A367974 - OEIS

%I #47 Dec 30 2023 23:38:39

%S 1,25,1024,6272,1953125,4117715,15813251,213797679,346146025,

%T 488281250,714592137,1719341824,3676531250,10510100501,10852734375,

%U 11214315503,17241013443,25421511971

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

%C 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.

%C 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

%C 3262027661312 is a term. - _Martin Ehrenstein_, Dec 08 2023

%C 10852734375 is a term. - _Michael S. Branicky_, Dec 08 2023

%C 10510100501 is a term. - _Michael S. Branicky_, Dec 09 2023

%e 1 is a term since it has no prime factors.

%e 25 is a term as 25 = 5*5, i.e., two 5's being "25", which appears in 25.

%e 1024 is a term as 1024 = 2^10, i.e., ten 2's being "102", which appears in 1024.

%e 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.

%e 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.

%t l[n_]:=ToString/@Reverse[Flatten[FactorInteger[n]]]; len[n_]:=Length[l[n]];

%t 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 *)

%o (Python)

%o from sympy import factorint

%o def ok(n):

%o     s = str(n)

%o     return all(str(e)+str(p) in s for p, e in factorint(n).items())

%o print([k for k in range(10**5) if ok(k)]) # _Michael S. Branicky_, Dec 08 2023

%Y Cf. A045918, A005150, A027746.

%K nonn,base,more

%O 1,2

%A _Scott R. Shannon_, Dec 07 2023

%E 1 prepended by _Martin Ehrenstein_, Dec 08 2023

%E a(14)-a(16) from _Michael S. Branicky_, Dec 13 2023

%E a(17)-a(18) from _Michael S. Branicky_, Dec 27 2023