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Numbers k such that the decimal encoding of the prime factorization of k (A067599) ends in k.
1

%I #25 Jul 10 2023 08:20:35

%S 11,8571,11371,190911,12711811,14713491,19090911,71119711,12531135391,

%T 15311195711,112717566411,158318548011,518914376931,7292811659931

%N Numbers k such that the decimal encoding of the prime factorization of k (A067599) ends in k.

%C a(13) > 2*10^11. 518914376931, 7292811659931, 19090909090909090911 and prime repunits (A004022) are also terms. - _Donovan Johnson_, Dec 04 2012

%C Are there any terms not ending in 1? Equivalently, are any terms also in A070003? - _Charles R Greathouse IV_, Dec 05 2012

%C a(15) > 10^13. If exponents equal to 1 are not represented (as in A080670), the corresponding sequence starts with 113113, 31373137, and 533517177839 = 853 * 3517 * 177839. - _Giovanni Resta_, Jun 26 2017

%e The prime factorization of 190911 is 3^1 * 7^1 * 9091^1 with decimal encoding 317190911, which ends in 190911. Hence 190911 is a term of the sequence.

%t (*returns true if a ends with b, false o.w.*) f[a_, b_] := Module[{c, d, e, g, h, i, r}, r = False; c = ToString[a]; d = ToString[b]; e = StringLength[c]; g = StringPosition[c, d]; h = Length[g]; If[h > 0, i = g[[h]]; If[i[[2]] == e, r = True]]; r]; (*gives the decimal encoding of the prime factorization of n*) g[n_] := FromDigits[Flatten[IntegerDigits[FactorInteger[n]]]]; Do[If[f[g[n], n], Print[n]], {n, 1, 10^6} ]

%o (PARI) {a067254(a,b) = local(n,v,k,j); for(n=max(2,a),b,v=factor(n); if(eval(concat(vector(matsize(v)[1],k, concat(vector(matsize(v)[2],j,Str(v[k,j]))))))%(10^length(Str(n)))==n,print1(n,",")))}

%o a067254(2,2*10^7) \\ _Klaus Brockhaus_, Feb 22 2002

%Y Cf. A004022, A067599, A070003, A080670.

%K base,nonn,more

%O 1,1

%A _Joseph L. Pe_, Feb 20 2002

%E a(5)-a(7) from _Klaus Brockhaus_, Feb 22 2002

%E a(8)-a(10) from _Donovan Johnson_, Mar 26 2010

%E a(11)-a(12) from _Donovan Johnson_, Dec 04 2012

%E a(13) from _Giovanni Resta_, Jun 09 2017

%E a(14) from _Giovanni Resta_, Jun 26 2017