Erich Friedman - cp's OEIS frontend

A381494 Smallest number with reciprocal of period length n in base 7.

1, 2, 4, 9, 5, 2801, 36, 29, 64, 27, 11, 1123, 13, 16148168401, 113, 31, 17, 14009, 108, 419, 55, 261, 23, 47, 73, 2551, 53, 81, 145, 59, 99, 311, 256, 3631, 56036, 81229, 135, 223, 1676, 486643, 41, 83, 1017, 166003607842448777, 115, 837, 188, 13722816749522711, 153, 3529, 10204

Offset: 0

Erich Friedman, Feb 25 2025

For n > 1, a(n) is the smallest positive d such that d divides 7^n - 1 and does not divide any of 7^k - 1 for 0 < k < n.

			a(3)=9 since 1/9 has period 3 in base 7 (.053053053...) and no smaller number has this property.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Cf. A003060, A379640.

f:= proc(n) local d,k;
      for d in sort(convert(numtheory:-divisors(7^n-1),list)) do
        if andmap(k -> 7^k-1 mod d <> 0, [$1 .. n-1]) then return d fi
      od
end proc:
f(0):= 1: f(1):= 2:
map(f, [$0..80]); # Robert Israel, Feb 28 2025

a[n_] := First[Select[Divisors[7^n - 1], MultiplicativeOrder[7, #] == n &, 1]];
a[0] = 1; a[1] = 2; Table[a[n], {n, 0, 50}]

from sympy import divisors
def A381494(n): return next(d for d in divisors(7**n-1) if d>1 and all(pow(7,k,d)!=1 for k in range(1,n))) if n else 1 # Chai Wah Wu, Feb 28 2025

Conjecture: a(n) = A218358(n) for n>=2. - R. J. Mathar, Mar 03 2025

A381493 Smallest number with reciprocal of period length n in base 8.

Original entry on oeis.org

1, 7, 3, 73, 5, 31, 19, 49, 17, 262657, 11, 23, 37, 79, 43, 631, 97, 103, 81, 32377, 25, 3577, 67, 47, 323, 601, 237, 2593, 29, 233, 209, 2147483647, 193, 199, 307, 71, 405, 223, 571, 937, 187, 13367, 817, 431, 115, 271, 139, 2351, 577, 343, 251

Offset: 0

Erich Friedman, Feb 25 2025

For n > 1, a(n) is the smallest positive d such that d divides 8^n - 1 and does not divide any of 8^k - 1 for 0 < k < n.

			a(3)=73 because 1/73 has period 3 in base 8 (.007007007...) and no smaller number has this property.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Cf. A003060, A379641.

a[n_] := First[Select[Divisors[8^n - 1], MultiplicativeOrder[8, #] == n &, 1]];
a[0] = 1; a[1] = 7; Table[a[n], {n, 0, 50}]

from sympy import divisors
def A381493(n):
    if n == 0: return 1
    for d in divisors(8**n-1):
        if d>1 and all(pow(8,k,d)!=1 for k in range(1,n)):
            return d # Chai Wah Wu, Feb 28 2025

A381370 Smallest number with reciprocal of period length n in base 9.

Original entry on oeis.org

1, 2, 5, 7, 32, 11, 35, 547, 17, 19, 25, 23, 224, 398581, 29, 31, 128, 103, 95, 1597, 352, 43, 115, 47, 97, 151, 53, 109, 928, 59, 155, 683, 256, 161, 515, 71, 608, 18427, 7985, 79, 187, 83, 203, 431, 89, 181, 235, 1223, 896, 491, 101

Offset: 0

Erich Friedman, Feb 25 2025

For n > 1, a(n) is the smallest positive d such that d divides 9^n - 1 and does not divide any of 9^k - 1 for 0 < k < n.

			a(3)=7 because 1/7 has period 3 in base 9 (.125125125...) and no smaller number has this property.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.

Cf. A003060, A379642.

a[n_] := First[Select[Divisors[9^n - 1], MultiplicativeOrder[9, #] == n &, 1]];
a[0] = 1; a[1] = 2; Table[a[n], {n, 0, 50}]

from sympy import divisors
def A381370(n): return next(d for d in divisors(9**n-1) if d>1 and all(pow(9,k,d)!=1 for k in range(1,n))) if n else 1 # Chai Wah Wu, Feb 28 2025

A381216 Number of isomers of C_n H_{2n+2} O_2.

Original entry on oeis.org

1, 2, 5, 11, 28, 69, 179, 463, 1225, 3246, 8697, 23366, 63137, 171051, 465002, 1266831, 3459262, 9462393, 25926939, 71139400, 195451500, 537608802, 1480316960, 4079977874, 11254956840, 31072771980, 85850016944, 237356027117, 656657132953, 1817758531055, 5034725293449

Offset: 0

Erich Friedman, Feb 17 2025

nonn

			a(2)=5 because the following compounds are possible:
     | |       |   |       | |       |     |       |
-O-O-C-C-   -O-C-O-C-   -O-C-C-O-   -C-O-O-C-   -O-C-O-
     | |       |   |       | |       |     |       |
                                                  -C-
                                                   |

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000598, A000599, A000600, A000602, A000635, A000636.

\\ here R(n) gives g.f. of A000598.
R(n)={my(g=O(x)); for(n=0, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x*x^n)); g}
seq(n)={my(A=O(x*x^n), p=R(n), p2=subst(p,x,x^2) + A, q=(p^2+p2)/2, q2=subst(q,x,x^2) + A); Vec((p^2/(1 - x^2*q^2) + p2/(1 - x^2*q2))*(1 + x*q)/2)} \\ Andrew Howroyd, Feb 17 2025

a(10) onwards from Andrew Howroyd, Feb 17 2025

A378557 Powers of 3 that do not contain the digit 3.

Original entry on oeis.org

1, 9, 27, 81, 729, 2187, 6561, 59049, 177147, 4782969, 1162261467, 7625597484987, 22876792454961, 16677181699666569, 12157665459056928801, 717897987691852588770249, 174449211009120179071170507, 11972515182562019788602740026717047105681

Offset: 1

Erich Friedman, Nov 30 2024

Any additional terms have exponent at least 10^5.

Cf. A000244, A131629.

Select[3^Range[0,100000],Not[MemberQ[IntegerDigits[#],3]]&]

a(n) = 3^A131629(n).

A378556 Powers of 2 that do not include the digit 2.

Original entry on oeis.org

1, 4, 8, 16, 64, 4096, 16384, 65536, 1048576, 4194304, 8388608, 67108864, 17179869184, 34359738368, 68719476736, 549755813888, 4398046511104, 70368744177664, 18014398509481984, 18446744073709551616, 18889465931478580854784, 9671406556917033397649408, 374144419156711147060143317175368453031918731001856

Offset: 1

Erich Friedman, Nov 30 2024

Any additional terms have exponent at least 10^5.

Cf. A000079, A034293.

Select[2^Range[0,100000],Not[MemberQ[IntegerDigits[#],2]]&]

a(n) = 2^A034293(n).

A378558 Powers of 4 that do not include the digit 4.

Original entry on oeis.org

1, 16, 256, 65536, 16777216, 1099511627776

Offset: 1

Erich Friedman, Nov 30 2024

Any additional terms have an exponent of at least 10^5.

Cf. A000302.

Select[4^Range[0,100000],Not[MemberQ[IntegerDigits[#],4]]&]

A378492 Squares where larger digits have larger multiplicity.

Original entry on oeis.org

0, 1, 4, 9, 144, 441, 1444, 29929, 55225, 166464, 255025, 299209, 633616, 646416, 767376, 4999696, 9696996, 34433424, 228281881, 414041104, 414488881, 424442404, 536663556, 969699600, 1649496996, 1929229929, 2636206336, 2666999449, 2929299129, 2996029696, 4664343616

Offset: 1

Erich Friedman, Nov 28 2024

Alois P. Heinz, Table of n, a(n) for n = 1..500

Cf. A000290, A018884, A052046, A235718, A378498.

filter:= proc(n) local L,S;
   L:= convert(n,base,10);
   S:= Statistics:-Tally(L,output=list);
   S:= sort(S, (a,b) -> lhs(a) < lhs(b));
   andmap(t -> rhs(S[t])Robert Israel, Nov 29 2024

increasingQ[L_]:=Min[Rest[(L-RotateRight[L])]]>0;
sortedQ[n_]:=increasingQ[Sort[Tally[IntegerDigits[n]]][[All,2]]]
Select[Range[575000000]^2,sortedQ]

A378400 Number of subsets of {1,2,3,...,n}^2 with the property that every point has exactly two other closest points.

Original entry on oeis.org

1, 1, 2, 8, 52, 676, 14463

Offset: 0

Erich Friedman, Nov 24 2024

			For n=3, the a(3)=8 subsets are
  ...  oo.  .oo  ...  ...  ooo  .o.  o.o
  ...  oo.  .oo  oo.  .oo  o.o  o.o  ...
  ...  ...  ...  oo.  .oo  ooo  .o.  o.o
For n=4, the a(4)=52 subsets include
  .o..  .o..  oooo  oo.o  ..oo  ooo.
  o.o.  ...o  o..o  oo..  o.oo  o.o.
  .o..  o...  oo.o  ..oo  ....  o.o.
  ...o  ..o.  .ooo  ..oo  o.o.  ooo.

Cf. A297664.

d[p1_,p2_]:=(p1-p2).(p1-p2)
dists[L_,pt_]:=Sort[Map[d[pt,#]&,Complement[L,{pt}]]]
check[L_,pt_]:=(L =={})||((Length[L]>= 3)&&(dists[L,pt][[1]]==dists[L,pt][[2]])&&(dists[L,pt][[2]]< dists[L,pt][[3]]))
check[L_]:=Apply[And,Map[check[L, #]&,L]]
Table[Length[Select[Tuples[Tuples[{0,1},n],n],check[Position[#, 1]]&]],{n,0,5}]

a(6) from Michael S. Branicky, Jan 01 2025

A378498 Squares where larger digits have smaller multiplicity.

Original entry on oeis.org

1, 4, 9, 100, 121, 225, 400, 484, 676, 900, 10000, 11881, 40000, 44944, 69696, 90000, 111556, 202500, 220900, 225625, 232324, 261121, 265225, 300304, 442225, 444889, 695556, 1000000, 1002001, 1020100, 1210000, 2250000, 2295225, 4000000, 4008004, 4080400, 4840000, 5112121, 6760000, 8008900, 9000000

Offset: 1

Erich Friedman, Nov 28 2024

Conjecture: a(n) ≍ n^2. - Charles R Greathouse IV, Nov 29 2024

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000290, A018884, A052046, A235718, A378492.

decreasingQ[L_]:=Max[Rest[(L-RotateRight[L])]]<0;
sortedQ[n_]:=decreasingQ[Sort[Tally[IntegerDigits[n]]][[All,2]]];
Select[Range[10000]^2, sortedQ]

has(n)=my(d=matreduce(digits(n))[,2]); for(i=2,#d, if(d[i]>=d[i-1], return(0))); 1
list(lim)=my(v=List()); for(n=1,sqrtint(lim\1), if(has(n^2), listput(v,n^2))); Vec(v) \\ Charles R Greathouse IV, Nov 29 2024

n^2 << a(n) << 1.001^n. - Charles R Greathouse IV, Nov 29 2024

cp's OEIS Frontend

User: Erich Friedman

A381494 Smallest number with reciprocal of period length n in base 7.

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A381493 Smallest number with reciprocal of period length n in base 8.

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A381370 Smallest number with reciprocal of period length n in base 9.

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A381216 Number of isomers of C_n H_{2n+2} O_2.

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A378557 Powers of 3 that do not contain the digit 3.

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A378556 Powers of 2 that do not include the digit 2.

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A378558 Powers of 4 that do not include the digit 4.

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A378492 Squares where larger digits have larger multiplicity.

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