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A016041 - OEIS
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>A016041 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A016041" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fA016041">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="A016041 - OEIS"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A016041 </div> <div class=seqname> Primes that are palindromic in base 2 (but written here in base 10). </div> </div> <div class=scorerefs> 35 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>3, 5, 7, 17, 31, 73, 107, 127, 257, 313, 443, 1193, 1453, 1571, 1619, 1787, 1831, 1879, 4889, 5113, 5189, 5557, 5869, 5981, 6211, 6827, 7607, 7759, 7919, 8191, 17377, 18097, 18289, 19433, 19609, 19801, 21157, 22541, 22669, 22861, 23581, 24029</div> <div class=seqdatalinks> (<a href="/A016041/list">list</a>; <a href="/A016041/graph">graph</a>; <a href="/search?q=A016041+-id:A016041">refs</a>; <a href="/A016041/listen">listen</a>; <a href="/history?seq=A016041">history</a>; <a href="/search?q=id:A016041&fmt=text">text</a>; <a href="/A016041/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>See <a href="/A002385" title="Palindromic primes: prime numbers whose decimal expansion is a palindrome.">A002385</a> for palindromic primes in base 10, and <a href="/A256081" title="Non-palindromic balanced primes in base 2.">A256081</a> for primes whose binary expansion is "balanced" (see there) but not palindromic. - <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Mar 14 2015</div> <div class=sectline>Number of terms less than 4^k, k=1,2,3,...: 1, 3, 5, 8, 11, 18, 30, 53, 93, 187, 329, 600, 1080, 1936, 3657, 6756, 12328, 23127, 43909, 83377, 156049, 295916, 570396, 1090772, 2077090, 3991187, 7717804, 14825247, 28507572, 54938369, 106350934, ..., partial sums of <a href="/A095741" title="Number of base-2 palindromic primes (A016041) in range [2^2n,2^(2n+1)].">A095741</a> plus 1. - <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Feb 23 2018, corrected by <a href="/wiki/User:Jeppe_Stig_Nielsen">Jeppe Stig Nielsen</a>, Jun 17 2023</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Robert G. Wilson v, <a href="/A016041/b016041.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Zak Seidov, terms 1001..3000 from Michael De Vlieger)</div> <div class=sectline>Kevin S. Brown, <a href="http://www.mathpages.com/home/kmath359.htm">On General Palindromic Numbers</a></div> <div class=sectline>Patrick De Geest, <a href="http://www.worldofnumbers.com/palpri.htm">World!Of Palindromic Primes</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>Sum_{n>=1} 1/a(n) = <a href="/A194097" title="Decimal expansion of the sum of the reciprocals of A016041 (primes that are binary palindromes).">A194097</a>. - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Mar 19 2021</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>lst = {}; Do[ If[ PrimeQ@n, t = IntegerDigits[n, 2]; If[ FromDigits@t == FromDigits@ Reverse@ t, AppendTo[lst, n]]], {n, 3, 50000, 2}]; lst (* syntax corrected by <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Aug 10 2009 *)</div> <div class=sectline>pal2Q[n_] := Reverse[x = IntegerDigits[n, 2]] == x; Select[Prime[Range[2800]], pal2Q[#] &] (* <a href="/wiki/User:Jayanta_Basu">Jayanta Basu</a>, Jun 23 2013 *)</div> <div class=sectline>genPal[n_Integer, base_Integer: 10] := Block[{id = IntegerDigits[n, base], insert = Join[{{}}, {# - 1} & /@ Range[base]]}, FromDigits[#, base] & /@ (Join[id, #, Reverse@id] & /@ insert)]; k = 0; lst = {}; While[k < 100, AppendTo[lst, Select[ genPal[k, 2], PrimeQ]]; lst = Flatten@ lst; k++]; lst (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Feb 23 2018 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) is(n)=isprime(n)&&Vecrev(n=binary(n))==n \\ <a href="/wiki/User:M._F._Hasler">M. F. Hasler</a>, Feb 23 2018</div> <div class=sectline>(Magma) [NthPrime(n): n in [1..5000] | (Intseq(NthPrime(n), 2) eq Reverse(Intseq(NthPrime(n), 2)))]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Feb 24 2018</div> <div class=sectline>(Python)</div> <div class=sectline>from sympy import isprime</div> <div class=sectline>def ok(n): return isprime(n) and (b:=bin(n)[2:]) == b[::-1]</div> <div class=sectline>print([k for k in range(10**5) if ok(k)]) # <a href="/wiki/User:Michael_S._Branicky">Michael S. Branicky</a>, Apr 20 2024</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Intersection of <a href="/A000040" title="The prime numbers.">A000040</a> and <a href="/A006995" title="Binary palindromes: numbers whose binary expansion is palindromic.">A006995</a>.</div> <div class=sectline>First row of <a href="/A095749" title="Square array A(row>=1, col>=1) by antidiagonals: A(r,c) contains the c:th prime p for which A037888(p)=(r-1).">A095749</a>.</div> <div class=sectline><a href="/A095741" title="Number of base-2 palindromic primes (A016041) in range [2^2n,2^(2n+1)].">A095741</a> gives the number of terms in range [2^(2n), 2^(2n+1)].</div> <div class=sectline>Cf. <a href="/A095730" title="Primes p whose Zeckendorf-expansion A014417(p) is palindromic.">A095730</a> (primes whose Zeckendorf expansion is palindromic), <a href="/A029971" title="Palindromic primes in base 3.">A029971</a> (primes whose ternary (base-3) expansion is palindromic).</div> <div class=sectline>Cf. <a href="/A117697" title="Palindromic primes in base 2 (written in base 2).">A117697</a> (written in base 2), <a href="/A002385" title="Palindromic primes: prime numbers whose decimal expansion is a palindrome.">A002385</a>, <a href="/A194097" title="Decimal expansion of the sum of the reciprocals of A016041 (primes that are binary palindromes).">A194097</a>, <a href="/A256081" title="Non-palindromic balanced primes in base 2.">A256081</a>.</div> <div class=sectline>Sequence in context: <a href="/A370686" title="a(n) is the number of 132-avoiding permutations p so that p^3 is the identity permutation.">A370686</a> <a href="/A174394" title="Fourth root of largest n-digit number with exactly five divisors">A174394</a> <a href="/A057476" title="Numbers k such that x^k + x^6 + 1 is irreducible over GF(2).">A057476</a> * <a href="/A140797" title="Numbers of the form (2^p^N-1)/(2^p^(N-1)-1), where N>0, p is prime.">A140797</a> <a href="/A245730" title="Primes of the form 1+2^k+2^(2*k)+...+2^((n-1)*k) for some k>0, n>0.">A245730</a> <a href="/A038893" title="Odd primes p such that 21 is a square mod p.">A038893</a></div> <div class=sectline>Adjacent sequences: <a href="/A016038" title="Strictly non-palindromic numbers: n is not palindromic in any base b with 2 <= b <= n-2.">A016038</a> <a href="/A016039" title="Inverse of 2030th cyclotomic polynomial.">A016039</a> <a href="/A016040" title="Integer part of Chebyshev's theta function: floor( log(Product_{k=1..n} prime(k)) ).">A016040</a> * <a href="/A016042" title="Inverse of 2033rd cyclotomic polynomial.">A016042</a> <a href="/A016043" title="2^(2^n) +- 1 without repeats.">A016043</a> <a href="/A016044" title="Inverse of 2035th cyclotomic polynomial.">A016044</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a sequence of nonnegative numbers">nonn</span>,<span title="it is very easy to produce terms of sequence">easy</span>,<span title="dependent on base used for sequence">base</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>More terms from <a href="/wiki/User:Patrick_De_Geest">Patrick De Geest</a></div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified February 26 17:46 EST 2025. Contains 381235 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>