A244623 -id:A244623 - cp's OEIS frontend

A079290 Composite numbers satisfying A073078(n)=(n+1)/2.

9, 15, 25, 27, 49, 81, 121, 125, 169, 243, 289, 343, 361, 529, 625, 729, 841, 961, 1331, 1369, 1681, 1849, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 9409, 10201, 10609, 11449

Offset: 1

Benoit Cloitre, Apr 09 2003

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..94

Cf. A073078, A244623.

A073078 := proc(n)
    local bink,k ;
    bink := 1 ;
    for k from 1 do
        bink := 2*bink*(2-1/k) ;
        if modp(bink,n) = 0 then
            return k;
        end if;
    end do:
end proc:
A079290 := proc(n)
    option remember;
    local a;
    if n = 1 then
        9;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and 2*A073078(a) = a+1 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Aug 20 2014

b[n_] := For[k=1, True, k++, If[Divisible[Binomial[2k, k], n], Return[k]]];
Select[Select[Range[12000], CompositeQ], b[#] == (# + 1)/2&] (* Jean-François Alcover, Oct 31 2019 *)

p=5;forprime(q=7,1e4,forstep(n=p+2,q-2,2, for(s=2,n\2, if(binomial(2*s,s)%n==0,next(2)));print1(n", ")); p=q) \\ Charles R Greathouse IV, May 24 2013

a(21)-a(43) from Charles R Greathouse IV, May 24 2013

A380410 Numbers with greater product of prime indices than sum of prime factors.

Original entry on oeis.org

1, 45, 49, 63, 75, 77, 81, 91, 99, 105, 117, 119, 121, 125, 126, 133, 135, 143, 147, 150, 153, 161, 162, 165, 169, 171, 175, 182, 187, 189, 195, 198, 203, 207, 209, 210, 217, 221, 225, 231, 234, 238, 242, 243, 245, 247, 250, 253, 255, 259, 261, 266, 270, 273

Offset: 1

Gus Wiseman, Jan 25 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, with product A003963.

			126 has prime indices {1,2,2,4} and prime factors {2,3,3,7}, and 16 > 15, so 126 is in the sequence.
The terms together with their prime indices begin:
     1: {}
    45: {2,2,3}
    49: {4,4}
    63: {2,2,4}
    75: {2,3,3}
    77: {4,5}
    81: {2,2,2,2}
    91: {4,6}
    99: {2,2,5}
   105: {2,3,4}
   117: {2,2,6}
   119: {4,7}
   121: {5,5}
   125: {3,3,3}
   126: {1,2,2,4}
   133: {4,8}
   135: {2,2,2,3}

For factors instead of indices we have A002808.
The case of prime powers is A244623.
For indices instead of factors we have A325037, see also A325038.
The version for equality is A331384, counted by A331383.
Positions of positive terms in A380344.
Partitions of this type are counted by A380411.
A000040 lists the primes, differences A001223.
A001222 counts prime factors with multiplicity.
A055396 gives least prime index, greatest A061395.
Triangles:
- A027746 = prime factors
- A112798 = prime indices
Statistics:
- A000027 = product of prime factors = row products of A027746
- A001414 = sum of prime factors = row sums of A027746
- A003963 = product of prime indices = row products of A112798
- A056239 = sum of prime indices = row sums of A112798
Combinations:
- A075254 = product of factors + sum of factors = A000027 + A001414
- A075255 = product of factors - sum of factors = A000027 - A001414
- A178503 = product of factors - sum of indices = A000027 - A056239
- A325036 = product of indices - sum of indices = A003963 - A056239
- A379681 = product of indices + sum of indices = A003963 + A056239
- A380344 = product of indices - sum of factors = A003963 - A001414
- A380345 = product of factors + sum of indices = A000027 + A056239
- A380409 = product of indices + sum of factors = A003963 + A001414
Cf. A000720, A175508, A301987, A319000, A325040, A325041, A325044, A379682, A380220.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@prix[#]>Plus@@Prime/@prix[#]&]

A003963(a(n)) > A001414(a(n)).

A158248 Composite numbers with primitive root 10.

Original entry on oeis.org

49, 289, 343, 361, 529, 841, 2209, 2401, 3481, 3721, 4913, 6859, 9409, 11881, 12167, 12769, 16807, 17161, 22201, 24389, 27889, 32041, 32761, 37249, 49729, 52441, 54289, 66049, 69169, 72361, 83521, 97969

Offset: 1

Robert Hutchins, Mar 15 2009

Previous name was: Numbers m whose reciprocal generates a repeating decimal fraction with period phi(m) and m/2 < phi(m) < m-1.
All terms are proper powers of full reptend primes (A001913).
This sequence does not contain every proper power of every term in A001913, for example, A001913 has 487 as its 26th term, but since 10 is not a primitive root of 487^2, 487^2 is not a term of this sequence. - Robert Hutchins, Oct 14 2021
A shorter description appears to be "Composite numbers with primitive root 10". - Arkadiusz Wesolowski, Jul 04 2012 (The two definitions certainly produce the same terms up through 83521. - N. J. A. Sloane, Jul 05 2012)

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A007732, A001913, A046145, A046146.
Subsequence of A244623.
Subsequence of A167797.
Cf. A108989 (for base 2), A346316 (for base 6).

select(n -> not isprime(n) and numtheory:-primroot(9,n) = 10,[$2..10000]);
# N. J. A. Sloane, Jul 05 2012

Select[Range[10^5], GCD[10, #] == 1 && #/2 < MultiplicativeOrder[10, #] < # - 1 &] (* Ray Chandler, Oct 17 2012 *)

More terms from Robert Hutchins, Mar 21 2009
Entry revised by N. J. A. Sloane, Jul 05 2012
New name (using comment by Arkadiusz Wesolowski) from Joerg Arndt, Nov 22 2021

A346316 Composite numbers with primitive root 6.

Original entry on oeis.org

121, 169, 289, 1331, 1681, 2197, 3481, 3721, 4913, 6241, 6889, 7921, 10609, 11449, 11881, 12769, 14641, 16129, 17161, 18769, 22801, 24649, 28561, 32041, 39601, 49729, 51529, 52441, 54289, 63001, 66049, 68921, 73441, 76729, 83521, 120409, 134689, 139129, 157609

Offset: 1

Robert Hutchins, Jul 13 2021

nonn

An alternative description: Numbers k such that 1/k in base 6 generates a repeating fraction with period phi(n) and n/2 < phi(n) < n-1.

For example, in base 6, 1/121 has repeat length 110 = phi(121) which is > 121/2 but less than 121-1.

Robert Hutchins, PrimRoot.c

Subsequence of A244623.
Subsequence of A167794.
Cf. A108989 (for base 2),  A158248 (for base 10).
Cf. A157502.

isA033948 := proc(n)
    if n in {1,2,4} then
        true;
    elif type(n,'odd') and nops(numtheory[factorset](n)) = 1 then
        true;
    elif type(n,'even') and type(n/2,'odd') and nops(numtheory[factorset](n/2)) = 1 then
        true;
    else
        false;
    end if;
end proc:
isA167794 := proc(n)
    if not isA033948(n) or n = 1 then
        false;
    elif numtheory[order](6,n) = numtheory[phi](n) then
        true;
    else
        false;
    end if;
end proc:
A346316 := proc(n)
    option remember;
    local a;
    if n = 1 then
        121;
    else
        for a from procname(n-1)+1 do
            if not isprime(a) and isA167794(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A346316(n),n=1..20) ; # R. J. Mathar, Sep 15 2021

Select[Range[160000], CompositeQ[#] && PrimitiveRoot[#, 6] == 6 &] (* Amiram Eldar, Jul 13 2021 *)

isok(m) = (m>1) && !isprime(m) && (gcd(m, 6)==1) && (znorder(Mod(6, m))==eulerphi(m)); \\ Michel Marcus, Aug 12 2021

A167794 INTERSECT A002808.

A378028 Positions of records in A377059.

Original entry on oeis.org

1, 4, 9, 17, 22, 25, 46, 49, 81, 118, 121, 169, 243, 334, 337, 343, 361, 529, 841, 961, 1331, 1369, 2187, 2197, 2209, 2809, 3481, 3721, 4489, 5041, 6241, 6859, 6889, 7921, 10201, 11449, 12167, 14641, 16129, 17161, 19321, 22201, 24389, 26569, 27889, 29791, 29929, 32041, 32761, 38809, 39601, 44521, 49729

Offset: 1

Robert Israel, Nov 14 2024

nonn

Numbers k such that A377059(k) > A377059(j) for all j < k.
The record values are in A378029.
It appears that in most cases a(n) is in A244623 (Odd prime powers that are not primes) and A378029(n) = A000010(a(n)).
If p is in A001122 and is not a Wieferich prime (A001220), then p^2 is a term with A377059(p^2) = p*(p-1).

			a(3) = 9 is a term because A377059(9) = 6 > A377059(k) for all k < 9.

Cf. A000010, A001122, A001220, A244623, A377059, A378029.

f:= proc(n) local x,r;
  for x from 2 to n do
    if igcd(x,n) <> 1 then next fi;
    r:= numtheory:-order(x,n);
    if r::even and r < n-1 then return r fi
  od;
  0
end proc:
J:= 1: m:= 0: count:= 0:
for k from 2 while count < 100 do
v:= f(k);
if v > m then m:= v; J:= J,k; count:= count+1 fi;
od:
J;

A378029(n) = A377059(a(n)).

cp's OEIS Frontend

A079290 Composite numbers satisfying A073078(n)=(n+1)/2.

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A380410 Numbers with greater product of prime indices than sum of prime factors.

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A158248 Composite numbers with primitive root 10.

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A346316 Composite numbers with primitive root 6.

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A378028 Positions of records in A377059.

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Maple

Formula