A118118 -id:A118118 - cp's OEIS frontend

A143641 Odd prime-proof numbers (A118118) not ending in 5.

212159, 595631, 872897, 1203623, 1293671, 1566691, 1702357, 1830661, 3716213, 3964169, 4103917, 4134953, 4173921, 4310617, 4376703, 4586509, 4703801, 4749187, 4801387, 4928909, 5005353, 5051179, 5231739, 5258901, 5317573

Offset: 1

M. F. Hasler, Aug 27 2008, Sep 04 2008

Most "prime-proof" numbers are even or multiples of 5, cf. A118118.

Nicol & Selfridge proved that this sequence is infinite. - Charles R Greathouse IV, Jan 27 2014

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..3655 from Klaus Brockhaus)
Michael Filaseta, Mark Kozek, Charles Nicol and John Selfridge, Composites that remain composite after changing a digit, Journal of Combinatorics and Number Theory 2 (2011), pp. 25-36.
Project Euler, Problem 200: Prime-proof Squbes (2008).

Cf. A118118.

IsA143641:=function(n); D:=Intseq(n); return Intseq(n)[1] ne 5 and forall{ : k in [1..#D], j in [0..9] | not IsPrime(Seqint(Insert(D, k, k, [j]))) }; end function; [ n: n in [1..4000000 by 2] | IsA143641(n) ]; // Klaus Brockhaus, Mar 03 2011

forstep( i=1,10^7,2, i%5 || next; isA118118(i) && print1(i","))

from sympy import isprime
from itertools import count, islice
def selfplusneighs(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))
def agen():
    for n in count(1, 2):
        if n%5 == 0: continue
        if all(not isprime(k) for k in selfplusneighs(n)):
            yield n
print(list(islice(agen(), 8))) # Michael S. Branicky, Aug 16 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 *)

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A143641 Odd prime-proof numbers (A118118) not ending in 5.

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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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Author

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Comments

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Links

Crossrefs

Programs

Magma

Mathematica

PARI

Python

Extensions

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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Haskell

Mathematica