A097393 -id:A097393 - cp's OEIS frontend

A099954 Numbers k such that Fibonacci(k) and its reversal are two distinct semiprimes.

19, 22, 31, 41, 59, 107, 193, 199, 227, 467

Offset: 1

Jonathan Vos Post and G. L. Honaker, Jr., Nov 13 2004

a(11) > 1000. - Donovan Johnson, Jun 06 2009

a(11) >= 1801. Inclusion of 1801 depends on the factorization of Fibonacci(1801), a 377-digit composite number. - Tyler Busby, Jan 14 2023

			F(19) = 4181 = 37 * 113, reverse(F(19)) = 1814 = 2 * 907.

factordb, Status of Fibonacci(1801).

F(a(n)) is the intersection of A053409 and A097393

with(combinat): with(numtheory): rev:=proc(n) local nn: nn:=convert(n,base,10): add(nn[nops(nn)+1-j]*10^(j-1),j=1..nops(nn)) end: a:=proc(n): if rev(fibonacci(n))<>fibonacci(n) and bigomega(fibonacci(n))=2 and bigomega(rev(fibonacci(n)))=2 then n else fi end: seq(a(n),n=1..200); # Emeric Deutsch, Jul 26 2006

fspQ[n_]:=Module[{f=Fibonacci[n]},f!=IntegerReverse[f]&&PrimeOmega[f] == PrimeOmega[IntegerReverse[f]]==2]; Select[Range[470],fspQ] (* Harvey P. Dale, Jul 24 2016 *)

is(k) = {(fib=fibonacci(k))!=(fibrev=fromdigits(Vecrev(digits(fib)))) && (bigomega(fib)==2 && bigomega(fibrev)==2)} \\ Tyler Busby, Jan 07 2023

More terms from Emeric Deutsch, Jul 26 2006

a(10) from Donovan Johnson, Jun 06 2009

A217252 Lucky numbers whose digital reversal is another lucky number.

Original entry on oeis.org

13, 15, 31, 37, 51, 73, 115, 133, 163, 169, 189, 193, 195, 327, 331, 339, 361, 385, 391, 399, 511, 529, 537, 579, 583, 591, 723, 729, 735, 739, 925, 927, 933, 937, 961, 975, 981, 993

Offset: 1

M. F. Hasler, Mar 16 2013

Inspired by the error in A140291 (forgotten palindromes 33 and 99), pointed out by L. Edson Jeffery in a post to the SeqFan list.
This sequence is obtained from A118561 by removal of the palindromes A031161 = (1, 3, 7, 9, 33, 99, 111, 141, 151, 171, 303, 393, 535, 717, 727, 777, 787, 979, ...)
By analogy with the "emirps" A006567 and "emirpimes" A097393, these might be called "ykcul"s, for lucky numbers which, when reversed, are different lucky numbers.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000959, A002385, A006567, A097393, A029742, A031161, A002113, A118561, A140291.

A217252 = A118561 \ A031161 = A118561 \ A002113 = A118561 intersect A029742 = { n in A000959 | R(n) is in A000959 and different from n }, where R = A004086.

A264815 Semirps: a semirp (or semi-r-p) is a semiprime r*p with r and p both reversed primes.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 39, 49, 51, 55, 62, 65, 74, 77, 85, 91, 93, 111, 119, 121, 142, 143, 146, 155, 158, 169, 185, 187, 194, 202, 213, 214, 217, 219, 221, 226, 237, 259, 262, 289, 291, 298, 302, 303, 314, 321, 334, 339, 341, 355

Offset: 1

Danny Rorabaugh, Nov 25 2015

A semiprime (A001358) is the product of two prime, not necessarily distinct. A semiprime is in this list if those two primes (A000040) are reversed primes (A004087).

Since A007500 is the intersection of A000040 and A004087, this sequence is also the sorted list of all r*p with r and p in A007500.

			9 is in the list because 9 = 3*3 is a semiprime and reverse(3) = 3 is prime.
143 is in the list because 143 = 11*13 is a semiprime and both reverse(11) = 11 and reverse(13) = 31 are prime.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000

Cf. A001358, A006567, A007500, A097393, A109019, A115670.

With[{nn=250},Take[Union[Times@@@Select[Tuples[IntegerReverse/@Prime[Range[nn]],2],AllTrue[#,PrimeQ]&]],60]] (* Harvey P. Dale, Apr 27 2025 *)

reverse = lambda n: sum([10^i*int(str(n)[i]) for i in range(len(str(n)))])
def is_semirp(n):
  F = factor(n)
  if sum([f[1] for f in F])==2:
    r, p = F[0][0], F[-1][0]
    if is_prime(reverse(r)) and is_prime(reverse(p)): return True
[a for a in range(1,356) if is_semirp(a)] # Danny Rorabaugh, Nov 25 2015

[A007500]^2, sorted.

A373731 Semiprimes k such that the digit reversal of k is a semiprime > k.

Original entry on oeis.org

15, 26, 39, 49, 58, 115, 122, 123, 129, 143, 155, 158, 159, 169, 177, 178, 183, 185, 187, 203, 205, 226, 265, 289, 314, 319, 326, 327, 329, 335, 339, 355, 394, 398, 415, 437, 497, 538, 559, 586, 589, 629, 667, 718, 899, 1006, 1011, 1027, 1041, 1043, 1046, 1047, 1057, 1059, 1067, 1079, 1115, 1119

Offset: 1

Zak Seidov and Robert Israel, Jun 17 2024

			a(3) = 39 is a term because 39 = 3*13 is a semiprime, its reversal 93 = 3*31 is a semiprime, and 93 > 39.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A001358, A004086, A050700, A097393.

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
filter:= proc(n) local r;
  r:= rev(n);
  r > n and numtheory:-bigomega(n) = 2 and numtheory:-bigomega(r) = 2
end proc:
select(filter, [$1..2000]);

s = {}; Do[fd = FromDigits[Reverse[IntegerDigits[k]]]; If[{2, 2} ==PrimeOmega[{fd, k}] && fd > k, AppendTo[s, k]], {k, 1000}]; s

A085777 Semiprimes such that when they are concatenated with their reversal are semiprime.

Original entry on oeis.org

14, 34, 38, 74, 91, 94, 115, 146, 169, 185, 194, 305, 334, 341, 346, 355, 361, 362, 386, 394, 395, 398, 706, 721, 731, 745, 755, 766, 794, 799, 901, 923, 926, 955, 974, 985, 1027, 1046, 1094, 1138, 1141, 1147, 1174, 1195, 1202, 1238, 1247, 1261, 1262, 1271

Offset: 1

Jason Earls, Jul 23 2003

This is to A001358 = semiprimes (or biprimes): products of two primes, as A054217 is to primes A000040. Note that a subsequence is the emirpimes (A097393) which, when concatenated with their digital reversals, become different semiprimes. Examples of this subsequence: 94, 115, 169, 185, ... because 9449 = 11 * 859, 115511 = 11 * 10501, 169961 = 11 * 15451, 185581 = 11 * 16871. [From Jonathan Vos Post, Apr 05 2010]

			a(7)=115 because 115511 = 11*10501.

Cf. A001358.

Cf. A000040, A001358, A004086, A097393 [From Jonathan Vos Post, Apr 05 2010]

{positive integers s = p*q in A001358, p and q primes, such that Concatenate(s, R(s)) = u*v is in A001358, u and v primes} = {positive integers s = p*q in A001358, p and q primes, such that Concatenate(s, A004086(s)) = u*v is in A001358, u and v primes}. [From Jonathan Vos Post, Apr 05 2010]

A097435 Numbers n such that n and its reversal are distinct brilliant numbers (A078972).

Original entry on oeis.org

143, 169, 187, 319, 341, 781, 913, 961, 1273, 1343, 1691, 1843, 1961, 3431, 3481, 3721, 10807, 11413, 12769, 15049, 15347, 15707, 15857, 16171, 16837, 16867, 17161, 18203, 18437, 19939, 30227, 30281, 31411, 31439, 31979, 32639, 33017, 33109

Offset: 1

Jason Earls, Aug 22 2004

A subset of A097393.

Sometimes called "Tnaillirb" numbers. - Jonathan Vos Post, Nov 22 2004

			30227 is in the sequence because 30227=167*181 and 72203=103*701.

Cf. A001358.

A114015 Quaternary emirpimes.

Original entry on oeis.org

12, 21, 1022, 1102, 1113, 1222, 1233, 1303, 1313, 1323, 2011, 2012, 2032, 2102, 2201, 2221, 2302, 3031, 3111, 3131, 3231, 3321, 10102, 10213, 10231, 10232, 10233, 10322, 11012, 11033, 11103, 11231, 11321, 11331, 12013, 12022, 12023, 12032

Offset: 1

Jonathan Vos Post, Jun 13 2006

These are semiprimes when read as base-4 numbers and their reversals are different semiprimes when read as base-4 numbers. Base-4 analog of what for base 3 is A119684 and for base 10 is A097393. The base-10 representation of this sequence is 6, 9, 74, 87, 106, 111, 115, 119, 123, 133, 134, 142, 146, 161, 169, 178, 205, 213, 221, 237.

			a(1) = 12 because 12 (base 4) = 6 (base 10) is semiprime and R(12) = 21, where 21 (base 4) = 9 (base 10) is a different semiprime.
a(19) = 3131 because 3131 (base 4) = 221 (base 10) = 13 * 17 (base 10) is semiprime and R(3131) = 1313, where 1313 (base 4) = 119 (base 10) = 7 * 17 (base 10) is a different semiprime.

Eric Weisstein, Jonathan Vos Post, et al., Emirpimes.
Eric Weisstein's World of Mathematics, Quaternary.

Cf. A001358, A004086, A007090, A097393.

A007090 := proc(n) local b4; b4 := convert(n,base,4) ; add( op(i,b4)*10^(i-1),i=1..nops(b4)) ; end proc:
isA114015 := proc(n) local b4;b4rev; if isA001358(n) then b4 := convert(n,base,4) ; b4rev := add(op(-i,b4)*4^(i-1),i=1..nops(b4)) ; if n = b4rev then false; else isA001358(b4rev) ; end if; else false; end if; end proc:
for n from 1 to 400 do if isA114015(n) then printf("%d,",A007090(n)) ; end if; end do: # R. J. Mathar, Dec 22 2010

a(n) = A007090(i) for some i in A001358 and R(a(n)) = A007090(j) for some j =/= i in A001358. a(n) = A007090(i) for some i in A001358 and A004086(a(n)) = A007090(j) for some j =/= i in A001358.

1102 inserted by R. J. Mathar, Dec 22 2010

A159238 Primes which are digital reversals of products of "emirpimes" pairs.

Original entry on oeis.org

2161, 39841, 242491, 414553, 513691, 555871, 830551, 854431, 904261, 1663951, 1706473, 2060803, 3201643, 5543029, 6075379, 6497509, 6830797, 6846787, 7245937, 7558297, 9300043, 9339439, 12248779, 23175751, 23793631, 24769057

Offset: 1

Jonathan Vos Post, Apr 06 2009

			a(1) = 2161 because this is prime, and R(A158126(2)) = R(1612) = 2161. a(2) = 39841 because this is prime, and R(A158126(20)) = R(148930) = 039841 which in OEIS form strips away leading zero to become 39841. a(7)is the prime 854431 = R(A158126(16)) = R(134458).

Cf. A000040, A001358, A004086, A014613, A083815, A097393, A158126.

{A004086(A158126(i)) such that A004086(A158126(i)) is an element of A000040}.

Edited by N. J. A. Sloane, Apr 07 2009

414553 inserted by R. J. Mathar, Dec 06 2009

A181931 Lesser of emirpimes pairs the product of whose members has prime digital sum.

Original entry on oeis.org

115, 205, 226, 289, 335, 497, 667, 718, 1027, 1057, 1079, 1135, 1141, 1154, 1195, 1234, 1243, 1247, 1286, 1315, 1322, 1343, 1357, 1379, 1387, 1402, 1415, 1466, 1469, 1502, 1513, 1514, 1538, 1658, 1679, 1691, 1703, 1765, 1769, 1774, 1817, 1843, 1882, 1927, 1937, 1942

Offset: 1

Jonathan Vos Post, Apr 02 2012

This is to A210547 (Lesser of emirp pairs whose members have prime digital products) as emirpimes A097393 are to emirps A006567 and as A007953 (digital sums) are to A007954 (digital products).

			The smallest emirpimes, 15, is not an element, because 15 * 51 = 765 and 7 + 6 + 5 = 18, which is composite.
a(1) = 115 because 115 * 511 = 58765 and 5+8+7+6+5 = 31 is prime.
a(2) = 205 because 205 * 502 = 102910 and 1+0+2+9+1+0 = 13 is prime.
a(3) = 226 because 226 * 622 = 140572 and 1+4+0+5+7+2 = 19 is prime.

Cf. A001358, A004086, A007953, A007954, A097393, A210547.

read("transforms");
# insert A097393 code here
isA181931 := proc(n)
    local R ;
    R := digrev(n) ;
    if n < R then
        if isA097393(n) then
            isprime(digsum(n*R)) ;
        else
            false;
        end if;
    else
        false;
    end if;
end proc:
for n from 1 to 2000 do
    if isA181931(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Apr 05 2012

{k in A097393, k < R(k), such that A007953(k * R(k)) is prime}, where R(k) = A004086(k).

More terms from Robert G. Wilson v, Apr 04 2012

A189484 Numbers that can be factored into semiprimes which, when concatenated in increasing order, produce a palindrome.

Original entry on oeis.org

1, 4, 6, 9, 16, 22, 33, 36, 55, 56, 64, 77, 81, 111, 121, 136, 141, 156, 161, 202, 216, 256, 262, 276, 296, 303, 323, 351, 376, 393, 441, 454, 484, 505, 515, 516, 535, 545, 560, 565, 621, 626, 707, 717, 729, 737, 765, 767, 784, 818, 838, 878, 898, 939, 949

Offset: 1

Jonathan Vos Post, Apr 22 2011

This is to semiprimes A001358 as A046447 is to primes A000040.

The initial 1 represents the empty product.

			The first value not itself a semiprime palindrome (A046328) or power of semiprimes (i.e., 16 = 4 * 4 which concatenate to the palindrome 44, 484 = 22^2) is 56 = 4 * 14. The first where additionally the first factor is not a single digit is 765 = 15 * 51 = 3^2 * 5 * 17 since (15, 51) are a pair of emirpimes A097393, and 765 = A158126(1).

Cf. A001358, A002113, A046328, A046447, A097393, A158126 Products of emirpimes pairs, sorted.

ok[n_] := n == 1 || Block[{d, p = Join @@ mu /@ FactorInteger[n]}, EvenQ@ Length[p] && AnyTrue[ Union[ Sort /@ ((Times @@@ #) & /@ Union[ (Sort /@ Partition[#, 2]) & /@ Permutations[p]])], (d = Join @@ IntegerDigits[#]; d == Reverse[d]) &]]; Select[ Range[1000], ok] (* Giovanni Resta, Sep 15 2018 *)

Additional terms from Franklin T. Adams-Watters, Apr 28 2011

More terms from Giovanni Resta, Sep 15 2018

cp's OEIS Frontend

A099954 Numbers k such that Fibonacci(k) and its reversal are two distinct semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A217252 Lucky numbers whose digital reversal is another lucky number.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A264815 Semirps: a semirp (or semi-r-p) is a semiprime r*p with r and p both reversed primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Formula

A373731 Semiprimes k such that the digit reversal of k is a semiprime > k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A085777 Semiprimes such that when they are concatenated with their reversal are semiprime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A097435 Numbers n such that n and its reversal are distinct brilliant numbers (A078972).

Views

Author

Keywords

Comments

Examples

Crossrefs

A114015 Quaternary emirpimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A159238 Primes which are digital reversals of products of "emirpimes" pairs.

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A181931 Lesser of emirpimes pairs the product of whose members has prime digital sum.