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

2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89, 101, 227, 229, 281, 401, 443, 449, 467, 601, 607, 647, 661, 683, 809, 821, 863, 881, 2221, 2267, 2281, 2447, 4001, 4027, 4229, 4463, 4643, 6007, 6067, 6803, 8009, 8221, 8821, 20261, 24407, 26881, 28429, 40427

Offset: 1

Author

Amarnath Murthy, Nov 10 2003

Keywords

• base
• nonn

Comments

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.

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.

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

Primes among the magnanimous numbers A252996. - M. F. Hasler, Dec 25 2014

Examples

			2267 is a member which gives primes 2+267 = 269, 22+67 = 89, 226+7 = 233 and 2267 itself.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..84
• Eric Angelini et al., Insert "+" and always get a prime, Dec
• 2014
• Giovanni Resta, magnanimous numbers, 2013.
• Carlos Rivera, Puzzle 401. Magnanimous primes, 2007, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A089393, A089394, A089695, A028834, A182175, A088134, A221699, A227823, A252996.

Programs

Maple

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

Mathematica

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 *)

PARI

is_A089392(n)={!for(i=1,#Str(n),ispseudoprime([1,1]*(divrem(n,10^i)))||return)} \\ M. F. Hasler, Dec 25 2014

Python

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    return all(isprime(int(s[:i])+int(s[i:])) for i in range(1, len(s)))
print([k for k in range(357) if ok(k)]) # Michael S. Branicky, Oct 14 2024

Extensions

Corrected and extended by C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 25 2004

Comments edited by Zak Seidov, Jan 29 2013

Edited by M. F. Hasler, Dec 25 2014