A380811 -id:A380811 - 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).

A380884 Primes p such that there is an m < 10 for which m*p contains every decimal digit of p.

Original entry on oeis.org

2, 5, 41, 43, 59, 97, 191, 197, 251, 263, 373, 401, 443, 491, 499, 599, 653, 691, 967, 991, 997, 1481, 1901, 1913, 1997, 2549, 2551, 2591, 3067, 3491, 4001, 4013, 4493, 4793, 4931, 4943, 4967, 4973, 4993, 4999, 5021, 5443, 5647, 6053, 6361, 6521, 6703, 6991, 7489, 7901, 7951, 7993

Offset: 1

David James Sycamore, Feb 07 2025

			6*2 = 12 therefore 2 is a term (m = 6 < 10).
499*6 = 2994 therefore 499 is a term (m = 6 < 10).
599*5 = 2995 therefore 599 is a term (m = 5 < 10).
37 is not a term since 10*37 = 370 is the smallest multiple of 37 containing 3 and 7

Michael De Vlieger, Table of n, a(n) for n = 1..5000

Cf. A000040, A380811, A380883.

nn = 10000; Reap[Do[p = Prime[n]; d = DigitCount[p]; k = 2; While[! AllTrue[DigitCount[#] - d, # >= 0 &] &[p*k], k++]; If[k < 10, Sow[p]], {n, nn}]][[-1, 1]] (* Michael De Vlieger, Feb 20 2025 *)

lista(nn) = forprime(p=2, nn, if (A380883(p) != 10*p, print1(p, ", "))); \\ Michel Marcus, Feb 20 2025

59, 491, and more terms added by Michel Marcus, Feb 20 2025

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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