A121342 -id:A121342 - 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.

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

Deron Stewart, Feb 19 2019

Subsequence of A083359.

Overlap of the distinct primes is allowed per A324257. Terms without overlap form the subsequence A121342.

			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.

Deron Stewart, C# methods to evaluate candidate terms given the distinct prime factors.

Cf. A324257, A083359, A121342.

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

Deron Stewart, Feb 19 2019

Generalization of A083360 with multiplicity of the distinct prime factors (not limited by the number of times a prime factor appears in the factorization of the number).

			5117695 = 5 * 11^2 * 769, formed by 5||11||769||5. The prime factor 5 is used twice.

Deron Stewart, C# methods to evaluate candidate terms given the distinct prime factors.

Cf. A324257, A083360, A121342.

A306474 Composite numbers that are anagrams of the concatenation of their prime factors.

Original entry on oeis.org

735, 1255, 3792, 7236, 11913, 12955, 13175, 17276, 17482, 19075, 19276, 23535, 25105, 32104, 34112, 37359, 42175, 100255, 101299, 104392, 105295, 107329, 117067, 117873, 121325, 121904, 121932, 123544, 123678, 124483, 127417, 129595, 131832, 132565, 139925

Offset: 1

Michel Lagneau, Feb 18 2019

The sequence contains two subsequences:
Subsequence 1: numbers with distinct digits. This finite subsequence begins with the numbers 735, 3792, 7236, 17482, 19075, 19276, 32104, ...
Subsequence 2: numbers with non-distinct digits. This subsequence begins with the numbers 1255, 11913, 12955, 13175, 17276, 23535, ...

			3792 is in the sequence because the concatenation of the prime distinct divisors {2, 3, 79} is 2379, anagram of 3792.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A023086, A209799.

A121342 is a subsequence.

with(numtheory):
for n from 1 to 140000 do:
if type(n,prime)=false
  then
  x:=factorset(n):n1:=nops(x): s:=0:s0:=0:
    for i from n1 by -1 to 1 do:
     a:=x[i]:b:=length(a):s:=s+a*10^s0:s0:=s0+b:
    od:
      if sort(convert(n, base, 10)) = sort(convert(s, base, 10))
       then
        printf(`%d, `, n):
        else
      fi:fi:
     od:

Select[Range[2,140000],If [!PrimeQ[#],Sort@IntegerDigits@#==Sort[Join@@IntegerDigits[First/@FactorInteger[#]]]]&]

A287916 Numbers m such that the decimal digits of m are exactly the same as those of all the indices corresponding to the prime factors of m.

Original entry on oeis.org

12, 14, 123, 154, 2127, 2391, 3614, 4031, 5318, 7174, 8491, 11142, 12435, 12830, 18126, 20314, 23514, 24612, 25201, 28731, 31934, 42158, 50314, 51124, 61411, 62116, 65315, 72401, 73201, 81254, 81372, 92315, 93243, 112924, 123126, 123612, 123861, 124341, 125102

Offset: 1

Michel Lagneau, Jul 11 2017

The sequence contains two subsequences:
Subsequence 1: numbers with distinct digits. This finite subsequence begins with the numbers 12, 14, 123, 154, 2391, 3614, 4031, 5318, 8491, 12435, 12830, 23514, ... (see example 1)
Subsequence 2: numbers with non-distinct digits (see example 2). This subsequence begins with the numbers 2127, 7174, 11142, 18126, ...

			12435 is in the sequence because the prime factors are {3, 5, 829} with 3 = prime(2), 5 = prime(3) and 829 = prime(145). The decimal digits corresponding to the indices {2, 3, 145} of the prime divisors are the same as the digits of the number 12435.
61411 is in the sequence because the prime factors are {7, 31, 283} with 7 = prime(4), 31 = prime(11) and 283 = prime(61). The decimal digits corresponding to the indices {4, 11, 61} are the same as the digits of the number 61411.

Cf. A121342, A209799.

with(numtheory):nn:=200000:
for n from 10 to nn do:
  x:=convert(n,base,10):n0:=nops(x):U:=array(0..9,[0$10]):
   for a from 1 to n0 do:
    U[x[a]]:=U[x[a]]+1:
   od:
   y:=factorset(n):n1:=nops(y):V:=array(0..9,[0$10]):
  for i from 1 to n1 do :
    p:=y[i]:ii:=0:
      for k from 1 to 10000 while(ii=0) do:
        if ithprime(k) = p
        then
        ii:=1:z:=convert(k,base,10):nz:=nops(z):
           for b from 1 to nz do:
            V[z[b]]:=V[z[b]]+1:
           od:
        else
        fi:
      od:
     od:
       jj:=0:
       for b from 0 to 9 do:
        if U[b]<>V[b] then jj:=1:
        else fi:
       od:
        if jj=0
         then print(n):
         else fi:
     od:

Select[Range[125102], Sort@ IntegerDigits@ # == Sort[Join @@ IntegerDigits[ PrimePi[ First /@ FactorInteger[#]]]] &] (* Giovanni Resta, Jul 11 2017 *)

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.

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A324260 Subsequence of A324257 (Conceited Numbers) where the distinct prime factors are concatenated without multiplicity.

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A324258 Subsequence of A324257 (Conceited Numbers) where the prime factors are concatenated without overlap.

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A306474 Composite numbers that are anagrams of the concatenation of their prime factors.

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Maple

Mathematica

A287916 Numbers m such that the decimal digits of m are exactly the same as those of all the indices corresponding to the prime factors of m.

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Programs

Maple

Mathematica