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A340449 Nonsquare composites n whose smallest prime factor is greater than or equal to n^(2/5).

15, 35, 55, 77, 91, 119, 143, 187, 209, 221, 247, 253, 299, 319, 323, 341, 377, 391, 403, 437, 481, 493, 527, 533, 551, 559, 589, 629, 667, 697, 703, 713, 731, 779, 799, 817, 851, 893, 899, 901, 943, 989, 1003, 1007, 1037, 1073, 1081, 1121, 1139, 1147, 1159, 1189

Offset: 1

Phil Scovis, Jan 07 2021

nonn

Referred to as "freak-show composites" by Doug Massey (see link), who noted that they are exceptionally difficult to factor by hand using trial division.
4181, 4183, 4187, and 4189 are the first four consecutive terms that differ only in the last decimal digit.  The next such set of four consecutive terms begins at 806621.
Ankit Bisain says that for sufficiently large x, there are fewer values in this sequence less than x than there are primes less than x.

			The smallest prime factor of 4189 is 59, which is greater than 4189^(2/5).

Quora, What's the definition of a freak-show composite number?

Superset of A006094, except for initial 6 in that sequence.

Subsequence of A089229.

q:= n-> not (isprime(n) or issqr(n) or min(numtheory[factorset](n))^5Alois P. Heinz, Jan 07 2021

nscQ[n_]:=CompositeQ[n]&&!IntegerQ[Sqrt[n]]&&FactorInteger[n][[1,1]]>= Surd[n^2,5]; Select[Range[1200],nscQ] (* Harvey P. Dale, Jul 25 2021 *)

isok(n) = (n>1) && !isprime(n) && !issquare(n) && (factor(n)[1,1]^5 >= n^2); \\ Michel Marcus, Jan 07 2021

A274046 a(n) is the smallest positive integer which can be represented as the sum of distinct positive triangular numbers in exactly n ways, or 0 if no such integer exists.

Original entry on oeis.org

1, 10, 25, 31, 49, 46, 55, 67, 70, 76, 82, 117, 102, 91, 97, 107, 101, 135, 110, 112, 116, 115, 119, 128, 0, 131, 133, 130, 148, 145, 136, 0, 137, 149, 154, 146, 0, 169, 152, 157, 155, 168, 171, 158, 174, 161, 0, 183, 184, 167, 0, 0, 173, 0, 175, 181, 190

Offset: 1

Phil Scovis, Jun 07 2016

nonn

46 is the smallest number that can be expressed as the sum of distinct triangular numbers in five ways, but 49 is the smallest that can be so expressed in exactly five ways. There are further examples of this phenomenon.

			25 = 1 + 3 + 6 + 15 = 10 + 15 = 1 + 3 + 21. This is the smallest number that can be written as the sum of distinct triangular numbers in three different ways. So a(3)=25.
The first null values of a(n) occur for n = 25, 32, 37, 47, 51, 52, 54, 61,... - _Giovanni Resta_, Jun 08 2016

Cf. A007294, A060773, A024940, A064816.

nT[n_, m_: 0] := nT[n,m] = If[n == 0, 1, Block[{t, i=m+1, s=0}, While[(t = i*(i+1)/2) <= n, s += nT[n-t, i]; i++]; s]]; a[n_] := Block[{k=0, t}, While[(t = nT[++k]) != n && t < Max[2*n, 30]]; If[t == n, k, 0]]; Array[a, 57] (* Giovanni Resta, Jun 08 2016 *)

a(15)-a(20) from Tom Edgar, Jun 08 2016

a(21)-a(57) from Giovanni Resta, Jun 08 2016

A213918 a(n) = smallest possible element of a set of n positive integers s_1, s_2, ..., s_n such that for i != j, |s_i - s_j| = gcd(s_i, s_j), where |x| denotes absolute value.

Original entry on oeis.org

1, 1, 2, 6, 36, 210, 14976, 552720, 309582000

Offset: 1

Phil Scovis, Mar 04 2013

			Examples of sets for the first few cases:
{1},
{1,2},
{2, 3, 4},
{6, 8, 9, 12},
{36, 40, 42, 45, 48},
{210, 216, 220, 224, 225, 240},
{14976, 14980, 14994, 15000, 15008, 15015, 15120},
{552720, 552825, 552960, 553000, 553014, 553140, 553280, 554400},
{309582000, 309583680, 309583800, 309583872, 309583890, 309584000, 309584025, 309584100, 309584160}.

Cf. A061799, A214799.

ok[v_, n_] := v == Select[v, GCD[#, n] == Abs[n - #] &];
ric[p_, cc_, k_] :=
If[Length@p == k, sol = p; True,
  Block[{c = cc, x, r = False},
   While[c != {}, x = First@c; c = Rest@c;
    If[p == Select[p, GCD[#, x] == Abs[x - #] &] &&
     ric[Append[p, x], c, k], r = True; Break[]]]; r]];
a[k_] := Block[{n = 1, d}, While[Length[d = Divisors@n] < k - 1 ||
!ric[{n}, n + d, k], n++]; n];
Do[Print[n, " ", a[n], " ", sol], {n, 7}]

Corrected (with Mathematica program) by Giovanni Resta, Mar 05 2013. Entry revised by N. J. A. Sloane, Mar 05 2013
a(8) from Robert Gerbicz, Mar 05 2013
a(9) from Robert Gerbicz, Mar 06 2013

A144049 Number of different cycles of digits in the hexadecimal (base-16) expansions of 1/p, 2/p, ..., (p-1)/p where p = n-th prime different from 2.

Original entry on oeis.org

2, 4, 2, 2, 4, 8, 2, 2, 4, 6, 4, 8, 6, 2, 4, 2, 4, 2, 2, 8, 2, 2, 8, 8, 4, 2, 2, 12, 16, 18, 2, 8, 2, 4, 10, 12, 2, 2, 4, 2, 4, 2, 8, 4, 2, 2, 6, 2, 12, 8, 2, 40, 10, 64, 2, 4, 2, 12, 8, 6, 4

Offset: 1

Phil Scovis, Sep 08 2008

			For n=3, p=7. 1/7 in hexadecimal = 0.249249249... with a period of 3. (p-1)/3 = 2. a(3)=2.

A006556 for a similar sequence using decimal expansions.

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A340449 Nonsquare composites n whose smallest prime factor is greater than or equal to n^(2/5).

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A274046 a(n) is the smallest positive integer which can be represented as the sum of distinct positive triangular numbers in exactly n ways, or 0 if no such integer exists.

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A213918 a(n) = smallest possible element of a set of n positive integers s_1, s_2, ..., s_n such that for i != j, |s_i - s_j| = gcd(s_i, s_j), where |x| denotes absolute value.

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A144049 Number of different cycles of digits in the hexadecimal (base-16) expansions of 1/p, 2/p, ..., (p-1)/p where p = n-th prime different from 2.

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