cp's OEIS Frontend

A089392 Magnanimous primes: primes with the property that inserting a "+" in any place between two digits yields a sum which is prime.

%I A089392 #48 Dec 23 2024 14:53:42
%S A089392 2,3,5,7,11,23,29,41,43,47,61,67,83,89,101,227,229,281,401,443,449,
%T A089392 467,601,607,647,661,683,809,821,863,881,2221,2267,2281,2447,4001,
%U A089392 4027,4229,4463,4643,6007,6067,6803,8009,8221,8821,20261,24407,26881,28429,40427
%N A089392 Magnanimous primes: primes with the property that inserting a "+" in any place between two digits yields a sum which is prime.
%C A089392 Original definition: Let the digits of n be abcd. Then bcd+a, cd+ab, d+abc, abcd, etc. must all be primes. If n is a k-digit number then it must produce k such primes.
%C A089392 Partition the digits of n into two groups by placing a '+' sign anywhere inside; the result of the expression is prime in every case. Conjecture: sequence is infinite. 11 is the largest term with all odd digits. 2 is the only member with all even digits. Observation: all two-digit primes with the most significant digit even are members.
%C A089392 In contradiction to the above conjecture, it is rather expected that this sequence is finite, cf. the link to C. Rivera's "Puzzle 401", and G. Resta's web page. Concerning the statement about 2 and 11, one can say that all terms except 2, 11 and 101 consist of even digits followed by a final odd digit. - _M. F. Hasler_, Dec 25 2014
%C A089392 Primes among the magnanimous numbers A252996. - _M. F. Hasler_, Dec 25 2014
%H A089392 Zak Seidov, <a href="/A089392/b089392.txt">Table of n, a(n) for n = 1..84</a>
%H A089392 Eric Angelini et al., <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2014-December/014168.html">Insert "+" and always get a prime</a>, Dec
%H A089392 2014
%H A089392 Giovanni Resta, <a href="http://www.numbersaplenty.com/set/magnanimous_number/">magnanimous numbers</a>, 2013.
%H A089392 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_401.htm">Puzzle 401. Magnanimous primes</a>, 2007, The Prime Puzzles & Problems Connection.
%e A089392 2267 is a member which gives primes 2+267 = 269, 22+67 = 89, 226+7 = 233 and 2267 itself.
%p A089392 with(combinat): ds:=proc(s) local j: RETURN(add(s[j]*10^(j-1),j=1..nops(s))):end: for d from 1 to 6 do sch:=[seq([1,op(i),d+1],i=[[],seq([j],j=2..d)])]: for n from 10^(d-1) to 10^d-1 do sn:=convert(n,base,10): fl:=0: for s in sch do m:=add(j,j=[seq(ds(sn[s[i]..s[i+1]-1]),i=1..nops(s)-1)]): if not isprime(m) then fl:=1: break fi od: if fl=0 then printf("%d, ",n) fi od od: # C. Ronaldo
%t A089392 mpQ[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];And@@PrimeQ[ Table[ FromDigits[Take[idn,i]]+FromDigits[Take[idn,-(len-i)]],{i,len}]]]; Select[Range[41000],mpQ] (* _Harvey P. Dale_, Nov 06 2013 *)
%o A089392 (PARI) is_A089392(n)={!for(i=1,#Str(n),ispseudoprime([1,1]*(divrem(n,10^i)))||return)} \\ _M. F. Hasler_, Dec 25 2014
%o A089392 (Python)
%o A089392 from sympy import isprime
%o A089392 def ok(n):
%o A089392     if not isprime(n): return False
%o A089392     s = str(n)
%o A089392     return all(isprime(int(s[:i])+int(s[i:])) for i in range(1, len(s)))
%o A089392 print([k for k in range(357) if ok(k)]) # _Michael S. Branicky_, Oct 14 2024
%Y A089392 Cf. A089393, A089394, A089695, A028834, A182175, A088134, A221699, A227823, A252996.
%K A089392 base,nonn
%O A089392 1,1
%A A089392 _Amarnath Murthy_, Nov 10 2003
%E A089392 Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 25 2004
%E A089392 Comments edited by _Zak Seidov_, Jan 29 2013
%E A089392 Edited by _M. F. Hasler_, Dec 25 2014