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A329271 Numbers k such that k multiplied by the product of its divisors contains k as a substring.

%I A329271 #32 Sep 08 2022 08:46:24
%S A329271 1,5,6,10,16,24,25,30,36,40,50,51,60,70,76,90,92,100,125,176,195,240,
%T A329271 249,250,363,375,376,430,490,500,501,510,546,556,560,568,570,600,620,
%U A329271 624,625,648,680,730,749,750,760,810,875,909,930,972,975,976,990,999,1000,1001,1010,1636,1680,1930,2354,2400,2490,2500,2510,2512,2943,3000
%N A329271 Numbers k such that k multiplied by the product of its divisors contains k as a substring.
%C A329271 Inspired by A328095. To avoid all primes being in the sequence the divisors of k includes k itself.
%C A329271 Contains 10^k, 5*10^k and 6*10^k for all k, 3*10^k, 4*10^k, 7*10^k and 9*10^k for all odd k. - _Robert Israel_, Nov 11 2019
%H A329271 Scott R. Shannon, <a href="/A329271/b329271.txt">Table of n, a(n) for n = 1..1000</a>. Note when searching for these numbers one needs to use arbitrary precision packages; the product for 24570000 has 1486 digits.
%e A329271 16 is in the sequence as the divisors of 16 are 1,2,4,8,16, and 16*(1*2*4*8*16) = 16*1024 = 16384, and '16384' contains '16' as a substring.
%e A329271 30 is in the sequence as the divisors of 30 are 1,2,3,5,6,10,15,30, and 30*(1*2*3*5*6*10*15*30) = 30*810000 = 24300000, and '24300000' contains '30' as a substring.
%t A329271 f[n_] := n^(1+DivisorSigma[0, n]/2); aQ[n_] := SequenceCount[IntegerDigits[f[n]], IntegerDigits[n]] > 0; Select[Range[3000], aQ] (* _Amiram Eldar_, Nov 10 2019 *)
%o A329271 (Magma) a:=[]; for k in [1..3000] do t:=IntegerToString(k*(&*Divisors(k))); s:=IntegerToString(k); if s in t then Append(~a,k); end if; end for; a; // _Marius A. Burtea_, Nov 10 2019
%Y A329271 Cf. A007955, A328095, A326806, A328544.
%Y A329271 The sequence of primes contained in their squares is A115738.
%K A329271 nonn,base
%O A329271 1,2
%A A329271 _Scott R. Shannon_, Nov 10 2019