A380885 -id:A380885 - cp's OEIS frontend

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

David James Sycamore, Feb 07 2025

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.

			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)).

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.

Cf. A000040, A087217, A380811.

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 *)

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

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

A381700 a(n) is the smallest multiple k*n that contains all the distinct digits of n, where k > 0 must have at least one digit > 1.

Original entry on oeis.org

12, 12, 36, 24, 15, 36, 147, 48, 189, 120, 132, 192, 312, 714, 105, 416, 714, 108, 912, 120, 126, 132, 322, 432, 125, 624, 702, 728, 928, 360, 713, 832, 132, 1394, 315, 936, 703, 836, 936, 240, 164, 294, 344, 264, 405, 644, 1457, 384, 294, 150, 153, 1352, 1325, 1458, 165, 1456, 3705, 1508, 295

Offset: 1

Ali Sada and M. F. Hasler, Mar 04 2025

The restriction on k is to avoid trivial solutions where k is a sum of distinct powers of 10. Even if we exclude that k*n starts with the digits of n, most terms would correspond to k = 10^L+1 where L is about half the number of digits of n, so that the initial and final digits of the product are "trivially" those of n.
Asymptotically 100% of positive integers n have a(n) = 2*n. In particular, for n <= x, a(n) > 2*n occurs at most O(x^k) where k = log 9/log 10 = 0.954.... - Charles R Greathouse IV, Mar 20 2025
Is a(n)/n bounded? Define b(n) = a(n)/n; then b(42941736805) = 5609 is the largest I know. - Charles R Greathouse IV, Apr 24 2025

			36 is the least nontrivial multiple of 6 that contains the digit 6, so a(6) = 36.

Vincenzo Librandi, Table of n, a(n) for n = 1..3500

Cf. A380885 (without different start constraint).

A381700 := function(n)
    S := Set(Intseq(n));
    k := 1;
    repeat
        k +:= 1;
        digitsKn := Set(Intseq(k*n));
    until (S subset digitsKn) and (Max(Intseq(k)) ge 2);
    return k*n;
end function;
[ A381700(n) : n in [1..60] ]; // Vincenzo Librandi, Mar 17 2025

A381700[n_Integer]:=Module[{S,k,digitsKn},S=DeleteDuplicates[IntegerDigits[n]]; (* Set of digits of n *) k=1;
While[k++;
digitsKn=DeleteDuplicates[IntegerDigits[k n]];
!(SubsetQ[digitsKn,S]&&Max[IntegerDigits[k]]>=2)];
k n] (* Vincenzo Librandi, Mar 17 2025 *)

A381700(n)=my(S=Set(digits(n))); for(k=2,oo, #setminus(S,Set(digits(k*n))) || vecmax(digits(k))<2 || return(k*n));

cp's OEIS Frontend

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

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