A380883 a(n) is the smallest multiple of prime(n) which contains every decimal digit of prime(n), including repetitions.

12, 30, 15, 70, 110, 130, 170, 190, 230, 290, 310, 370, 164, 344, 470, 530, 295, 610, 670, 710, 730, 790, 830, 890, 679, 1010, 1030, 1070, 1090, 1130, 1270, 1310, 1370, 1390, 1490, 1510, 1570, 1630, 1670, 1730, 1790, 1810, 1719, 1930, 1379, 1990, 2110, 2230, 2270, 2290, 2330, 2390, 2410, 1255, 2570, 2367, 2690

Offset: 1

Author

David James Sycamore, Feb 07 2025

Keywords

• nonn
• base

Comments

Smallest number of the form m*prime(n) such that every decimal digit d in prime(n) (including repetitions) is also a digit in m*prime(n). For every n, m is in {3,4,5,6,7,8,9,10}. The graph displays 8 parallel straight lines, each corresponding to a different value of m (the uppermost being m = 10).

For all n, 10*prime(n) (m = 10) contains all the digits of prime(n), but there are some cases where for m < 10 every digit of prime(n) is found in m*prime(n). The first of these is when n = 1, m = 6; see Example.

This sequence is not the same as A087217(prime(n)) since here the order of digits in m*prime(n)is unimportant; see Example.

Examples

			a(1) = 6*prime(1) = 12.
a(109) = 2995 since prime(109) = 599 and 5*599 = 2995.
For n = 13, prime(13) = 41, a(n) = 164 = 4*31, whereas A097217(41) = 410. This is the first departure from A087217(prime(n)).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^5.

Crossrefs

Cf. A000040, A087217, A380811.

Programs

Mathematica

Reap[Do[p = Prime[n]; d = DigitCount[p]; k = 2; While[! AllTrue[DigitCount[#] - d, # >= 0 &] &[p*k], k++]; Sow[k *= p], {n, 120}]][[-1, 1]] (* Michael De Vlieger, Feb 20 2025 *)

Formula

a(n) = A380885(prime(n)).

A000040(n) < A380811(n) <= a(n) <= 10*A000040(n).