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

1, 5, 6, 10, 16, 24, 25, 30, 36, 40, 50, 51, 60, 70, 76, 90, 92, 100, 125, 176, 195, 240, 249, 250, 363, 375, 376, 430, 490, 500, 501, 510, 546, 556, 560, 568, 570, 600, 620, 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

Offset: 1

Author

Scott R. Shannon, Nov 10 2019

Keywords

• nonn
• base

Comments

Inspired by A328095. To avoid all primes being in the sequence the divisors of k includes k itself.

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

Examples

			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.
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.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..1000. Note when searching for these numbers one needs to use arbitrary precision packages; the product for 24570000 has 1486 digits.

Crossrefs

Cf. A007955, A328095, A326806, A328544.

The sequence of primes contained in their squares is A115738.

Programs

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

Mathematica

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 *)