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A252996 Magnanimous numbers: numbers such that the sum obtained by inserting a "+" anywhere between two digits gives a prime. 5
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 110, 112, 116, 118, 130, 136, 152, 158, 170, 172, 203, 209, 221, 227, 229, 245, 265, 281, 310, 316, 334, 338, 356 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Inclusion of the single-digit terms is conventional: here the property is voidly satisfied since no sum can be constructed by inserting a + sign between two digits, therefore all possible sums are prime. (It is not allowed to prefix a leading zero (e.g., to forbid 4 = 04 = 0+4) since in that case all terms must be prime and one would get A089392.)

All terms different from 20 and not of the form 10^k+1 have the last digit of opposite parity than that of all other digits.

The sequence is marked as "finite", although we do not have a rigorous proof for this, only very strong evidence (numerical and probabilistic). G. Resta has checked that up to 5e16 the only magnanimous numbers with more than 11 digits are 5391391551358 and 97393713331910, the latter being probably the largest element of this sequence. In that case the 10+33+79+104+112+96+71+35+18+6+5+0+1+1 = 571 terms listed in Wilson's b-file are the complete list, which is what the keyword "full" stands for.

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 1..571

H. Havermann, in reply to E. Angelini, Insert "+" and always get a prime, Dec 2014

G. Resta, magnanimous numbers, 2013.

C. Rivera, Puzzle 401. Magnanimous primes, 2007.

EXAMPLE

245 is in the sequence because the numbers 2 + 45 = 47 and 24 + 5 = 29 are both prime. See the first comment for the single-digit terms.

MAPLE

filter:= proc(n) local d;

  for d from 1 to ilog10(n)-1 do

    if not isprime(floor(n/10^d)+(n mod 10^d)) then return false fi

  od:

  true

end proc:

select(filter, [$0..10^5]); # Robert Israel, Dec 25 2014

MATHEMATICA

fQ[n_] := Block[{idn = IntegerDigits@ n, lng = Floor@ Log10@ n}, Union@ PrimeQ@ Table[ FromDigits[ Take[ idn, i]] + FromDigits[ Take[ idn, -lng + i -1]], {i, lng}] == {True}]; (* or *)

fQ[n_] := Block[{lng = Floor@ Log10@ n}, Union@ PrimeQ[ Table[ Floor[n/10^k] + Mod[n, 10^k], {k, lng}]] == {True}];

fQ[2] = fQ[3] = fQ[5] = fQ[7] = True; Select[ Range@ 500, fQ]

(* Robert G. Wilson v, Dec 26 2014 *)

mnQ[n_]:=AllTrue[Total/@Table[FromDigits/@TakeDrop[IntegerDigits[n], i], {i, IntegerLength[n]-1}], PrimeQ]; Join[Range[0, 9], Select[Range[ 10, 400], mnQ]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 26 2017 *)

PROG

(PARI) is(n)={!for(i=1, #Str(n)-1, ispseudoprime([1, 1]*(divrem(n, 10^i)))||return)}

t=0; vector(100, i, until(is(t++), ); t)

CROSSREFS

Cf. A089392, A089393, A089394, A028834, A182175, A088134, A221699, A227823.

Sequence in context: A180478 A062461 A178843 * A252495 A182175 A254329

Adjacent sequences:  A252993 A252994 A252995 * A252997 A252998 A252999

KEYWORD

nonn,base,nice,fini,full

AUTHOR

M. F. Hasler, Dec 25 2014

STATUS

approved

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Last modified July 8 22:01 EDT 2020. Contains 335537 sequences. (Running on oeis4.)