A182175 -id:A182175 - 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

A182178 Beginning with 1, smallest positive integer not yet in the sequence such that two adjacent digits of the sequence (also ignoring commas between terms) sum to a prime.

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 5, 8, 9, 20, 21, 11, 12, 14, 16, 50, 23, 25, 29, 41, 43, 47, 49, 83, 85, 61, 65, 67, 411, 111, 112, 30, 32, 34, 38, 52, 56, 58, 92, 94, 70, 74, 76, 114, 98, 302, 116, 120, 202, 121, 123, 89, 203, 205, 207, 412, 125, 211, 129, 212, 141, 143

Offset: 1

Jim Nastos and Eric Angelini, Apr 16 2012

See A219110 for the numbers which do not occur in this sequence. See A219250 for the analog when "sum" is replaced with "absolute difference", and A219248-A219251 for related sequences. - M. F. Hasler, Apr 11 2013

			20 follows 9 since 9+2 and 2+0 is prime, and no number less than 20 (not already in the sequence) satisfies the stated property.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A182175, A182177.

a[1] = 1; a[n_] := a[n] = For[id = IntegerDigits[a[n-1]]; k = 1, True, k++, If[FreeQ[Array[a, n-1], k], dd = Join[id, IntegerDigits[k]]; If[And @@ PrimeQ /@ Plus @@@ Transpose[{Most[dd], Rest[dd]}], Return[k]]]]; Array[a, 62] (* Jean-François Alcover, Apr 17 2013 *)

A182178_vec={(n, a=[1], u=0)->while(#aM. F. Hasler, Apr 11 2013

A219248 Numbers such that the absolute difference of any two adjacent (decimal) digits is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 14, 16, 18, 20, 24, 25, 27, 29, 30, 31, 35, 36, 38, 41, 42, 46, 47, 49, 50, 52, 53, 57, 58, 61, 63, 64, 68, 69, 70, 72, 74, 75, 79, 81, 83, 85, 86, 92, 94, 96, 97, 130, 131, 135, 136, 138, 141, 142, 146, 147, 149, 161, 163, 164

Offset: 1

M. F. Hasler, Apr 12 2013

Numbers which may (and do) occur in A219250 and A219249 (union {0}).

This is to A219250 and A219249 what A182175 is to A182177 and A182178.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
E. Angelini, Any digit-pair in S sums to a prime, SeqFan list, Apr 11 2013

Select[Range[0,200],And@@PrimeQ[Abs[Differences[IntegerDigits[#]]]]&] (* Harvey P. Dale, Jun 06 2014 *)

is_A219248(n)={!for(i=2,#n=digits(n),isprime(abs(n[i-1]-n[i]))||return)}

def ok(n):
    d = list(map(int, str(n)))
    return all(abs(d[i]-d[i-1]) in {2,3,5,7} for i in range(1, len(d)))
print([k for k in range(164) if ok(k)]) # Michael S. Branicky, Sep 11 2024

from itertools import count, islice
def A219248gen(seed=None): # generator of terms
    nxt = {None:"123456789", "0":"2357", "1":"3468", "2":"04579",
        "3":"01568", "4":"12679", "5":"02378", "6":"13489",
        "7":"02459", "8":"1356", "9":"2467"}
    def bgen(d, seed=None):
        if d == 0: yield tuple(); return
        yield from ((i,)+t for i in nxt[seed] for t in bgen(d-1, seed=i))
    yield 0
    for d in count(1):
        yield from (int("".join(t)) for t in bgen(d, seed=seed))
print(list(islice(A219248gen(), 65))) # Michael S. Branicky, Sep 11 2024

A219250 Lexicographically earliest sequence of nonnegative integers such that the absolute difference of any two adjacent digits is prime.

Original entry on oeis.org

0, 2, 4, 1, 3, 5, 7, 9, 6, 8, 13, 14, 16, 18, 30, 20, 24, 25, 27, 29, 41, 31, 35, 36, 38, 50, 52, 42, 46, 47, 49, 61, 63, 53, 57, 58, 64, 68, 69, 70, 72, 74, 75, 79, 202, 92, 94, 96, 81, 83, 85, 86, 97, 203, 130, 205, 207, 241, 302, 413, 131, 303, 135, 242, 414, 136, 138, 141, 305, 246, 142, 416, 146, 147, 247, 249

Offset: 1

M. F. Hasler, Apr 11 2013

See A219249 for the version allowing only positive integers, i.e., starting with a(1)=1.
See A219248 (= range of A219250) for the numbers which occur in this sequence, and A219251 for the complement.
A182177 is the analog of this sequence for replacing "absolute difference" by "sum", A182178 is the same analog for A219249; A182175 is the analog of A219248 and A219110 corresponds to A219251.

M. F. Hasler in reply to E. Angelini, Re: Any digit-pair in S sums to a prime, SeqFan list, Apr 11 2013

PARI
```
{(n,a=[0],u=0)->while(#a
				
```

A182177 Beginning with 0, smallest positive integer not yet in the sequence such that two adjacent digits of the sequence (also ignoring commas between terms) sum to a prime.

Original entry on oeis.org

0, 2, 1, 4, 3, 8, 5, 6, 7, 41, 11, 12, 9, 20, 21, 14, 16, 50, 23, 25, 29, 43, 47, 49, 83, 85, 61, 65, 67, 411, 111, 112, 30, 32, 34, 38, 52, 56, 58, 92, 94, 70, 74, 76, 114, 98, 302, 116, 120, 202, 121, 123, 89, 203, 205, 207, 412, 125, 211, 129, 212, 141, 143, 214, 147, 414, 149, 216, 161, 165, 230, 232, 167, 416, 502, 303, 234, 305, 238, 307, 430, 250, 252, 320, 256, 503, 258, 321, 292, 323, 294, 325, 298, 329, 432, 341, 434, 343, 438, 347, 470, 349

Offset: 1

Jim Nastos and Eric Angelini, Apr 16 2012

A219250 is the analog of this sequence, replacing "sum" by "absolute difference". A219249 is the same analog for A182178. A219248 is the analog of A182175 and A219251 corresponds to A219110 = terms which do not occur in this sequence, i.e., the complement of its range. - M. F. Hasler, Apr 12 2013

			41 appears after 7 because 7+4 is prime and 4+1 is prime, and no other number less than 41 (not already in the sequence) satisfies this property. Example: 11 does not directly follow 7 since 7+1 is not prime.

Cf. A182175.

A182177_vec={(n, a=[0], u=0)->while(#aM. F. Hasler, Apr 11 2013

A219110 Numbers for which at least one sum of two adjacent digits is not prime.

Original entry on oeis.org

10, 13, 15, 17, 18, 19, 22, 24, 26, 27, 28, 31, 33, 35, 36, 37, 39, 40, 42, 44, 45, 46, 48, 51, 53, 54, 55, 57, 59, 60, 62, 63, 64, 66, 68, 69, 71, 72, 73, 75, 77, 78, 79, 80, 81, 82, 84, 86, 87, 88, 90, 91, 93, 95, 96, 97, 99, 100, 101, 102, 103, 104, 105

Offset: 1

M. F. Hasler, Apr 11 2013

Different from A104211 where ("only") the sum of all digits is considered; of course exactly the two-digit terms coincide.

Numbers missing in A182177 and A182178. Otherwise said, complement of the range of A182177 (in the set of nonnegative integers) and of the range of A182178 (in the set of positive integers) and of A182175 in the set of integers > 9.

			102 is here because 1+0 is not prime (even though 0+2 is).

M. F. Hasler in reply to E. Angelini, Re: Any digit-pair in S sums to a prime, SeqFan list, Apr 11 2013

Select[Range[10, 105], MemberQ[PrimeQ[Total /@ Partition[IntegerDigits[#], 2, 1]], False] &] (* T. D. Noe, Apr 16 2013 *)

is(n)=for(i=2,#n=digits(n),isprime(n[i-1]+n[i])||return(1))

A221699 Numbers with property that each sum any pair of digits is prime.

Original entry on oeis.org

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, 111, 112, 114, 116, 121, 141, 161, 203, 205, 211, 230, 250, 302, 320, 411, 502, 520, 611, 1111, 1112, 1114, 1116, 1121, 1141, 1161

Offset: 1

Jaroslav Krizek, Jan 22 2013

Supersequence of A091939. Subsequence of A182175.

			Number 203 is a term because 2+3=5, 2+0=2, 3+0=3 (primes).

Vincenzo Librandi, Table of n, a(n) for n = 1..145

Cf. A172508, A091939, A182175.

fQ[n_] := Module[{d = IntegerDigits[n]}, Union[PrimeQ[Plus @@@ Union[Subsets[d, {2}]]]] == {True}]; Select[Range[6000], fQ] (* T. D. Noe, Jan 24 2013 *)
Select[Range[10,1200],AllTrue[Total/@Subsets[IntegerDigits[#],{2}],PrimeQ]&] (* Harvey P. Dale, Jan 31 2024 *)

A219249 Lexicographically earliest sequence of positive integers such that the absolute difference of any two adjacent digits is prime.

Original entry on oeis.org

1, 3, 5, 2, 4, 6, 8, 13, 14, 7, 9, 20, 24, 16, 18, 30, 25, 27, 29, 41, 31, 35, 36, 38, 50, 52, 42, 46, 47, 49, 61, 63, 53, 57, 58, 64, 68, 69, 70, 72, 74, 75, 79, 202, 92, 94, 96, 81, 83, 85, 86, 97, 203, 130, 205, 207, 241, 302, 413, 131, 303, 135, 242, 414, 136, 138, 141, 305, 246, 142, 416, 146, 147, 247, 249, 250

Offset: 1

Eric Angelini and M. F. Hasler, Apr 11 2013

See A219250 for the version allowing nonnegative integers, i.e., starting with a(1)=0.

See A219248 for the numbers which occur in this sequence, and A219251 for the complement.

E. Angelini, Any digit-pair in S sums to a prime, SeqFan list, Apr 11 2013

Cf. A182175, A182177, A182178.

PARI
```
{A219249(n,a=[1],u=0)=while(#a
				
```

A252996 Magnanimous numbers: numbers such that the sum obtained by inserting a "+" anywhere between two digits gives a prime.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Dec 25 2014

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.

			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.

Robert G. Wilson v, Table of n, a(n) for n = 1..571
Hans Havermann, in reply to Eric Angelini, 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. A089392, A089393, A089394, A028834, A182175, A088134, A221699, A227823.

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

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 *)

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)

from sympy import isprime
def ok(n):
    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

A252495 Restricted magnanimous numbers: numbers such that the sum obtained by inserting a "+" anywhere between two digits gives a prime, but no "leading zeros" may appear.

Original entry on oeis.org

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, 110, 112, 116, 118, 130, 136, 152, 158, 170, 172, 221, 227, 229, 245, 265, 281, 310, 316, 334, 338, 356

Offset: 1

M. F. Hasler, Dec 28 2014

Inclusion of the single-digit terms is conventional: here the property is vacuously 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.)
The restriction on leading zeros means that numbers with digit 0 other than in the last position are excluded, so this sequence equals A252996 with terms in A252480 removed.
Since all primes > 2 are odd, all terms different from 11 and 20 have the last digit of opposite parity to that of all other digits.
As A252996, this sequence is "finite" (without rigorous proof), and up to 5e16 the only terms with more than 11 digits are 5391391551358 and 97393713331910, the latter being probably the largest element of this sequence (due to Giovanni Resta).
(See links for "intellectual ownership": The sequence (without single-digit terms) was suggested by Eric Angelini, a first list of terms computed by Lars Blomberg, then others. Hans Havermann observed that this is a variant of what had been termed "magnanimous numbers" at least 10 years ago by A. Murthy, G. Resta and/or C. Rivera, cf. A089392 and links.)

			110 is in the sequence since 1+10=11 and 11+0 = 11 are both prime.
101 is not in the sequence because although 10+1 = 11 and 1+01 = 2 are prime, the latter sum is forbidden since 01 has a leading zero.
Number, smallest and largest of the n-digit terms:
| n   #     min    max
| 1  10      0      9
| 2  33      11     98
| 3  69     110     998
| 4  90     1112    9910
| 5  81    11116    99998
| 6  71    111112   999994
| 7  54   1115756   9959374
| 8  25   11771992  95559998
| 9   9  117711170  995955112
|10   4  1777137770 9151995592
|11   4 22226226625 46884486265
|12   0  -
|13   1     5391391551358
|14   1     97393713331910
|15   0  -

M. F. Hasler, Table of n, a(n) for n = 1..452
E. Angelini and L. Blomberg, Insert "+" and always get a prime, Dec 2014
G. Resta, magnanimous numbers, 2013.
C. Rivera, Puzzle 401. Magnanimous primes, 2007.

Cf. A252996, A089392, A089393, A089394, A028834, A182175, A088134, A221699, A227823.

is(n)=!for(i=1,#Str(n)-1,ispseudoprime([1,1]*(divrem(n,10^i)))||return)&&(n<100||vecmin(digits(n\10)))
t=0;vector(100,i,until(is(t++),);t)

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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A182178 Beginning with 1, smallest positive integer not yet in the sequence such that two adjacent digits of the sequence (also ignoring commas between terms) sum to a prime.

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A219248 Numbers such that the absolute difference of any two adjacent (decimal) digits is prime.

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A219250 Lexicographically earliest sequence of nonnegative integers such that the absolute difference of any two adjacent digits is prime.

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A182177 Beginning with 0, smallest positive integer not yet in the sequence such that two adjacent digits of the sequence (also ignoring commas between terms) sum to a prime.

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A219110 Numbers for which at least one sum of two adjacent digits is not prime.

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A221699 Numbers with property that each sum any pair of digits is prime.

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A219249 Lexicographically earliest sequence of positive integers such that the absolute difference of any two adjacent digits is prime.

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