A158124 -id:A158124 - cp's OEIS frontend

A158125 Weakly prime numbers in the sense of A158124 but not A050249.

929573, 3070663, 5285767, 5974249, 7810223, 9262697, 9663683, 9700429, 10532453, 12968519, 19106729, 19158221, 19579907, 21825337, 23196157, 24328567, 29617897, 31181461, 31746383, 31839427, 36438379, 36745811, 37763641, 38585039, 38774851, 38888137, 39600559, 45412331, 45743483, 47500217, 47632271, 54127231, 56242891, 59816347

Offset: 1

Eric W. Weisstein, Mar 13 2009

A158124: initial digit may not be changed to a zero (and hence give a number with fewer digits).
A050249: initial digit may be changed to a zero.
For the following values 5, 6, 7, 8, 9, 10 of k, the number of terms < 10^k in this sequence is 0, 1, 8, 72, 589, 4977. - Jean-Marc Rebert, Nov 10 2015
Intersection of A227916 and A158124. So primes p that give another prime when the first digit is removed, but give a composite number when any one digit is modified in a way that does not change the digit count. - Jeppe Stig Nielsen, Jan 16 2022

Jean-Marc Rebert, Table of n, a(n) for n = 1..589
Eric Weisstein's World of Mathematics, Weakly Prime

Cf. A050249, A158124, A227916.

forprime(p=2,10^10,d=digits(p);!isprime(fromdigits(d[2..#d]))&&next();for(k=1,#d,for(j=(k==1),9,d[k]==j&&next();e=d;e[k]=j;isprime(fromdigits(e))&&next(3)));print1(p,", ")) \\ Jeppe Stig Nielsen, Jan 16 2022

A039818 Erroneous version of A158124.

Original entry on oeis.org

4409, 67307, 86573, 111857, 198349, 259507

Offset: 0

dead

A048853 Number of primes (different from n) that can be produced by altering one digit of decimal expansion of n (without changing the number of digits).

Original entry on oeis.org

4, 3, 3, 4, 3, 4, 3, 4, 4, 4, 7, 4, 8, 4, 4, 4, 7, 4, 7, 2, 7, 2, 6, 2, 2, 2, 7, 2, 5, 2, 5, 2, 8, 2, 2, 2, 5, 2, 7, 3, 6, 3, 7, 3, 3, 3, 6, 3, 8, 2, 7, 2, 6, 2, 2, 2, 7, 2, 5, 2, 5, 2, 8, 2, 2, 2, 5, 2, 7, 3, 6, 3, 7, 3, 3, 3, 8, 3, 6, 2, 7, 2, 6, 2, 2, 2, 7, 2, 5, 1, 6, 1, 7, 1, 1, 1, 4, 1, 6, 4, 10, 4, 8, 4, 4

Offset: 1

G. L. Honaker, Jr. and Patrick De Geest

a(A192545(n)) = 0. - Reinhard Zumkeller, Jul 05 2011

			Altering the number 13 gives eight primes: 11, 17, 19, 23, 43, 53, 73, 83, so a(13)=8.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A050652-A050673.

Cf. A010051, A158124.

import Data.List (inits, tails, nub)
a048853 n = (sum $ map (a010051 . read) $ tail $ nub $ concat $ zipWith
  (\its tls -> map ((\xs ys d -> xs ++ (d:ys)) its tls) "0123456789")
    (map init $ tail $ inits $ show n) (tail $ tails $ show n)) - a010051 n
-- Reinhard Zumkeller, Jul 05 2011

A048853 := proc(n::integer) local resul,ddigs,d,c,tmp ; resul := 0 ; ddigs := convert(n,base,10) ; for d from 1 to nops(ddigs) do for c from 0 to 9 do if c = 0 and d = nops(ddigs) then continue ; else if c <> op(d,ddigs) then tmp := [op(1..d-1,ddigs),c,op(d+1..nops(ddigs),ddigs)] ; tst := sum(op(i,tmp)*10^(i-1),i=1..nops(tmp)) ; if isprime(tst) then resul := resul+1 ; fi ; fi ; fi ; od : od ; RETURN(resul) ; end: for n from 1 to 90 do printf("%d,",A048853(n)) ; od ; # R. J. Mathar, Apr 25 2006

a[n_] := Module[{idn = IntegerDigits[n], id, np = 0}, Do[id = idn; If[ id[[j]] != k, id[[j]] = k; If[ id[[1]] != 0 && PrimeQ[ FromDigits[id]], np = np + 1]], {j, 1, Length[idn]}, {k, 0, 9}]; np]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Dec 01 2011 *)

from sympy import isprime
def h1(n): # hamming distance 1 neighbors of n, not starting with 0
    s = str(n); d = "0123456789"; L = len(s)
    yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L) if c!=s[i] and not (i==0 and c=="0"))
def a(n): return sum(1 for k in h1(n) if isprime(k))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jul 31 2022

A050249 Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes.

Original entry on oeis.org

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139, 5152507, 5564453, 5575259, 6173731, 6191371, 6236179, 6463267, 6712591, 7204777, 7469789, 7469797

Offset: 1

Eric W. Weisstein

Tao proved that this sequence is infinite. - T. D. Noe, Mar 01 2011

For k = 5, 6, 7, 8, 9, 10, the number of terms < 10^k in this sequence is 0, 5, 35, 334, 3167, 32323. - Jean-Marc Rebert, Nov 10 2015

Michael Filaseta and Jeremiah Southwick, Primes that become composite after changing an arbitrary digit, Math. Comp. (2021) Vol. 90, 979-993. doi:10.1090/mcom/3593

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (terms 1..1317 from Klaus Brockhaus, terms 1318..3167 from Jean-Marc Rebert).
Michael Filaseta and Jacob Juillerat, Consecutive primes which are widely digitally delicate, arXiv:2101.08898 [math.NT], 2021.
Jon Grantham, Finding a Widely Digitally Delicate Prime, arXiv:2109.03923 [math.NT], 2021.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Jackson Hopper and Paul Pollack, Digitally delicate primes, arXiv:1510.03401 [math.NT], 2015.
Dana Jacobsen, Digitally delicate primes up to 1e11
Matt Parker, How do you prove a prime is infinitely fragile?, Stand-up Maths YouTube video, 2022.
Jeremiah T. Southwick, Two Inquiries Related to the Digits of Prime Numbers, Ph. D. Dissertation, University of South Carolina (2020).
Terence Tao, A remark on primality testing and decimal expansions, arXiv:0802.3361 [math.NT], 2008-2010; Journal of the Australian Mathematical Society 91:3 (2011), pp. 405-413.
Eric Weisstein's World of Mathematics, Weakly Prime

Cf. A118118, A158124 (weakly primes), A158125 (weakly primes).

Cf. A137985 (analogous base-2 sequence), A186995 (weak primes in base n).

IsA118118:=function(n); D:=Intseq(n); return forall{ : k in [1..#D], j in [0..9] | j eq D[k] or not IsPrime(Seqint(S)) where S:=Insert(D, k, k, [j]) }; end function; [ p: p in PrimesUpTo(8000000) | IsA118118(p) ]; // Klaus Brockhaus, Feb 28 2011

fQ[n_] := Block[{d = IntegerDigits@ n, t = {}}, Do[AppendTo[t, FromDigits@ ReplacePart[d, i -> #] & /@ DeleteCases[Range[0, 9], x_ /; x == d[[i]]]], {i, Length@ d}]; ! AnyTrue[Flatten@ t, PrimeQ]] ; Select[Prime@ Range[10^5], fQ] (* Michael De Vlieger, Nov 10 2015, Version 10 *)

isokp(n) = {v = digits(n); for (k=1, #v, w = v; for (j=0, 9, if (j != v[k], w[k] = j; ntest = subst(Pol(w), x, 10); if (isprime(ntest), return(0));););); return (1);}
lista(nn) = {forprime(p=2, nn, if (isokp(p), print1(p, ", ")););} \\ Michel Marcus, Dec 15 2015

from sympy import isprime
def h1(n): # hamming distance 1 neighbors of n
    s = str(n); d = "0123456789"; L = len(s)
    yield from (int(s[:i]+c+s[i+1:]) for c in d for i in range(L) if c!=s[i])
def ok(n): return isprime(n) and all(not isprime(k) for k in h1(n) if k!=n)
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Jun 19 2022

Edited by Charles R Greathouse IV, Aug 02 2010

A192545 Numbers such that all numbers are composite when replacing exactly one digit with another, except the leading digit with zero.

Original entry on oeis.org

200, 202, 204, 205, 206, 208, 320, 322, 324, 325, 326, 328, 510, 512, 514, 515, 516, 518, 530, 532, 534, 535, 536, 538, 620, 622, 624, 625, 626, 628, 840, 842, 844, 845, 846, 848, 890, 892, 894, 895, 896, 898, 1070, 1072, 1074, 1075, 1076, 1078, 1130, 1132

Offset: 1

Reinhard Zumkeller, Jul 05 2011

A048853(a(n)) = 0;
Intersection of this sequence and A000040 is A158124. - Evgeny Kapun, Dec 13 2016
If the last digit of an element is 0, 2, 4, 5, 6 or 8, then replacing it with 0, 2, 4, 5, 6 or 8 also yields an element. - David A. Corneth and corrected by Evgeny Kapun, Dec 13 2016

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Subsequences: A050249, A118118, A143641, A158124, A158125.

import Data.List (elemIndices)
a192545 n = a192545_list !! (n-1)
a192545_list = map (+ 1) $ elemIndices 0 $ map a048853 [1..]

Select[Range@ 1200, Function[w, Total@ Boole@ Flatten@ Map[Function[d, PrimeQ@ FromDigits@ ReplacePart[w, d -> #] & /@ If[d == 1, #, Prepend[#, 0]] &@ Range@ 9], Range@ Length@ w] == 0]@ IntegerDigits@ # &] (* Michael De Vlieger, Dec 13 2016 *)

A253269 Weakly Twin Primes in base 10: Can only reach one other prime by single-decimal-digit changes.

Original entry on oeis.org

89391959, 89591959, 519512471, 519512473, 531324041, 561324041, 699023791, 699023891, 874481011, 874487011, 1862537503, 2232483271, 2232483871, 2608559351, 3127181789, 3157181789, 3928401949, 3928401989, 4070171669, 4070171969, 5225628323, 5309756339, 5525628323

Offset: 1

Michael Kleber, May 01 2015

Each pair of twins here form a size-two connected component in the graph considered in A158576.
A naive heuristic argument based on the density of primes claims that this sequence should be infinite, and in fact that a positive proportion of all primes should have this property.  A prime p has 9*log_10(p) neighbors, each prime with "probability" 1/log(p), and with all the other 2*9*log_10(p) neighbors being composite with "probability" (1-1/log(p))^(2*9*log_10(p)). For a large prime p, this goes to the limit 9/(exp(18/log(10))*log(10)), or about 0.16%.  The fact that base-10 primes need to end with digit 1/3/7/9 will change the value of this probability, but won't change the fact that it is nonzero.
This is analogous to a theorem about weakly prime numbers; see the Terence Tao paper referenced in A050249.

Michael Kleber, Table of n, a(n) for n = 1..66

Cf. A158576, A050249, A158124.

NeighborsAndSelf[n_] := Flatten[MapIndexed[Table[ n + (i - #)*10^(#2[[1]] - 1), {i, 0, 9}] &, Reverse[IntegerDigits[n, 10]]]]
PrimeNeighbors[n_] := Complement[Select[NeighborsAndSelf[n],PrimeQ],{n}]
WeaklyTwinPrime[p_] := (Length[#] == 1 && PrimeNeighbors[#[[1]]] == {p}) &[PrimeNeighbors[p]]
For[k = 0, k <= PrimePi[10^10], k++, If[WeaklyTwinPrime[Prime[k]], Print[Prime[k]]]]

A347424 Digitally delicate truncatable primes: every suffix is prime, changing any one decimal digit always produces a composite number, except the first to zero.

Original entry on oeis.org

7810223, 19579907, 909001523, 984960937, 78406036607, 90124536947, 99020400307, 190002706337, 393086079907, 500708906197, 509000702017, 600180367883, 780430098443, 3534900290107, 5046024021013, 6006006800743, 6009000432797, 9001924501223, 12090900340283

Offset: 1

Marc Morgenegg, Sep 01 2021

These prime numbers are both:
- digitally delicate primes (also called weakly prime numbers) A158124: changing any one decimal digit always produces a composite number, with restriction that first digit may not be changed to a 0 (that means no change of the number of significant digits from its original value).
- left-truncatable primes A033664:  every suffix is prime, means repeatedly deleting the most significant digit gives a prime at every step until a single-digit prime remains.

Michael S. Branicky, Table of n, a(n) for n = 1..9175 (all terms with <= 29 digits)

Cf. A158124, A033664.

from sympy import isprime, primerange
def is_digitally_delicate(p):
    s = str(p)
    for i in range(len(s)):
        for d in "0123456789":
            if d != s[i] and not (i == int(d) == 0):
                if isprime(int(s[:i] + d + s[i+1:])): return False
    return True
def A033664gen(maxdigits):
    yield from [2, 3, 5, 7]
    primestrs, digits, d = ["2", "3", "5", "7"], "0123456789", 1
    while len(primestrs) > 0 and d < maxdigits:
        cands = (d+p for p in primestrs for d in "0123456789")
        primestrs = [c for c in cands if c[0] == "0" or isprime(int(c))]
        yield from sorted(map(int, (p for p in primestrs if p[0] != "0")))
        d += 1
def afind(maxdigits):
    for p in A033664gen(maxdigits):
        if is_digitally_delicate(p): print(p, end=", ")
afind(12) # Michael S. Branicky, Sep 01 2021

a(3)-a(4) from Amiram Eldar, Sep 01 2021

a(5)-a(19) from Michael S. Branicky, Sep 01 2021

A158641 Strong primes p: adding 2 to any one digit of p produces a prime number (no digits 8 & 9 in p).

Original entry on oeis.org

3, 5, 11, 17, 41, 107, 137, 347, 2111, 2657, 3527, 4421, 6761, 21011, 24371, 32057

Offset: 1

Zak Seidov, Mar 23 2009

All terms are lesser of twin pairs.

The sequence is finite. In particular, there are no terms longer than 5 digits since 2*10^k for k=0,1,...,5 give all nonzero residues modulo 7, implying that adding 2 to some digit yields a multiple of 7. - Max Alekseyev, Apr 25 2010

			2111 is in this sequence because all 2111, 4111, 2311, 2131 and 2113 are prime numbers.
32057 is in this sequence because all 32057, 52057, 34057, 32257, 32077 and 32059 are prime numbers.

Cf. A050249, A158124, A158125 Weakly prime numbers (changing any one digit always produces a composite number).

Lton := proc(L) local i; add(op(i,L)*10^(i-1),i=1..nops(L)) ; end: isA158641 := proc(p) local pdgs,pplus,i ; if isprime(p) then pdgs := convert(p,base,10) ; if convert(pdgs,set) intersect {8,9} <> {} then false; else for i from 1 to nops(pdgs) do pplus := subsop(i=2+op(i,pdgs),pdgs) ; if not isprime(Lton(pplus)) then RETURN(false); fi; od: true; fi; else false; fi; end: for n from 1 do p := ithprime(n) ; if isA158641(p) then print(p) ; fi; od: # R. J. Mathar, Apr 16 2009

spQ[p_]:=Max[IntegerDigits[p]]<8&&AllTrue[FromDigits/@Table[MapAt[ 2+#&,IntegerDigits[ p],n],{n,IntegerLength[p]}],PrimeQ]; Select[Prime[ Range[ 3500]],spQ] (* Harvey P. Dale, Nov 26 2022 *)

test(p)=my(v=eval(Vec(Str(p)))); for(i=1,#v, if(v[i]>7,return(0))); for(i=0,#v-1, if(!isprime(p+2*10^i), return(0))); 1
forprime(p=2,4e9, if(isprime(p+2) && test(p), print1(p","))) \\ Charles R Greathouse IV, Sep 09 2009

has(n)=if(vecmax(Set(digits(n)))>7, return(0)); for(i=0,#digits(n)-1, if(!isprime(n+2*10^i), return(0))); 1
select(has, primes(3438)) \\ Charles R Greathouse IV, Mar 11 2016

Keywords full, fini from Max Alekseyev, Apr 25 2010

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A158125 Weakly prime numbers in the sense of A158124 but not A050249.

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A039818 Erroneous version of A158124.

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A048853 Number of primes (different from n) that can be produced by altering one digit of decimal expansion of n (without changing the number of digits).

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A050249 Weakly prime numbers (changing any one decimal digit always produces a composite number). Also called digitally delicate primes.

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A192545 Numbers such that all numbers are composite when replacing exactly one digit with another, except the leading digit with zero.

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A253269 Weakly Twin Primes in base 10: Can only reach one other prime by single-decimal-digit changes.

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A347424 Digitally delicate truncatable primes: every suffix is prime, changing any one decimal digit always produces a composite number, except the first to zero.

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A158641 Strong primes p: adding 2 to any one digit of p produces a prime number (no digits 8 & 9 in p).

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