A180489 -id:A180489 - cp's OEIS frontend

A274328 a(n) is the sum of a sequence of multiples of the n-th prime such that it contains each of the digits from 0 to 9 exactly once and with the least sum possible, or 0 if there is no satisfying sequence.

270, 135, 38475, 252, 1881, 702, 918, 684, 1656, 2349, 1953, 7326, 2952, 2322, 2961, 3339, 3717, 3843, 3015, 3195, 3285, 5688, 8217, 5607, 4365, 95445, 6489, 4815, 3924, 37629, 35433, 10611, 9864, 5004, 41571, 4077, 39564, 2934, 34569, 42039

Offset: 1

Claudio Meller, Jun 21 2016

From Ryan Hitchman, Sep 15 2017: (Start)
a(172) = 1023847569, prime(172) = 1021 is the first entry with one multiple.
a(1884) = 145953, prime(1884) = 16217 is last with more than one multiple.
a(10545) = 0, prime(10545) = 111119 is the first zero. (End)

			For n = 7, a(7) = 918 because prime(7) = 17, sequence 34, 85, 102, 697, sum 918.

Ryan Hitchman, Table of n, a(n) for n = 1..10544
Claudio Meller Blog, Multiples with all the digits, June 13 2016.

Cf. A180489 for n>1884. Superset of A050288. - Ryan Hitchman, Sep 15 2017

(m = Select[#*Range[10000], Max[DigitCount[#]] == 1 &];
   Total[m*LinearProgramming[m, Thread[DigitCount /@ m],
      ConstantArray[{1, 0}, 10], 0, Integers]]) & /@ Prime[Range[40]] (* Ryan Hitchman, Sep 15 2017 *)

Terms a(9) and beyond, zero case from Ryan Hitchman, Sep 15 2017

A217535 Least number having in its decimal representation each digit n times.

Original entry on oeis.org

1023456789, 10012233445566778899, 100011222333444555666777888999, 1000011122223333444455556666777788889999, 10000011112222233333444445555566666777778888899999, 100000011111222222333333444444555555666666777777888888999999

Offset: 1

M. F. Hasler, Oct 05 2012

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A050278, A051018, A049363, A074205, A154566, A204047, A020666, A020667, A061604, A180489.

A subsequence of A171102.

a:= n-> parse(cat(1,0$n,1$(n-1),seq(i$n, i=2..9))):
seq(a(n), n=1..10);  # Alois P. Heinz, Jun 25 2017

A217535(n)=sum(d=1,9,10^(n-(d==1))\9*d*10^(n*(9-d)))+10^(10*n-1)

a(n) ~ 10^(10n-1). See PARI code for an exact formula.

A292471 Primes that do not divide any 10-digit pandigital number (i.e. any value in A050278).

Original entry on oeis.org

111119, 123457, 178889, 199999, 224467, 246913, 325477, 333337, 333367, 333667, 336667, 345679, 359147, 361909, 387403, 394549, 411113, 419753, 443221, 444449, 449161, 470551, 473219, 476647, 476659, 504323, 506173, 509053, 512683, 513269, 514289, 514357

Offset: 1

David J. Seal, Sep 21 2017

This is the complement in A000040 of the finite list of primes that divide one or more 10-digit pandigital numbers. That finite list has been obtained by computer; it contains 1102173 primes, with the first prime that is not in the list being prime(10545) = 111119 and the last that is in the list being prime(55537259) = 1097393447.

			a(1) = 111119 because 111119 is prime and does not divide any of the 10-digit pandigital numbers 1023456789, 1023456798, ..., 9876543210, and all smaller primes do divide at least one of them.

Cf. A050278, A061604, A180489, A274328, A292703.

A292703 Values of n such that prime(n) does not divide any 10-digit pandigital number (i.e. any value in A050278).

Original entry on oeis.org

10545, 11602, 16237, 17984, 19978, 21788, 28046, 28666, 28669, 28693, 28928, 29629, 30698, 30896, 32869, 33438, 34699, 35373, 37198, 37300, 37639, 39273, 39477, 39755, 39756, 41859, 42003, 42219, 42490, 42538, 42619, 42624

Offset: 1

David J. Seal, Sep 21 2017

This is the complement of the finite list of n such that prime(n) divides one or more 10-digit pandigital numbers. That finite list has been obtained by computer; it contains 1102173 numbers, with the first number that is not in the list being 10545 and the last that is in the list being 55537259.
A292471 is the corresponding list of primes.
These are the values of n for which A180489(n) has more than 10 digits, and also the values of n for which A274328(n) = 0.

			a(1) = 10545 because prime(10545) = 111119 does not divide any of the 10-digit pandigital numbers 1023456789, 1023456798, ..., 9876543210, and all smaller primes do divide at least one of them.

Cf. A050278, A061604, A180489, A274328, A292471.

cp's OEIS Frontend

A274328 a(n) is the sum of a sequence of multiples of the n-th prime such that it contains each of the digits from 0 to 9 exactly once and with the least sum possible, or 0 if there is no satisfying sequence.

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A217535 Least number having in its decimal representation each digit n times.

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A292471 Primes that do not divide any 10-digit pandigital number (i.e. any value in A050278).

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A292703 Values of n such that prime(n) does not divide any 10-digit pandigital number (i.e. any value in A050278).

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