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

1,1

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.

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.

Formula

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

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

Example

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)).

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 *)

Crossrefs

Cf. A000040, A087217, A380811.

Sequence in context: A183723 A247152 A380811 * A299041 A334989 A331367

Adjacent sequences: A380880 A380881 A380882 * A380884 A380885 A380886

Keyword

nonn,base

Author

David James Sycamore, Feb 07 2025

Status

approved