A046397
Palindromes which are the product of exactly 7 distinct primes.
Original entry on oeis.org
22444422, 24266242, 26588562, 35888853, 36399363, 43777734, 47199174, 51066015, 53588535, 53888835, 55233255, 59911995, 60066006, 62588526, 62700726, 62888826, 81699618, 87788778, 89433498, 122434221, 202040202
Offset: 1
The first two palindromes with 7 distinct prime factors are 20522502 = 2 * 3^2 * 7 * 11 * 13 * 17 * 67 and 21033012 = 2^2 * 3 * 7 * 11 * 13 * 17 * 103, but these are excluded since one of the prime factors occurs to a higher power.
a(1) = 22444422 = 2 * 3 * 7 * 11 * 13 * 37 * 101, which is squarefree, is therefore the first term of this sequence.
Cf.
A046333 (similar but prime factors counted with multiplicity),
A373467 (similar but counting just the distinct prime divisors).
Cf.
A002113 (palindromes),
A123321 (products of 7 distinct primes),
A176655 (numbers with omega = 7 distinct prime divisors).
-
digrev:= 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 F;
F:= ifactors(n)[2];
nops(F) = 7 and map(t -> t[2],F)=[1$7]
end proc:
Res:= NULL:
count:= 0:
for d from 2 while count < 100 do
if d::even then
m:= d/2;
for n from 10^(m-1) to 10^m-1 while count < 100 do
v:= n*10^m+digrev(n);
if filter(v) then count:= count+1; Res:= Res, v; fi;
od;
else
m:= (d-1)/2;
for n from 10^(m-1) to 10^m-1 while count < 100 do
for y from 0 to 9 while count < 100 do
v:= n*10^(m+1)+y*10^m+digrev(n);
if filter(v) then count:= count+1; Res:= Res, v; fi;
od od
fi
od:
Res; # Robert Israel, Jan 20 2020
-
A046397_upto(N, start=vecprod(primes(7)), num_fact=7)={ my(L=List()); is_A002113(start)&& start--; while(N >= start = nxt_A002113(start), omega(start)==num_fact && issquarefree(start) && listput(L, start)); L} \\ M. F. Hasler, Jun 06 2024
A373467
Palindromes with exactly 7 (distinct) prime divisors.
Original entry on oeis.org
20522502, 21033012, 22444422, 23555532, 24266242, 25777752, 26588562, 35888853, 36399363, 41555514, 41855814, 42066024, 43477434, 43777734, 44888844, 45999954, 47199174, 51066015, 51666615, 52777725, 53588535, 53888835, 55233255, 59911995, 60066006, 60366306, 61777716, 62588526, 62700726
Offset: 1
Obviously all terms must be palindromic; let us consider the prime factorization:
a(1) = 20522502 = 2 * 3^2 * 7 * 11 * 13 * 17 * 67 has exactly 7 distinct prime divisors, although the factor 3 appears twice in the factorization. (Without the second factor 3 the number would not be palindromic.)
a(2) = 21033012 = 2^2 * 3 * 7 * 11 * 13 * 17 * 103 has exactly 7 distinct prime divisors, although the factor 2 appears twice in the factorization. (Without the second factor 2 the number would not be palindromic.)
a(3) = 22444422 = 2 * 3 * 7 * 11 * 13 * 37 * 101 is the product of 7 distinct primes (cf. A123321), hence the first squarefree term of this sequence.
Cf.
A046333 (same with bigomega = 7: counting prime factors with multiplicity),
A046397 (same but only squarefree terms),
A373465 (same with omega = 5),
A046396 (same with omega = 6).
-
A373467_upto(N, start=vecprod(primes(7)), num_fact=7)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && listput(L, start)); L}
A046381
Palindromes with exactly 7 palindromic prime factors (counted with multiplicity).
Original entry on oeis.org
82728, 279972, 657756, 29077092, 85688658, 89288298, 561363165, 579828975, 582636285, 819828918, 2209559022, 5154334515, 5637337365, 6824774286, 8904664098, 19585758591, 42372027324, 62663836626, 70151815107, 167985589761, 4650120210564, 6552226222556
Offset: 1
The palindrome 7051034301507 is a term since it has 7 factors 3^2 7 101^2 313 35053, all palindromic.
A348050
Palindromes setting a new record of their number of prime divisors A001222.
Original entry on oeis.org
1, 2, 4, 8, 88, 252, 2112, 4224, 8448, 44544, 48384, 405504, 4091904, 405909504, 677707776, 4285005824, 21128282112, 29142024192, 4815463645184, 445488555884544, 27874867776847872, 40539458585493504, 63556806860865536, 840261068860162048, 4870324782874230784
Offset: 1
-
m=0;lst=Union@Flatten[Table[{FromDigits@Join[s=IntegerDigits@n,Reverse@s],FromDigits@Join[w=IntegerDigits@n,Rest@Reverse@w]},{n,10^5}]];Do[t=PrimeOmega@lst[[n]];If[t>m,Print@lst[[n]];m=t],{n,Length@lst}] (* Giorgos Kalogeropoulos, Oct 25 2021 *)
-
from sympy import factorint
from itertools import product
def palsthru(maxdigits):
midrange = [[""], [str(i) for i in range(10)]]
for digits in range(1, maxdigits+1):
for p in product("0123456789", repeat=digits//2):
left = "".join(p)
if len(left) and left[0] == '0': continue
for middle in midrange[digits%2]:
yield int(left+middle+left[::-1])
def afind(maxdigits):
record = -1
for p in palsthru(maxdigits):
f = factorint(p, multiple=True)
if p > 0 and len(f) > record:
record = len(f)
print(p, end=", ")
afind(10) # Michael S. Branicky, Oct 25 2021
Showing 1-4 of 4 results.
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