A077723 -id:A077723

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A077722

Primes which can be expressed as sums of distinct powers of 8.

+10
13

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

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

OFFSET

1,1

COMMENTS

Primes whose base 8 representations contain only 0's and 1's.

Intersection of A000040 and A033045. - Michel Marcus, Sep 14 2013

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

PROG

(PARI) 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

(PARI) forstep(n=7, 999, 2, t=fromdigits(binary(n), 8); if(isprime(t), print1(t", "))) \\ Charles R Greathouse IV, Jun 08 2015

CROSSREFS

Cf. A020449, A000695, A033045, A077717, A077718, A077719, A077720, A077721, A077723.

KEYWORD

nonn

AUTHOR

Amarnath Murthy, Nov 19 2002

EXTENSIONS

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

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

A235479

Primes whose base-2 representation also is the base-9 representation of a prime.

+10
6

11, 13, 19, 41, 79, 109, 137, 151, 167, 191, 193, 199, 227, 239, 271, 307, 313, 421, 431, 433, 457, 487, 491, 521, 563, 613, 617, 659, 677, 709, 727, 757, 929, 947, 1009, 1033, 1051, 1249, 1483, 1693, 1697, 1709, 1721, 1831, 1951, 1979, 1987, 1993

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

OFFSET

1,1

COMMENTS

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.

A subsequence of A027697, A050150, A062090 and A176620.

LINKS

Robert Price, Table of n, a(n) for n = 1..1958

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

EXAMPLE

11 = 1011_2 and 1011_9 = 6571 are both prime, so 11 is a term.

PROG

(PARI) is(p, b=9)=isprime(vector(#d=binary(p), i, b^(#d-i))*d~)&&isprime(p)

CROSSREFS

Cf. A235466 ⊂ A077723, A235266, A152079, A235475 - A235478, A065720 ⊂ A036952, A065721 - A065727, A089971 ⊂ A020449, A089981, A090707 - A091924, A235394, A235395, A235461 - A235482. See the LINK for further cross-references.

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Jan 12 2014

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

A235466

Primes whose base-9 representation also is the base-2 representation of a prime.

+10
2

739, 811, 6571, 59779, 532261, 591301, 4783699, 4789621, 4842109, 4849399, 5314411, 5314501, 5373469, 5374279, 43047541, 43112341, 43113061, 47888821, 47889559, 47895301, 48361861, 48420271, 48420919, 387421219, 387486109, 388011061, 388011709, 392210029, 392262589, 392734981

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

OFFSET

1,1

COMMENTS

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=9, thus a subsequence of A077723.

LINKS

Zak Seidov, Table of n, a(n) for n = 1..1400

M. F. Hasler, Primes whose base c expansion is also the base b expansion of a prime

EXAMPLE

739 = 1011_9 and 1011_2 = 11 are both prime, so 739 is a term.

MATHEMATICA

fQ[n_, j_, k_] := Block[{id = IntegerDigits[n, j]}, Max[id] < k && PrimeQ[ FromDigits[ id, k]]]; lst = {}; p = 2; While[p < 4*10^9, If[ fQ[p, 9, 2], AppendTo[lst, p]; Print[p]]; p = NextPrime@ p] (* Robert G. Wilson v, Oct 09 2014 *)

pr9Q[n_]:=Module[{idn9=IntegerDigits[n, 9]}, Max[idn9]<2&&PrimeQ[ FromDigits[ idn9, 2]]]; Select[Prime[Range[21*10^6]], pr9Q] (* Harvey P. Dale, Aug 25 2015 *)

PROG

(PARI) is(p, b=2, c=9)=vecmax(d=digits(p, c))<b&&isprime(vector(#d, i, b^(#d-i))*d~)&&isprime(p)

(PARI) forprime(p=1, 1e3, is(p, 9, 2)&&print1(vector(#d=digits(p, 2), i, 9^(#d-i))*d~, ", ")) \\ To produce the terms, this is much more efficient than to select them using straightforwardly is(.)=is(., 2, 9)

CROSSREFS

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

KEYWORD

nonn,base

AUTHOR

M. F. Hasler, Jan 11 2014

STATUS

approved

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