A068188 -id:A068188 - cp's OEIS frontend

A119742 Partial sum of tetradic primes (A068188).

11, 112, 293, 18474, 1026475, 2207286, 4088167, 5970048, 106081049, 206969050, 315070851, 425181862, 536191973, 647373084, 766191895, 946292976, 1127404157, 1309292338, 1497303219, 1686192100, 11694372101, 21776190102

Offset: 1

Jonathan Vos Post, Jun 16 2006

Tetradic primes are primes that are palindromes and use only the digits 0, 1 and 8, so they read the same backwards and upside down. a(1) = 11, a(3) = 293 and a(21) = 11694372101 are primes.

			a(21) = 11 + 101 + 181 + 18181 + 1008001 + 1180811 + 1880881 + 1881881 + 100111001 + 100888001 + 108101801 + 110111011 + 111010111 + 111181111 + 118818811 + 180101081 + 181111181 + 181888181 + 188010881 + 188888881 + 10008180001 = 11694372101.

Eric Weisstein's World of Mathematics, Tetradic Number.

Cf. A000040, A006072, A068188.

a(n) = SUM[i=1..n] A068188(i). a(n) = SUM[i=1..n] {A006072(k) such that A006072(k) is in A000040}.

Corrected by Jens Kruse Andersen, Apr 27 2010

A068187 a(n) is the smallest number such that the product of its decimal digits equals n^n, or 0 if no solutions exist.

Original entry on oeis.org

1, 4, 39, 488, 55555, 88999, 7777777, 88888888, 999999999, 25555555555888, 0, 88888888999999, 0, 4777777777777778888, 35555555555555559999999, 2888888888888888888888, 0, 888888999999999999999999, 0, 2555555555555555555558888888888888, 37777777777777777777779999999999

Offset: 1

Labos Elemer, Feb 18 2002

a(n) = 0 if and only if n has a prime factor > 7. If n > 1 has no prime factor > 7, let n^n = 2^a*3^b*5^c*7^d. Let m(x) denote the number of digit x in a(n). Then a(n) is a number whose digits are nondecreasing and defined as follows. m(2) = 1 if a mod 3 == 1 and 0 otherwise, m(3) = 1 if b mod 2 == 1 and 0 otherwise, m(4) = 1 if a mod 3 == 2 and 0 otherwise, m(5) = c, m(6) = 0, m(7) = d, m(8) = floor(a/3), m(9) = floor(b/2). - Chai Wah Wu, Aug 12 2017

Chai Wah Wu, Table of n, a(n) for n = 1..200

Cf. A000312, A067734, A068183, A068184, A068185, A068186, A068188, A068190.

from sympy import factorint
def A068187(n):
    if n == 1:
        return 1
    pf = factorint(n)
    return 0 if max(pf) > 7 else int(''.join(sorted(''.join(str(a)*(n*b) for a,b in pf.items()).replace('222','8').replace('22','4').replace('33','9')))) # Chai Wah Wu, Aug 13 2017

Edited by Dean Hickerson and Henry Bottomley, Mar 07 2002

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

A155801 Nontrivial "Strobogrammatic" primes, the same "upside-down" in at least one base b with 2 <= b <= 10.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 31, 37, 43, 73, 101, 107, 127, 181, 257, 313, 443, 619, 757, 1093, 1193, 1297, 1453, 1571, 1619, 1787, 1831, 1879, 2801, 4889, 5113, 5189, 5557, 5869, 5981, 6211, 6827, 7607, 7759, 7919, 8191

Offset: 1

Jonathan Vos Post, Jan 27 2009

I have to say "nontrivial" because every nonnegative integer is strobogrammatic in base 1. Strobogrammatic binary primes == primes in A006995 == A016041. Strobogrammatic primes in base 3 = 13, 757, 1093, 9103, ... == primes strobogrammatic in bases 2 and 3. For bases 2 < k < 8 we have that every strobogrammatic prime in base k must also be strobogrammatic in base 2 and hence palindromic in base 2. Hence we have, for example, strobogrammatic base-4 primes = A056130 = "Palindromic primes in bases 2 and 4."

Strobogrammatic primes in base 5 = 31, 19531, 394501, 472631, ... == primes strobogrammatic in base 2 and base 5. Strobogrammatic primes base 6 = 7, 37, 43, 1297, 55987, ... == primes strobogrammatic in base 2 and base 6. Note that 1101011 (base 6) = 18881 (base 10) which is strobogrammatic base 10 but not prime base 6 nor 10 (though prime base 2). Strobogrammatic primes base 7 = 2801, 134807, this last being strobogrammatic prime in bases 2, 4 and 7. Strobogrammatic primes base 8 = 73, 262657, 295433, ... Strobogrammatic primes base 9 break the above pattern, as they can have the digit 8 and are A068188 (tetradic primes). Strobogrammatic primes base 10 == A007597. Except sometimes for the first element, these (for the same range of k) must all have an odd number of digits.

			5189 = 1101011 (base 6) which numeral string is the same upside-down (and backwards). 11, 101, 181 and 619 are strobogrammatic base 10, the conventional interpretation of the word.

Cf. A000040, A006995, A016041, A056130, A007597, A133207, A155584.

A000040 INTERSECTION A155584[1 < k < 11, n].

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A119742 Partial sum of tetradic primes (A068188).

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A068187 a(n) is the smallest number such that the product of its decimal digits equals n^n, or 0 if no solutions exist.

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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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A155801 Nontrivial "Strobogrammatic" primes, the same "upside-down" in at least one base b with 2 <= b <= 10.

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