A084317 -id:A084317 - cp's OEIS frontend

A343158 a(n) is the smallest m such that A343156(m) = n, or -1 if no such m exists.

2, 4, 10, 35, 15, 34, 190, 290, 303, 395, 130, 465, 553, 265, 195, 663, 218, 582, 481, 858, 714, 418, 345, 530, 382, 1771, 1207, 2098, 3890, 1426, 2090, 4834, 4618, 627, 2321, 2163, 326, 866, 3302, 1298, 3886, 3094, 1086, 6130, 4807, 3646, 5181, 905, 3945, 5753

Offset: 0

N. J. A. Sloane, Apr 07 2021

			2 takes 0 steps to reach a prime, so a(0) = 2.
10 -> 25 -> 5 takes 2 steps to reach a prime (and no smaller number takes that many steps), so a(2) = 10.
35 -> 57 -> 319 -> 1129 takes 3 steps to reach a prime (and no smaller number takes that many steps), so a(3) = 35.

Eric Angelini, W. Edwin Clark, Hans Havermann, Frank Stevenson, Allan C. Wechsler, and others, Postings to Math Fun mailing list, April 2021.

Cf. A084317, A084319, A343156.

is(m, n) = my(k=m); for(i=1, n, if(isprime(k), return(0), k=eval(concat(apply(t->Str(t), factor(k)[, 1]~))))); isprime(k);
a(n) = for(m=2, oo, if(is(m, n), return(m))); \\ Jinyuan Wang, Jul 16 2022

a(32)-a(42) from Hans Havermann, Apr 07 2021
a(43)-a(48) from Hans Havermann, Apr 08 2021
a(49) from Jinyuan Wang, Jul 16 2022

A287637 a(n) = A249125(n)/concatenation of prime factors of A249125(n).

Original entry on oeis.org

2, 4, 3, 8, 5, 9, 16, 7, 2, 32, 27, 4, 11, 25, 64, 13, 8, 81, 10, 128, 17, 49, 19, 16, 20, 256, 23, 125, 243, 32, 29, 31, 40, 512, 50, 121, 37, 64, 41, 43, 80, 1024, 729, 169, 47, 343, 100, 53, 625, 128, 59, 61, 160, 2048, 67, 289, 200, 71, 73, 79, 250, 256

Offset: 1

Michel Lagneau, May 28 2017

The squares of the sequence are, in increasing order: 4, 9, 16, 25, 49, 64, 81, 100, 121, 169, 256, 289, 361, 400, 625, 729, 1024, 4096,... including the squares of the prime numbers.
The numbers p^n, p prime and n = 1, 2, 3, 4,... are in the sequence.
The twin primes (a(m), a(m+1)) of the sequence are (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139),...
The numbers whose prime factors are 2 and 5 (A033846) are in the sequence.

			a(9)=2 because A249125(9) = 50 and the concatenation of the prime factors of 50 is 25. Hence, 50/25 = 2.

Cf. A033846, A077800, A084317, A249125.

with(numtheory):
for n from 2 to 10000 do:
  if type(n,prime)=false
   then
    x:=factorset(n):n0:=nops(x):
    d:=sum('length(x[i])', 'i'=1..n0):
    l:=sum('x[i]*10^sum('length(x[j])', 'j'=i+1..n0)', 'i'=1..n0):
    z:=n/l:
     if floor(z)=z
      then
      printf(`%d, `,z):
      else
     fi:
   fi:
od:

cf[n_] := FromDigits@ Flatten[ IntegerDigits /@ First /@ FactorInteger@n]; Reap[ Do[If[ CompositeQ[n] && IntegerQ[rz = n/cf[n]], Sow[rz]], {n, 6400}]][[2, 1]] (* Giovanni Resta, May 29 2017 *)

A342999 a(n) is always followed by the concatenation of a(n)'s distinct prime factors in increasing order. If this concatenation is already in the sequence, a(n+1) is the smallest term not yet present.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 23, 7, 8, 9, 10, 25, 11, 12, 13, 14, 27, 15, 35, 57, 319, 1129, 16, 17, 18, 19, 20, 21, 37, 22, 211, 24, 26, 213, 371, 753, 3251, 28, 29, 30, 235, 547, 31, 32, 33, 311, 34, 217, 731, 1743, 3783, 31397, 36, 38, 219, 373, 39, 313, 40, 41, 42, 237, 379, 43, 44, 45, 46, 223, 47

Offset: 1

Eric Angelini and Carole Dubois, Apr 02 2021

This is a permutation of the positive terms.

			a(6) is not = 5, though the only prime factor of a(5) is precisely 5; but as 5 is already in the sequence we must take a(6) = 6, the smallest term not yet present in the sequence.
a(7) = 23 as the prime factors of a(6) = 6 are 2 and 3, which, concatenated in increasing order, give 23;
a(8) is not = 23, though the only prime factor of a(7) is precisely 23; but as 23 is already in the sequence we must take a(8) = 7, the smallest term not yet present in the sequence; etc.

Index entries for sequences that are permutations of the natural numbers

Cf. A084317 (concatenation of the prime factors of n, in increasing order), A037276 (replace n with the concatenation of its prime factors in increasing order).

a[1]=1;a[n_]:=a[n]=(g=FromDigits@Flatten[IntegerDigits@*First/@FactorInteger@a[n-1]];If[FreeQ[k=Array[a,n-1],g],g,Min@Complement[Range@Max[k+1],k]])
Array[a,100] (* Giorgos Kalogeropoulos, Apr 02 2021 *)

from sympy import primefactors
def aupton(terms):
  alst, aset = [1, 2], {1, 2}
  while len(alst) < terms:
    an = int("".join(map(str, primefactors(alst[-1]))))
    if an in aset:
      an = 1
      while an in aset: an += 1
    alst.append(an); aset.add(an)
  return alst[:terms]
print(aupton(100)) # Michael S. Branicky, Apr 02 2021

cp's OEIS Frontend

A343158 a(n) is the smallest m such that A343156(m) = n, or -1 if no such m exists.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

PARI

Extensions

A287637 a(n) = A249125(n)/concatenation of prime factors of A249125(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A342999 a(n) is always followed by the concatenation of a(n)'s distinct prime factors in increasing order. If this concatenation is already in the sequence, a(n+1) is the smallest term not yet present.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python