A007597 -id:A007597 - cp's OEIS frontend

A000787 Strobogrammatic numbers: the same upside down.

0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1111, 1691, 1881, 1961, 6009, 6119, 6699, 6889, 6969, 8008, 8118, 8698, 8888, 8968, 9006, 9116, 9696, 9886, 9966, 10001, 10101, 10801, 11011, 11111, 11811, 16091, 16191

Offset: 1

N. J. A. Sloane

Strobogrammatic numbers are a kind of ambigrams that retain the same meaning when viewed upside down. - Daniel Mondot, Sep 27 2016

"Upside down" here means rotated by 180 degrees (i.e., central symmetry), NOT "vertically flipped" (symmetry w.r.t. horizontal line, which are in A045574). - M. F. Hasler, May 04 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
J. M. Howell, Strobogrammatic years, Math. Mag., 34 (1961), p. 182 and 184.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Cf. A007597 (Primes in this sequence), A057770, A111065, A169731 (another version).

Subsequence of A045574. - M. F. Hasler, May 04 2012

fQ[n_] := Block[{s = {0, 1, 6, 8, 9}, id = IntegerDigits[n]}, If[ Union[ Join[s, id]] == s && (id /. {6 -> 9, 9 -> 6}) == Reverse[id], True, False]]; Select[ Range[0, 16190], fQ[ # ] &] (* Robert G. Wilson v, Oct 11 2005 *)

from itertools import count, islice, product
def ud(s): return s[::-1].translate({ord('6'):ord('9'), ord('9'):ord('6')})
def agen():
    yield from [0, 1, 8]
    for d in count(2):
        for start in "1689":
            for rest in product("01689", repeat=d//2-1):
                left = start + "".join(rest)
                right = ud(left)
                for mid in [[""], ["0", "1", "8"]][d%2]:
                    yield int(left + mid + right)
print(list(islice(agen(), 47))) # Michael S. Branicky, Mar 29 2022

More terms from Robert G. Wilson v, Oct 11 2005

A153806 Strobogrammatic cyclops numbers.

Original entry on oeis.org

0, 101, 609, 808, 906, 11011, 16091, 18081, 19061, 61019, 66099, 68089, 69069, 81018, 86098, 88088, 89068, 91016, 96096, 98086, 99066, 1110111, 1160911, 1180811, 1190611, 1610191, 1660991, 1680891, 1690691, 1810181, 1860981

Offset: 1

Omar E. Pol, Jan 15 2009

Intersection of A000787 and A134808.

			1680891 is a member because it is the same upside down (A000787) and also a cyclops number (A134808).

Kenny Lau, Table of n, a(n) for n = 1..20000

Cf. A000787, A007597, A134808, A134809, A138131, A138148, A153807.

Select[Range[10^7], And[OddQ@ Length@#, Part[#, Ceiling[Length[#]/2]] == 0, Times @@ Boole@ Map[MemberQ[{0, 1, 6, 8, 9}, #] &, Union@ #] == 1, Count[#, 0] == 1, (Take[#, Floor[Length[#]/2]] /. {6 -> 9, 9 -> 6}) ==
Reverse@ Take[#, -Floor[Length[#]/2]]] &@ IntegerDigits@ # &] (* Michael De Vlieger, Jul 05 2016 *)

import sys
f = open('b153806.txt', 'w')
i = 1
n = 0
a = [""]
r = [""]  #reversed strobogrammatically
while True:
    for x,y in zip(a,r):
        f.write(str(i)+" "+x+"0"+y+"\n")
        i += 1
        if i>20000:
            f.close()
            sys.exit()
    a = sum([[x+"1",x+"6",x+"8",x+"9"] for x in a],[])
    r = sum([["1"+x,"9"+x,"8"+x,"6"+x] for x in r],[])
# Kenny Lau, Jul 05 2016

Extended beyond 11011 by R. J. Mathar, Jan 17 2009

A018847 Strobogrammatic primes: the same upside down (calculator-style numerals).

Original entry on oeis.org

2, 5, 11, 101, 151, 181, 619, 659, 6229, 10501, 12821, 15551, 16091, 18181, 19861, 60209, 60509, 61519, 61819, 62129, 116911, 119611, 160091, 169691, 191161, 196961, 605509, 620029, 625529, 626929, 650059, 655559, 656959, 682289, 686989, 688889

Offset: 1

David W. Wilson

This is the subsequence of primes in A018846. - M. F. Hasler, May 05 2012

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..648 from M. F. Hasler)

Cf. A007597 (more restrictive version not allowing digits 2 or 5).

lst = {}; fQ[n_] := Block[{allset = {0, 1, 2, 5, 6, 8, 9}, id = IntegerDigits@n}, Union@ Join[id, allset] == allset && Reverse[id /. {6 -> 9, 9 -> 6}] == id]; Do[ If[ PrimeQ@n && fQ@n, AppendTo[lst, n]], {n, 700000}]; lst (* Robert G. Wilson v, Feb 27 2007 *)

{write("/tmp/b018847.txt","1 2\n2 5"); c=2; s2=[0,1,2,5,6,8,9]; s=[0,1,2,5,8]; s1=[0,1,2,5,9,8,6]; for(n=2,9, p1=vector( (n+1)\2, i, 10^(i-1)); p2=vector( (n+1)\2, i, 10^(n-i)); forvec( v=vector((n+1)\2,i,if( i>1, [ 1,if( i>n\2, #s, #s1)], [2,5])), v[1]==3 & v[1]=5; ispseudoprime( t=sum(i=1,n\2,p1[i]*s1[v[i]]+p2[i]*s2[v[i]] ) +if(n%2,p1[#p1]*s[v[#v]] )) & /* print1(t",") */ write("/tmp/b018847.txt",c++" "t)))} \\ - M. F. Hasler, Apr 26 2012

is_A018847(n,t=Vec("012..59.86"))={ isprime(n) & apply(x->t[eval(x)+1], n=Vec(Str(n)))==vecextract(n, "-1..1") } \\ - M. F. Hasler, May 05 2012

from itertools import count, islice
from sympy import isprime
def A018847_gen(): # generator of terms
    r, t, u = ''.maketrans('69','96'), set('0125689'), {0,1,2,5,8}
    for x in count(1):
        for y in range(10**(x-1),10**x):
            if y%10 in u:
                s = str(y)
                if set(s) <= t and isprime(m:=int(s+s[-2::-1].translate(r))):
                    yield m
        for y in range(10**(x-1),10**x):
            s = str(y)
            if set(s) <= t and isprime(m:=int(s+s[::-1].translate(r))):
                yield m
A018847_list = list(islice(A018847_gen(),20)) # Chai Wah Wu, Apr 09 2024

A048890 Primes that yield a different prime when rotated by 180 degrees.

Original entry on oeis.org

19, 61, 109, 199, 601, 661, 1019, 1061, 1091, 1109, 1181, 1601, 1609, 1669, 1699, 1811, 1901, 1999, 6011, 6091, 6101, 6199, 6619, 6661, 6689, 6691, 6899, 6991, 10061, 10069, 10091, 10691, 10861, 10909, 11069, 11681, 11909, 16001, 16619, 16661

Offset: 1

G. L. Honaker, Jr.

Also called invertible primes. [Lekraj Beedassy, Jan 03 2009]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
C. K. Caldwell, The Prime Glossary, strobogrammatic [_Lekraj Beedassy_, Jan 03 2009]

Cf. A007597, A006567, A046732.

import Data.List (unfoldr)
a048890 n = a048890_list !! (n-1)
a048890_list = filter f a000040_list where
   f x = all (`elem` [0,1,6,8,9]) ds && x' /= x && a010051 x' == 1
     where x' = foldl c 0 ds
           c v 6 = 10*v + 9; c v 9 = 10*v + 6; c v d = 10*v + d
           ds = unfoldr d x
           d z = if z == 0 then Nothing else Just $ swap $ divMod z 10
-- Reinhard Zumkeller, Nov 18 2011

lst = {}; fQ[n_] := Block[{allset = {0, 1, 6, 8, 9}, id = IntegerDigits@n}, rid = Reverse[id /. {6 -> 9, 9 -> 6}]; Union@ Join[id, allset] == allset && PrimeQ@ FromDigits@ rid && rid != id]; Do[ If[ PrimeQ@n && fQ@n, AppendTo[lst, n]], {n, 16900}]; lst (* Robert G. Wilson v, Feb 27 2007 *)

from itertools import product
from sympy import isprime
A048890_list = []
for d in product('01689',repeat=6):
    s = ''.join(d)
    p = int(s)
    if p > 0:
        q = int(s[::-1].rstrip('0').translate(''.maketrans('69','96')))
        if p != q and isprime(q) and isprime(p):
            A048890_list.append(p) # Chai Wah Wu, Sep 13 2021

Better definition and more terms from Robert G. Wilson v, Feb 27 2007

A153807 Strobogrammatic cyclops primes.

Original entry on oeis.org

101, 16091, 1160911, 1180811, 1190611, 1690691, 1880881, 1960961, 1990661, 6110119, 6610199, 6860989, 166906991, 168101891, 169609691, 188906881, 189808681, 196906961, 199609661, 616906919, 661609199, 666101999, 668609899

Offset: 1

Omar E. Pol, Jan 15 2009

Primes in A153806.

Cf. A000787, A007597, A134808, A134809, A138131, A138148, A153806.

Extended by Ray Chandler, May 20 2009

A080788 Primes that are still primes when turned upsided down.

Original entry on oeis.org

11, 19, 61, 101, 109, 181, 199, 601, 619, 661, 1019, 1061, 1091, 1109, 1181, 1601, 1609, 1669, 1699, 1811, 1901, 1999, 6011, 6091, 6101, 6199, 6619, 6661, 6689, 6691, 6899, 6991, 10061, 10069, 10091, 10691, 10861, 10909, 11069, 11681, 11909, 16001, 16091

Offset: 1

P. Giannopoulos (pgiannop1(AT)yahoo.com), Mar 12 2003

P. Giannopoulos, The Brainteasers, unpublished.

K. D. Bajpai, Table of n, a(n) for n = 1..11210 (First 1000 terms from Reinhard Zumkeller)

Cf. A007597, A057770, A080789.

import Data.List (unfoldr)
a048890 n = a048890_list !! (n-1)
a048890_list = filter f a000040_list where
   f x = all (`elem` [0,1,6,8,9]) ds && x' /= x && a010051 x' == 1
     where x' = foldl c 0 ds
           c v 6 = 10*v + 9; c v 9 = 10*v + 6; c v d = 10*v + d
           ds = unfoldr d x
           d z = if z == 0 then Nothing else Just $ swap $ divMod z 10
-- Reinhard Zumkeller, Nov 18 2011

from sympy import isprime
from itertools import product
def ud(s):
    return s[::-1].translate({ord('6'):ord('9'), ord('9'):ord('6')})
def auptod(maxdigits):
    alst = []
    for d in range(1, maxdigits+1):
        for p in product("01689", repeat=d-1):
            if d > 1 and p[0] == "0": continue
            for end in "19":
                s = "".join(p) + end
                t, udt = int(s), int(ud(s))
                if isprime(t) and isprime(udt): alst.append(t)
    return alst
print(auptod(5)) # Michael S. Branicky, Nov 19 2021

Missing 1669 and 6689 inserted by Reinhard Zumkeller, Nov 18 2011

A133207 Strobogrammatic non-palindromic primes.

Original entry on oeis.org

619, 16091, 19861, 61819, 116911, 119611, 160091, 169691, 191161, 196961, 686989, 688889, 1068901, 1160911, 1190611, 1191611, 1681891, 1690691, 1898681, 1908061, 1960961, 1990661, 6081809, 6100019, 6108019, 6110119, 6608099, 6610199

Offset: 1

Tanya Khovanova, Oct 10 2007

Primes which are invariant under a 180-degree rotation, but do not have a mirror symmetry.

Cf. The elements of this sequence are the elements of A007597 (Strobogrammatic primes) excluding the elements of A068188 (Tetradic primes).

Select[Range[10000000], PrimeQ[ # ] && Union[IntegerDigits[ # ], {0, 1, 6, 8, 9}] == {0, 1, 6, 8, 9} && Reverse[IntegerDigits[ # ] /. {6 -> 9, 9 -> 6} ] == IntegerDigits[ # ] && Reverse[IntegerDigits[ # ]] != IntegerDigits[ # ] &]

A155584 Array, read by antidiagonals, of n-th strobogrammatic number in base k.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 5, 4, 0, 1, 5, 10, 7, 5, 0, 1, 6, 17, 13, 9, 6, 0, 1, 7, 26, 21, 28, 15, 7, 0, 1, 8, 37, 31, 65, 40, 17, 8, 0, 1, 9, 50, 43, 126, 85, 82, 21, 9, 0, 1, 8, 65, 57, 217, 156, 257, 91, 27, 10, 0, 1, 8, 10, 73, 344, 259, 626, 273, 112, 31, 11, 0, 1, 8, 11, 80, 513, 400, 1297, 651, 325, 121, 33, 12

Offset: 1

Jonathan Vos Post, Jan 24 2009

If a binary number is palindromic, it is also strobogrammatic. In bases 3 through 7, this is not true, where only digits 0 and 1 can be used, because 8 is not a digit, nor are either of the inversion paid (6,9). I do not show bases beyond 10, although admittedly some letters as digits are other letters upside-down.

			A[2,4] = 5 because 4th strobogrammatic number base 2 = 101 = 5 (base 10). A[9,8] = 154 because 8th strobogrammatic number base 9 = 181 = 154 (base 10). The array begins: ===================================================================================
..n.|.1.|.2.|.3.|..4.|..5.|...6.|...7.|....8.|....9.|...10.|...11.|....12.|
===================================================================================
k=1.|.0.|.1.|.2.|..3.|..4.|...5.|...6.|....7.|....8.|....9.|...10.|....11.|
k=2.|.0.|.1.|.3.|..5.|..7.|...9.|..15.|...17.|...21.|...27.|...31.|....33.|A006995
k=3.|.0.|.1.|.4.|.10.|.13.|..28.|..40.|...82.|...91.|..112.|..121.|...244.|
k=4.|.0.|.1.|.5.|.17.|.21.|..65.|..85.|..257.|..273.|..325.|..341.|..1025.|
k=5.|.0.|.1.|.6.|.26.|.31.|.126.|.156.|..626.|..651.|..756.|..781.|..3126.|
k=6.|.0.|.1.|.7.|.37.|.43.|.217.|.259.|.1297.|.1333.|.1519.|.1555.|..7777.|
k=7.|.0.|.1.|.8.|.50.|.57.|.344.|.400.|.2402.|.2451.|.2752.|.2801.|.16808.|
k=8.|.0.|.1.|.9.|.65.|.73.|.513.|.585.|.4097.|.4161.|.4617.|.4681.|.32769.|
k=9.|.0.|.1.|.8.|.10.|.80.|..82.|..91.|..154.|..656.|..665.|..728.|...730.|
k=10|.0.|.1.|.8.|.11.|.69.|..88.|..96.|..101.|..111.|..181.|..609.|...619.|A000787
===================================================================================

Cf. A006995, A000787, A006072, A068188, A007597.

strobo := proc(b,n)
        option remember;
        local a;
        if n <=2 then
                return n-1 ;
        elif b = 1 then
                return n-1 ;
        else
                for a from procname(b,n-1)+1 do
                        isstrobo := true ;
                        dgsa := convert(a,base,b) ;
                        for d from 1 to nops(dgsa) do
                                if op(d,dgsa)=1 and op(-d,dgsa) <> 1 then
                                        isstrobo := false;
                                elif op(d,dgsa)=8 and op(-d,dgsa) <> 8 then
                                        isstrobo := false;
                                elif op(d,dgsa)=6 and op(-d,dgsa) <> 9 then
                                        isstrobo := false;
                                elif op(d,dgsa)=9 and op(-d,dgsa) <> 6 then
                                        isstrobo := false;
                                elif op(d,dgsa)=0 and op(-d,dgsa) <> 0 then
                                        isstrobo := false;
                                elif op(d,dgsa) in { 2,3,4,5,7} then
                                        isstrobo := false;
                                end if;
                        end do;
                        if isstrobo then
                                return a;
                        end if;
                end do:
        end if;
end proc: # R. J. Mathar, Sep 30 2011

A274831 Strobogrammatic (A000787) composite numbers whose prime factors are strobogrammatic as well.

Original entry on oeis.org

1111, 11088011, 16169191, 101808101, 616101919, 668808899, 686808989, 1016969101, 1106699011, 1199006611, 6068888909, 6699116699, 6699886699, 11990006611, 16168089191, 19190006161, 19198086161, 66110001199, 66880008899, 110668899011, 161600009191

Offset: 1

G. L. Honaker, Jr. and Giovanni Resta, Jul 08 2016

The smallest term which is the product of 3 primes is 1111008888001111 = 11 * 101 * 1000008000001. The smallest term which is not a multiple of 11 or 101 is a(790) = 100111801888108111001 = 100111001 * 1000008000001.

			16169191 is a term because it is equal to 101 * 160091.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A000787, A007597.

A370447 Palindromic prime numbers that consist only of the digits {0,1,6,8,9} and which remain palindromic primes when their digits are rotated by 180 degrees.

Original entry on oeis.org

11, 101, 181, 16661, 18181, 19991, 1008001, 1160611, 1180811, 1190911, 1688861, 1880881, 1881881, 1988891, 100111001, 100888001, 101616101, 101919101, 106111601, 106191601, 108101801, 109111901, 109161901, 110111011, 111010111, 111181111, 116010611, 116696611

Offset: 1

Jean-Marc Rebert, Feb 18 2024

10886968801 is the least palindromic prime of this sequence for which the set of digits is {0,1,6,8,9}.

Terms must start and end with digit 1 and be of odd length for n > 1. - Michael S. Branicky, Feb 19 2024

			16661 becomes 19991 under such a rotation, and both are palindromic primes.

Michael S. Branicky, Table of n, a(n) for n = 1..16934 (all terms < 10^20)
Carlos Rivera, Puzzle 1163. Palindromic primes that are semiprimes when turned upside down, The Prime Puzzles & Problems Connection.

Subsequence of palindromes in A007597.

Cf. A002385, A178316.

rot(u)=my(v=[]);for(i=1,#u,my(x=u[i]);if(x==6,v=concat(9,v),x==9,v=concat(6,v),vecsearch([0,1,8],x)>0,v=concat(x,v)));v
is(x)=my(u=digits(x),su=Set(u));if(setintersect(su,Set([0,1,6,8,9]))!=su||!isprime(x)||Vecrev(u)!=u,return(0));my(y=fromdigits(rot(u)));return(isprime(y))

from sympy import isprime
from itertools import product, count, islice
def flip180(s): return s[::-1].translate({54:57, 57:54})
def agen(): # generator of terms
    yield 11
    for digits in count(3, 2):
        for rest in product("01689", repeat=digits//2-1):
            for mid in "01689":
                s = "".join(("1",)+rest+(mid,)+rest[::-1]+("1",))
                if isprime(t:=int(s)) and isprime(int(flip180(s))):
                    yield t
print(list(islice(agen(), 28))) # Michael S. Branicky, Feb 19 2024

cp's OEIS Frontend

A000787 Strobogrammatic numbers: the same upside down.

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A153806 Strobogrammatic cyclops numbers.

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A018847 Strobogrammatic primes: the same upside down (calculator-style numerals).

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PARI

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A048890 Primes that yield a different prime when rotated by 180 degrees.

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A153807 Strobogrammatic cyclops primes.

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A080788 Primes that are still primes when turned upsided down.

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A133207 Strobogrammatic non-palindromic primes.

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A155584 Array, read by antidiagonals, of n-th strobogrammatic number in base k.

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