: terms of A036953 read in base 3 (and written in base 10). - M. F. Hasler, Jul 25 2015

Equivalently, k such that A007089(k), read literally as a decimal number, is a prime. - N. J. A. Sloane, Feb 17 2023

			a(n) = 313 is 102121{3}, and 102121{10} is prime.

Cf. A036952-A036964.

Indices of primes in A007089.

FromDigits[#,3]&/@Select[Tuples[{0,1,2},6],PrimeQ[FromDigits[#]]&] (* Harvey P. Dale, Mar 27 2021 *)

is(n)=(n%3==1||n==2)&&isprime((n=digits(n,3))*vectorv(#n,i,10^(#n-i))) \\ M. F. Hasler, Jul 25 2015

a(n) == 1 (mod 3) for all n > 1. - M. F. Hasler, Jul 25 2015

Offset corrected to 1 and minor edits by M. F. Hasler, Jul 25 2015

A036958 Primes containing only digits from the set (0,1,2,3,4,5).

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 31, 41, 43, 53, 101, 103, 113, 131, 151, 211, 223, 233, 241, 251, 311, 313, 331, 353, 401, 421, 431, 433, 443, 503, 521, 523, 541, 1013, 1021, 1031, 1033, 1051, 1103, 1123, 1151, 1153, 1201, 1213, 1223, 1231, 1301, 1303, 1321

Offset: 1

Patrick De Geest, Jan 04 1999

Harvey P. Dale, Table of n, a(n) for n = 1..1000 [offset shifted by _Georg Fischer_, Oct 14 2019]
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)

Subsequence of A036960.

Cf. A036952-A036964.

Select[Prime[Range[220]],Max[IntegerDigits[#]]<6&] (* Harvey P. Dale, Sep 23 2011 *)

Offset 1 from Michel Marcus, Oct 10 2019

A036960 Primes containing only digits from the set (0,1,2,3,4,5,6).

Original entry on oeis.org

2, 3, 5, 11, 13, 23, 31, 41, 43, 53, 61, 101, 103, 113, 131, 151, 163, 211, 223, 233, 241, 251, 263, 311, 313, 331, 353, 401, 421, 431, 433, 443, 461, 463, 503, 521, 523, 541, 563, 601, 613, 631, 641, 643, 653, 661, 1013, 1021, 1031, 1033, 1051, 1061, 1063

Offset: 1

Patrick De Geest, Jan 04 1999

James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)

Subsequence of A036962.

Cf. A036952-A036964.

Select[Prime[Range[200]],Max[IntegerDigits[#]]<7&] (* Harvey P. Dale, Feb 08 2019 *)

Offset 1 from Michel Marcus, Oct 10 2019

A036961 Primes with digits (0,...,6) taken as base 7 and converted to base 10.

Original entry on oeis.org

2, 3, 5, 8, 10, 17, 22, 29, 31, 38, 43, 50, 52, 59, 71, 85, 94, 106, 115, 122, 127, 134, 143, 155, 157, 169, 185, 197, 211, 218, 220, 227, 239, 241, 248, 260, 262, 274, 290, 295, 304, 316, 323, 325, 332, 337, 353, 358, 365, 367, 379, 386, 388, 395, 409, 428

Offset: 1

Patrick De Geest, Jan 04 1999

			a(n)=323 -> is 641{7} -> 641{10} is prime.

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

Cf. A036952-A036964.

pd7[n_]:=With[{c=IntegerDigits[n]},If[Max[c]<7,FromDigits[c,7],Nothing]]; pd7/@Prime[Range[300]] (* Harvey P. Dale, Mar 14 2025 *)

A036962 Primes without {8, 9} as digits.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 47, 53, 61, 67, 71, 73, 101, 103, 107, 113, 127, 131, 137, 151, 157, 163, 167, 173, 211, 223, 227, 233, 241, 251, 257, 263, 271, 277, 307, 311, 313, 317, 331, 337, 347, 353, 367, 373, 401, 421, 431, 433, 443, 457, 461

Offset: 1

Patrick De Geest, Jan 04 1999

Harvey P. Dale, Table of n, a(n) for n = 1..10000
James Maynard, Primes with restricted digits, arXiv:1604.01041 [math.NT], 2016.
James Maynard and Brady Haran, Primes without a 7, Numberphile video (2019)
Index to entries for primes with digits in a given set

Subsequence of A038617.

Cf. A036952-A036964.

Select[FromDigits/@Tuples[Range[0,7],3],PrimeQ] (* Harvey P. Dale, Apr 13 2017 *)

A036963 Primes with digits (0,...,7) taken as base 8 and converted to base 10.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 15, 19, 25, 31, 33, 35, 39, 43, 49, 55, 57, 59, 65, 67, 71, 75, 87, 89, 95, 105, 111, 115, 119, 123, 137, 147, 151, 155, 161, 169, 175, 179, 185, 191, 199, 201, 203, 207, 217, 223, 231, 235, 247, 251, 257, 273, 281, 283, 291, 303, 305, 307, 311

Offset: 1

Patrick De Geest, Jan 04 1999

			a(n)=191 -> is 277{8} -> 277{10} is prime.

Cf. A036952-A036964.

Offset 1 from Michel Marcus, Oct 10 2019

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A065727 Primes p such that the decimal expansion of its base-9 conversion is also prime.

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A036952 Numbers whose binary expansion is a decimal prime.

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A036953 Primes having only {0, 1, 2} as digits.

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A036956 Primes containing only digits from the set (0,1,2,3,4).

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A036954 Primes with digits in {0,1,2} taken as base 3 and converted to base 10.

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A036958 Primes containing only digits from the set (0,1,2,3,4,5).

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A036960 Primes containing only digits from the set (0,1,2,3,4,5,6).

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A036961 Primes with digits (0,...,6) taken as base 7 and converted to base 10.

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