A324260 -id:A324260 - cp's OEIS frontend

A324257 Conceited Numbers: Composite numbers that are a concatenation of their distinct prime factors with multiplicity in some order allowing overlap.

735, 3792, 13377, 21372, 51375, 67335, 119911, 229912, 290912, 537975, 1341275, 1713192, 2317312, 2333772, 2971137, 3719193, 4773132, 5117695, 7237755, 7747292, 11973192, 13115375, 13731373, 16749933, 19853575, 22940075, 29090912, 29373375

Offset: 1

Deron Stewart, Feb 19 2019

"Conceited Numbers" (they are full of themselves).
The decimal representation of these numbers can be formed typographically from their prime factors. Every distinct prime factor must appear at least once. Generalization of the sequence A083359.
Subsequences:
--No overlap: A324258
--Every prime factor appears in number (not just distinct prime factors): A324259
--No multiplicity: A324260
--Multiplicity only up to the exponent of the distinct prime factor: A083359
Other subsequences are formed by more than one constraint; e.g., A121342 is the intersection of A324258 and A324260, terms with no overlap and no multiplicity.

			67335 = 3*5*67^2 formed by 67||3|||3||5 (this term is not in A083359 because two 3's are required in the concatenation).
3719193 = 3*19*71*919 formed by 3||71||9(19)||3 where 19 and 919 overlap.

Giovanni Resta, Table of n, a(n) for n = 1..302 (first 103 terms from Deron Stewart)
Deron Stewart, C# method to evaluate candidate terms given the distinct prime factors.

Cf. A324258, A324259, A324260, A083359, A083360, A083361, A121342.

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

Gleb Ivanov, Mar 21 2022

			126_10 = 1111110_2 = 2*3^2*7, and 1111110 = 11.111.10, where "." represents concatenation.

Cf. A324260, A324257.

q[n_] := CompositeQ[n] && MemberQ[Join @@@ Permutations @ IntegerDigits[ FactorInteger[n][[;; , 1]], 2], IntegerDigits[n, 2]]; Select[Range[600000], q] (* Amiram Eldar, Mar 21 2022 *)

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 = ', ')

a(10)-a(15) from Amiram Eldar, Mar 21 2022

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A324257 Conceited Numbers: Composite numbers that are a concatenation of their distinct prime factors with multiplicity in some order allowing overlap.

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A352334 Composite numbers that when written in base 2 are a concatenation of their distinct prime factors without multiplicity in some order.

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