A089695 -id:A089695 - 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.

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

Amarnath Murthy, Nov 10 2003

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

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

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.

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

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

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 *)

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

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

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

A348674 Number of distinct values that can be produced by splitting n and adding the parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4

Offset: 0

Alois P. Heinz, Oct 29 2021

Differs from A055642 first at n=120: a(120) = 4 != 3 = A055642(120).

The number of split positions can vary from 0 to length(n)-1.

			a(0) = 1: 0.
a(10) = 2: 1 = 1+0, 10.
a(100) = 3: 1 = 1+0+0, 10 = 10+0, 100.
a(120) = 4: 3 = 1+2+0, 12 = 12+0, 21 = 1+20, 120.
a(2493690) = 62 = |{33, 51, 60, 69, 78, 87, 96, 105, 114, 123, 132, 141, 150, 159, 168, 177, 186, 195, 213, 267, 294, 321, 348, 375, 384, 402, 420, 510, 564, 591, 618, 708, 726, 744, 789, 807, 942, 951, 969, 1032, 1050, 1185, 2508, 2562, 2589, 3183, 3705, 3723, 3741, 3939, 4947, 5028, 9375, 9393, 24945, 25026, 49371, 93696, 93714, 249369, 493692, 2493690}|.

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Ordinal transform gives A349315.
Where records occur: A349316.
Cf. A000041, A002275, A004216, A007953, A011557, A055642, A089695.

b:= proc(s) option remember; (n-> {parse(s), seq(seq(seq(x+y,
      y=b(s[i+1..n])), x=b(s[1..i])), i=1..n-1)})(length(s))
    end:
a:= n-> nops(b(""||n)):
seq(a(n), n=0..120);

a(n) <= 2^floor(log_10(n)) = 2^A004216(n) for n>0.
a((10^n-1)/9) = a(A002275(n)) <= A000041(n) with equality only for n <= 23.
a(10^n) = a(A011557(n)) = n+1.

A089696 Numbers k such that the numbers obtained by placing as many '' signs as possible anywhere between the digits and then adding 1 yields a prime in every case: let abc.. be the digits of k, then abc+1, abc+1, abc+1, ab*c+1, ... must all be primes.

Original entry on oeis.org

1, 2, 4, 6, 12, 16, 22, 28, 36, 52, 58, 66, 82, 112, 136, 166, 256, 352, 556, 562, 586, 616, 652, 658

Offset: 0

Amarnath Murthy, Nov 10 2003

Though the first 14 terms match with that of A089395, the next term of A089395 306 is not a member of this sequence. Conjecture: Sequence is finite.

No more terms < 10^7. The first 13 terms match with that of A089395, but A089395(14) = 106 is not included because 1*0*6+1 = 1 is not prime. - David Wasserman, Oct 04 2005

			256 is a member 256+1, 2*56 +1, 25*6+1, 2*5*6 +1 are all prime.

Cf. A089695.

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=choose([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:=mul(j,j=[seq(ds(sn[s[i]..s[i+1]-1]),i=1..nops(s)-1)])+1: if not isprime(m) then fl:=1: break fi od: if fl=0 then printf("%d, ",n) fi od od: # C. Ronaldo

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

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.

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A348674 Number of distinct values that can be produced by splitting n and adding the parts.

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A089696 Numbers k such that the numbers obtained by placing as many '' signs as possible anywhere between the digits and then adding 1 yields a prime in every case: let abc.. be the digits of k, then abc+1, abc+1, abc+1, ab*c+1, ... must all be primes.

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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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Extensions

A348674 Number of distinct values that can be produced by splitting n and adding the parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A089696 Numbers k such that the numbers obtained by placing as many '*' signs as possible anywhere between the digits and then adding 1 yields a prime in every case: let abc.. be the digits of k, then abc+1, a*bc+1, ab*c+1, a*b*c+1, ... must all be primes.

Views

Author

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Comments

Examples

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Maple

Extensions

A089696 Numbers k such that the numbers obtained by placing as many '' signs as possible anywhere between the digits and then adding 1 yields a prime in every case: let abc.. be the digits of k, then abc+1, abc+1, abc+1, ab*c+1, ... must all be primes.