A324260
Subsequence of A324257 (Conceited Numbers) where the distinct prime factors are concatenated without multiplicity.
Original entry on oeis.org
735, 3792, 1341275, 1713192, 2971137, 4773132, 13115375, 13731373, 22940075, 29373375, 31373137, 71624133, 121719472, 183171409, 221397372, 241153517, 311997175, 319953792, 331191135, 1019127375, 1147983375, 1190911909, 1453312395
Offset: 1
4773132 = 2^2 * 3^2 * 7 * 13 * 31 * 47 formed by 47||7||[3(1][3)]||2. The 6 distinct prime factors are used once each, with 3, 13 and 31 overlapping.
A324258
Subsequence of A324257 (Conceited Numbers) where the prime factors are concatenated without overlap.
Original entry on oeis.org
735, 3792, 13377, 67335, 290912, 537975, 1341275, 2333772, 5117695, 7747292, 13115375, 19853575, 22940075, 29090912, 29373375, 37723392, 52979375, 71624133, 79241575, 311997175, 319953792, 367543575, 533334375, 1019127375, 1147983375, 1734009275
Offset: 1
5117695 = 5 * 11^2 * 769, formed by 5||11||769||5. The prime factor 5 is used twice.
A324259
Subsequence of A324257 (Conceited Numbers) where every prime factor is visible in the number, not just each distinct prime factor.
Original entry on oeis.org
21372, 119911, 229912, 2971137, 3719193, 13731373, 16749933, 31373137, 183171409, 221397372, 238903323, 241153517, 254724772, 271141332, 331191135, 1153115117, 1190911909, 1453312395, 2511176437, 2923699119, 2971193115, 3124516195
Offset: 1
119911 = 11^2 * 991, formed by 11||(99(1)1), with 11 appearing twice in the number as required. (991 and 11 overlap.)
A324262
a(n) is the smallest term of A324261 with 2n digits if it exists, otherwise 0.
Original entry on oeis.org
0, 0, 0, 0, 13731373, 1190911909, 0, 19090911909091, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7316763215964848081373167632159648480813, 111272689909091345969111272689909091345969, 10527889691056689261011052788969105668926101, 0, 0, 0, 0
Offset: 0
With m = 27 there are three prime p's: 114175966169705419295257913, 352579141759661697054192911 and 525791141759661697054192913. The smallest p concatenated with itself gives a(27) = 114175966169705419295257913114175966169705419295257913.
With m = 28 there are no solutions so a(28) = 0.
A324261
Subsequence of A083359 (Visual Factor Numbers) of the form (10^m+1)*p, where the decimal representation of prime p contains all the prime factors of 10^m+1.
Original entry on oeis.org
13731373, 31373137, 1190911909, 9091190911, 19090911909091, 7316763215964848081373167632159648480813, 111272689909091345969111272689909091345969, 111279090913268945969111279090913268945969, 112726894596919090913112726894596919090913
Offset: 1
With m = 4: 10^4 + 1 = 10001 = 73 * 137. We can form prime p = 1373 which concatenates with itself to give a(1) = 13731373 = 73 * 137 * 1373. We can also form the prime p = 3137 which gives a(2). The number 7313 also contains all the prime factors of 10001 but it is not prime.
With m = 33: 10^33 + 1 = 7*11*11*13*23*4093*8779*599144041*183411838171, there are 4932 m-digit numbers that contain all the factors, of which 227 of them are prime. Each of these primes generates a term in the sequence with 66 digits, the smallest of which is 112359914404134093877918341183817112359914404134093877918341183817. This is A324262(33).
A352334
Composite numbers that when written in base 2 are a concatenation of their distinct prime factors without multiplicity in some order.
Original entry on oeis.org
126, 7902, 58167, 63198, 119565, 505566, 507771, 2043825, 8249085, 12568150, 132992559, 183431550, 196196825, 258858950, 533713761
Offset: 1
126_10 = 1111110_2 = 2*3^2*7, and 1111110 = 11.111.10, where "." represents concatenation.
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q[n_] := CompositeQ[n] && MemberQ[Join @@@ Permutations @ IntegerDigits[ FactorInteger[n][[;; , 1]], 2], IntegerDigits[n, 2]]; Select[Range[600000], q] (* Amiram Eldar, Mar 21 2022 *)
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from sympy import primefactors
from itertools import permutations
for i in range(1, 10**12):
p = primefactors(i)
if len(p) != 1:
p = list(map(lambda x: format(x, 'b'),p))
if all(j in format(i,'b') for j in p) and any(format(i,'b')==''.join(t) for t in permutations(p)):
print(i, end = ', ')
Showing 1-6 of 6 results.
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