A097393 -id:A097393 - cp's OEIS frontend

A109030 Numbers that have exactly ten prime factors counted with multiplicity (A046314) whose digit reversal is different and also has 10 prime factors (with multiplicity).

46848, 84864, 217152, 219456, 232848, 251712, 257664, 259776, 274104, 276048, 401472, 415584, 422820, 428160, 428736, 447360, 466752, 485514, 637824, 650160, 654912, 677952, 808320, 840672, 846369, 848232, 963648

Offset: 1

Jonathan Vos Post, Jun 16 2005

This sequence is the k = 10 instance of the series which begins with k = 1 (emirps), k = 2, k = 3 (A109023), k = 4 (A109024), k = 5 (A109025), k = 6 (A109026), k = 7 (A109027), k = 8 (A109028), k = 9 (A109029).

			a(1) = 46848 is in this sequence because 46848 = 2^8 * 3 * 61 has exactly 10 prime factors counted with multiplicity and reverse(46848) = 84864 = 2^7 * 3 * 13 * 17 also has exactly 10 prime factors counted with multiplicity.

David A. Corneth, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein and Jonathan Vos Post, Emirpimes.

Cf. A046314, A006567, A097393, A109018, A109023, A109024, A109025, A109026, A109027, A109028, A109029, A109031.

taQ[n_]:=Module[{idn=IntegerDigits[n],rev},rev=Reverse[idn];rev!=idn&&PrimeOmega[n] == 10 == PrimeOmega[FromDigits[rev]]]; Select[Range[ 1000000], taQ] (* Harvey P. Dale, May 03 2013 *)

is(n) = {
	my(r = fromdigits(Vecrev(digits(n))));
	n!=r && bigomega(n) == 10 && bigomega(r) == 10
} \\ David A. Corneth, Mar 07 2024

A085751 Semiprimes whose reversal is also a semiprime.

Original entry on oeis.org

4, 6, 9, 15, 22, 26, 33, 39, 49, 51, 55, 58, 62, 77, 85, 93, 94, 111, 115, 121, 122, 123, 129, 141, 143, 155, 158, 159, 161, 169, 177, 178, 183, 185, 187, 202, 203, 205, 221, 226, 262, 265, 289, 302, 303, 314, 319, 321, 323, 326, 327, 329, 335, 339, 341, 355

Offset: 1

Jason Earls, Jul 21 2003

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A001358, A097393.

Select[Range[400],PrimeOmega[#]==PrimeOmega[IntegerReverse[#]]==2&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 09 2016 *)

rev(n)=subst(Polrev(digits(n)), 'x, 10)
issemi(n)=bigomega(n)==2
list(lim)=my(v=List()); forprime(p=2, lim\2, forprime(q=2, min(lim\p, p), if(issemi(rev(p*q)), listput(v, p*q)))); Set(v) \\ Charles R Greathouse IV, Jan 27 2015

A050699 Nonprime numbers n such that n and n-reversed (<> n and no leading zeros) have the same number of prime factors (counted with multiplicity).

Original entry on oeis.org

15, 26, 39, 49, 51, 58, 62, 85, 93, 94, 115, 117, 122, 123, 126, 129, 143, 147, 155, 158, 159, 165, 169, 177, 178, 183, 185, 187, 203, 205, 221, 225, 226, 244, 246, 265, 285, 286, 289, 294, 302, 314, 315, 319, 321, 326, 327, 329, 335, 338, 339, 341, 355, 366

Offset: 1

Patrick De Geest, Aug 15 1999

			E.g., 321 = 3*107 and 123 = 3*41 -> both 321 and 123 have two prime factors.

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A050700, A050701, A050702, A006567, A097393, A109018-A109031.

with(numtheory): read(transforms): for n from 12 to 366 do r:=digrev(n): if(not isprime(n) and not n=r and not n mod 10 = 0 and bigomega(n)=bigomega(r))then printf("%d, ", n); fi: od: # Nathaniel Johnston, Jun 23 2011

nrnQ[n_]:=Module[{idn=IntegerDigits[n],rev},rev=Reverse[idn];!PrimeQ[n] &&First[rev]!=0&&idn!=rev&&PrimeOmega[n]==PrimeOmega[FromDigits[rev]]]; Select[Range[400],nrnQ] (* Harvey P. Dale, Jun 23 2011 *)

Definition clarified by Harvey P. Dale, Jun 23 2011

A131371 Number of anagrams of n that are semiprimes.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 2, 0, 1, 0, 1, 0, 0, 1, 1, 0, 2, 0, 2, 1, 1, 0, 1, 1, 1, 2, 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 1, 0, 2, 2, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1

Offset: 1

Jonathan Vos Post, Sep 30 2007

An anagram of a k-digit number is one of the k! permutations of the digits that does not begin with 0.

			a(123) = 3 because 123 = 3 * 41 is semiprime, 213 = 3 * 71 is semiprime, 321 = 3 * 107 is semiprime, while the other anagrams 132, 231 and 312 have respectively 3, 3 and 5 prime factors with multiplicity.
a(129) = 4 because 129 = 3 * 43 is semiprime, 219 = 3 * 73 is semiprime, 291 = 3 * 97 is semiprime, 921 = 3 * 307 is semiprime, while 192 and 912 have 7 and 6 prime factors with multiplicity.
a(134) = 5 because 134 = 2 * 67 and 143 = 11 * 13 and 314 = 2 * 157 and 341 = 11 * 31 and 413 = 7 * 59 are semiprimes, while 431 is prime.

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

Cf. A000040, A001358, A002113, A046810, A097393.

f:= proc(n) local L,m,t,i;
  L:= convert(n,base,10); m:= nops(L);
  nops(select(t -> t[-1] <> 0 and numtheory:-bigomega(add(t[i]*10^(i-1), i=1..m))=2, combinat:-permute(L)));
end proc:
map(f, [$1..200]); # Robert Israel, Jun 11 2023

from sympy import factorint
from sympy.utilities.iterables import multiset_permutations as mp
def c(n):
    return sum(factorint(n).values()) == 2
def a(n):
    return sum(1 for p in mp(str(n)) if p[0]!="0" and c(t:=int("".join(p))))
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Jun 11 2023

A115670 Semiprimes (A001358) whose digit reversal is prime.

Original entry on oeis.org

14, 34, 35, 38, 74, 91, 95, 106, 118, 119, 133, 134, 142, 145, 146, 166, 194, 301, 305, 334, 346, 358, 361, 362, 365, 371, 377, 382, 386, 391, 395, 703, 706, 713, 721, 731, 745, 746, 749, 755, 758, 763, 778, 779, 785, 791, 793, 799, 901, 905, 914, 917, 922

Offset: 1

Giovanni Resta, Jan 31 2006

			35=5*7 is semiprime and 53 is prime.

Cf. A001358, A083815, A085778, A097393.

A109019 Numbers whose digit reversal is different and has the same number of prime factors (with multiplicity).

Original entry on oeis.org

13, 15, 17, 26, 31, 37, 39, 49, 51, 58, 62, 71, 73, 79, 85, 93, 94, 97, 107, 113, 115, 117, 122, 123, 126, 129, 143, 147, 149, 155, 157, 158, 159, 165, 167, 169, 177, 178, 179, 183, 185, 187, 199, 203, 205, 221, 225, 226, 244, 246, 265, 270, 285, 286, 289, 290

Offset: 1

Jonathan Vos Post, Jun 16 2005

Ray Chandler has coauthorship credit for this sequence.

Eric Weisstein's World of Mathematics, Almost Prime.
Eric Weisstein's World of Mathematics, Emirp.
Eric Weisstein's World of Mathematics, Emirpimes.

Cf. A006567, A097393, A109018, A109023-A109031.

Select[Range@1000, ! PalindromeQ@# && Differences@PrimeOmega@{#,IntegerReverse@#} == {0} &] (* Hans Rudolf Widmer, Jun 03 2022 *)

{A006567} U {A097393} U {A109023} U {A109024} U ... U {A109031} U ...

More terms from Stefan Steinerberger, Jun 16 2007

A119684 Ternary emirpimes.

Original entry on oeis.org

112, 211, 1021, 1102, 1201, 2011, 2022, 2202, 10111, 11101, 11112, 12102, 12202, 12212, 20121, 20212, 20221, 21111, 21202, 21221, 100102, 100201, 101011, 101122, 101221, 102001, 102002, 102012, 102022, 102122, 102222, 110101, 110102, 110122, 110211, 111102, 111202, 112011, 112121, 112122, 112202

Offset: 1

Jonathan Vos Post, Jun 08 2006

These are semiprimes when read as base 3 numbers and their reversals are different semiprimes when read as base 3 numbers. Base 10 these are: 14, 22, 34, 38, 46, 58, 62, 74, 94, 118, 122, 146, 155, 158, 178, 185, 187, ... See: A097393 Emirpimes: numbers n such that n and its reversal are distinct semiprimes. See: A004086 Read n backwards (referred to as R(n) in many sequences). See: A007089 Numbers in base 3.

Apparently numbers with trailing zeros (reversed with leading zeros), like 1220 and 10020, are not included. - R. J. Mathar, Dec 22 2010

			a(1) = 112 because 112 (base 3) = 14 (base 10) is semiprime and R(112) = 211, where 211 (base 3) = 22 (base 10) is a different semiprime.
a(13) = 12202 because 12202 (base 3) = 155 (base 10) is semiprime and R(12202) = 20221, where 20221 (base 3) = 187 (base 10) is a different semiprime.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein, Jonathan Vos Post, et al., Emirpimes.
Eric Weisstein, Vincenzo Origlio, et al., Ternary.

Cf. A001358, A004086, A007089, A097393.

R:= NULL: count:= 0:
for m from 2 while count < 100 do
for j from 1 to 2 while count < 100 do
  n:= 3*m+j;
  if numtheory:-bigomega(n) <> 2 then next fi;
  L:= convert(n,base,3);
  r:= add(L[-i]*3^(i-1),i=1..nops(L));
  if r <> n and numtheory:-bigomega(r) = 2 then
     count:= count+1; R:= R, add(L[i]*10^(i-1),i=1..nops(L))
  fi
od od:
R; # Robert Israel, Jun 07 2020

(* First run the program for A105999 *) SemiPrimeQ[n_Integer] := TrueQ[SemiPrimePi[n] > SemiPrimePi[n - 1]]; BaseForm[Select[Table[SemiPrime[n], {n, 100}], GCD[#, 3] == 1 && # != FromDigits[Reverse[IntegerDigits[#, 3]], 3] && SemiPrimeQ[FromDigits[Reverse[IntegerDigits[#, 3]], 3]] &], 3] (* From Alonso del Arte, Dec 22 2010 *)

a(n) = A007089(i) for some i in A001358 and R(a(n)) = A007089(j) for some j =/= i in A001358. a(n) = A007089(i) for some i in A001358 and A004086(a(n)) = A007089(j) for some j =/= i in A001358.

More terms from Robert Israel, Jun 07 2020

A158126 Products of emirpimes pairs, sorted.

Original entry on oeis.org

765, 1612, 3627, 4606, 4930, 26962, 39483, 48763, 58765, 61306, 69723, 85405, 102910, 107485, 118809, 129682, 134458, 136467, 140572, 146047, 148930, 151209, 155038, 162409, 178555, 194242, 196315, 203098, 213310, 236421, 283798, 291247

Offset: 1

Jonathan Vos Post, Mar 13 2009

This is to A083815 as A097393 is to A000040. A subsequence of A014613.

			a(1) = 15*51. a(2) = 26*62. a(3) = 39*93. a(4) is not 51*15 because that has already occurred as a(1). a(4) = 58*85.

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

a(n) = n-th emirpimes not used so far * R(n-th emirpimes not used so far).

{a(n)} = {A097393(n)*A004086(A097393(n))}.

4930 inserted by R. J. Mathar, Dec 06 2009

A182150 Semiprimes that are also semiprime when their digits are sorted into nondecreasing order.

Original entry on oeis.org

4, 6, 9, 14, 15, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 69, 77, 85, 93, 94, 111, 115, 118, 119, 122, 123, 129, 133, 134, 143, 145, 146, 155, 158, 159, 166, 169, 177, 178, 185, 187, 202, 205, 206, 213, 219, 221, 226, 235, 237, 247, 249, 253

Offset: 1

Jonathan Vos Post, Apr 18 2012

This is to A211654 primes that are also prime when their digits are sorted into nondecreasing order as A001358 semiprimes are to A000040 primes. There is an ambiguity arising from OEIS conventions, exemplified by the semiprime 303, which sorts to 033 and truncates to the semiprime 33.

			51 is in the sequence because, though it is a semiprime whose digits are in descending order, once the digits are sorted to be nondecreasing, it is the semiprime 15, whose digits are (left to right) nondecreasing.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A001358, A097393, A211654.

h:= proc(m) local k; for k from m+1 while isprime(k) or
            add(i[2], i=ifactors(k)[2])<>2 do od; k
    end:
a:= proc(n) option remember; local k, l, s;
      k:= h(a(n-1));
      do l:= sort(convert(k, base, 10));
         s:= add(l[i]*10^(nops(l)-i), i=1..nops(l));
         if h(s-1)=s then return k else k:=h(k) fi
      od
    end: a(0):=0:
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 22 2012

Select[Range[300],PrimeOmega[#]==2&&PrimeOmega[FromDigits[ Sort[ IntegerDigits[ #]]]]==2&] (* Harvey P. Dale, Nov 13 2014 *)

More terms from Alois P. Heinz, Apr 22 2012

A097394 Number of emirpimes < 10^n.

Original entry on oeis.org

0, 10, 90, 898, 7200, 60732, 506724, 4320516, 37171762, 323469590, 2842710084

Offset: 1

Jonathan Vos Post

Computed by Eric W. Weisstein, Aug 13 2004.

Eric Weisstein's World of Mathematics, Emirpimes.

Cf. A097393.

a(8)-a(9) from Donovan Johnson, Jun 06 2009
a(10) from Donovan Johnson, Sep 18 2010
a(11) from Amiram Eldar, Jul 19 2025

cp's OEIS Frontend

A109030 Numbers that have exactly ten prime factors counted with multiplicity (A046314) whose digit reversal is different and also has 10 prime factors (with multiplicity).

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A085751 Semiprimes whose reversal is also a semiprime.

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A050699 Nonprime numbers n such that n and n-reversed (<> n and no leading zeros) have the same number of prime factors (counted with multiplicity).

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A131371 Number of anagrams of n that are semiprimes.

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A115670 Semiprimes (A001358) whose digit reversal is prime.

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A109019 Numbers whose digit reversal is different and has the same number of prime factors (with multiplicity).

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A119684 Ternary emirpimes.

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A158126 Products of emirpimes pairs, sorted.

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