A367974 Numbers which contain the "Look and Say" description (cf. A045918) of all their prime factors, counted with multiplicity.

1, 25, 1024, 6272, 1953125, 4117715, 15813251, 213797679, 346146025, 488281250, 714592137, 1719341824, 3676531250, 10510100501, 10852734375, 11214315503, 17241013443, 25421511971

Offset: 1

Author

Scott R. Shannon, Dec 07 2023

Keywords

• nonn
• base
• more

Comments

Overlapping of the "Look and Say" prime factor description strings is allowed. It is likely, although unproven, that 25 = 5*5 = "two 5's" = "25" is the only number that "perfectly" describes its own "Look and Say" factorization, i.e., there are no overlapping factor description strings, and all digits of the number are used in the factor description strings. It is unknown if the sequence is infinite.

There are many terms of the form 5^k, where k is 2, 9, 55, 62, 71, 82, 84, 86, 125, etc. - Ivan N. Ianakiev, Dec 07 2023

3262027661312 is a term. - Martin Ehrenstein, Dec 08 2023

10852734375 is a term. - Michael S. Branicky, Dec 08 2023

10510100501 is a term. - Michael S. Branicky, Dec 09 2023

Examples

			1 is a term since it has no prime factors.
25 is a term as 25 = 5*5, i.e., two 5's being "25", which appears in 25.
1024 is a term as 1024 = 2^10, i.e., ten 2's being "102", which appears in 1024.
346146025 is a term as 346146025 = 5^2 * 61^4, i.e., two 5's and four 61's being "25" and "461", respectively, both of which appear in 346146025.
1719341824 is a term as 1719341824 = 2^8 * 719 * 9341, i.e., eight 2's and one 719 and one 9341 being "82", "1719" and "19341" respectively, all of which appear in 1719341824. Note that only the final digit 4 is not used in the string descriptions.
		

Crossrefs

Cf. A045918, A005150, A027746.

Programs

Mathematica

l[n_]:=ToString/@Reverse[Flatten[FactorInteger[n]]]; len[n_]:=Length[l[n]];
fQ[n_]:=AllTrue[Table[StringJoin[l[n][[i]],l[n][[i+1]]],{i,1,len[n],2}], StringPosition[ToString[n],#]!={}&]; Select[Range[6272],fQ[#]&] (* Ivan N. Ianakiev, Dec 07 2023 *)

Python

from sympy import factorint
def ok(n):
    s = str(n)
    return all(str(e)+str(p) in s for p, e in factorint(n).items())
print([k for k in range(10**5) if ok(k)]) # Michael S. Branicky, Dec 08 2023

Extensions

1 prepended by Martin Ehrenstein, Dec 08 2023

a(14)-a(16) from Michael S. Branicky, Dec 13 2023

a(17)-a(18) from Michael S. Branicky, Dec 27 2023