A071904 -id:A071904 - cp's OEIS frontend

A046344 Sum of the prime factors of the odd composite numbers (counted with multiplicity).

6, 8, 10, 10, 9, 14, 12, 16, 11, 14, 20, 16, 22, 13, 18, 26, 13, 18, 12, 22, 32, 20, 34, 24, 17, 15, 40, 28, 19, 24, 22, 44, 15, 46, 26, 14, 50, 24, 34, 17, 23, 36, 56, 30, 19, 26, 25, 17, 62, 64, 42, 28, 16, 21, 70, 36, 46, 29, 30, 74, 48, 38, 76, 30, 16, 21, 52, 82, 15, 19

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

			a(54)=21 because 195 = 3 * 5 * 13 and 21 = 3 + 5 + 13.

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A046343, A046345, A071904.

t={}; Do[If[!PrimeQ[n],AppendTo[t,Total[Times@@@FactorInteger[n]]]],{n,9,245,2}]; t (* Jayanta Basu, Jun 04 2013 *)
spf[n_]:=Total[Flatten[Table[#[[1]],#[[2]]]&/@FactorInteger[n]]]; spf/@ Select[Range[9,501,2],CompositeQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 29 2021 *)

A073763 Least number of unrelated set belonging to these numbers is odd.

Original entry on oeis.org

24, 48, 96, 120, 168, 192, 240, 264, 312, 336, 384, 408, 456, 480, 528, 552, 600, 624, 672, 696, 744, 768, 816, 840, 888, 912, 960, 984, 1032, 1056, 1104, 1128, 1176, 1200, 1248, 1272, 1320, 1344, 1392, 1416, 1464, 1488, 1536, 1560, 1608, 1632, 1680, 1704

Offset: 1

Labos Elemer, Aug 08 2002

nonn

			n=24: UnrelatedSet[24]={9, 10, 14, 15, 16, 18, 20, 21, 22}, Min=9, so 24 is here. In cases of all solutions (<50000) the odd number was always 9. This is not an accident. Primes are either divisors or primes to n. Thus a term here should be a composite odd number from A071904, whose first entry is 9; so next candidates are 15, 21, 25, 27... While 15 and 21 not [yet] found, prime powers 25 and 27 did arise.
Least odd unrelated number to 55440 is 25 and smallest unrelated (i.e. neither divisor, nor in RRS) to 3603600 is 27.
Question: can be a smallest odd unrelated number be other than a true power of odd prime?
Answer: no.  Proof: Suppose A073758(n) = k is odd and not a prime power.  Let k = g*u where g = gcd(n,k) > 1.  Since k does not divide n, u > 1.  Since 2*g < k is not unrelated to n, it must divide n, so n is even.  Let p be a prime factor of u.  Since 2*p is not unrelated to n, p must divide n.  But then p^d < k is unrelated to n, where p^d is the highest power of p dividing k. - _Robert Israel_, Sep 11 2014

Cf. A045763, A073757-A073762.

Cf. A071904.

A073758:= proc(n) local k;
  for k from 2 to n-2 do
    if igcd(k,n) > 1 and n mod k > 1 then return k fi
  od;
  0
end proc:
select(t -> A073758(t)::odd, [$1..1000]); # Robert Israel, Sep 11 2014

tn[x_] := Table[w, {w, 1, x}] di[x_] := Divisors[x] rrs[x_] := Flatten[Position[GCD[tn[x], x], 1]] nd[x_] := Complement[tn[x], di[x]] rs[x_] := Union[rrs[x], di[x]] urs[x_] := Complement[tn[x], rs[x]] Do[s=Min[urs[n]]; If[OddQ[s], Print[{n, s}]], {n, 1, 10000}]
unQ[n_] := OddQ[Min[Complement[r = Range[n - 1], Select[r, Divisible[n, #] || GCD[n, #] == 1 &]]]]; Select[Range[1710], unQ] (* Jayanta Basu, Jul 09 2013 *)

Solutions to Mod[A073758(x), 2]=1.
Conjecture: a(n) = 36*n - 18 - 6*(-1)^n = 24 * A001651(n). - Ralf Stephan, Oct 19 2013
The conjecture is false, first counterexample being a(1541) = 55440. - Robert Israel, Sep 11 2014

A129146 a(n) = n-th odd prime minus n-th odd composite number.

Original entry on oeis.org

-6, -10, -14, -14, -14, -16, -16, -16, -16, -18, -14, -14, -14, -16, -12, -10, -14, -10, -10, -12, -8, -8, -4, 2, 2, -2, -4, -6, -4, 8, 10, 14, 14, 20, 18, 22, 22, 24, 28, 32, 28, 36, 34, 36, 34, 42, 52, 52, 52, 50, 54, 54, 62, 62, 62, 66, 66, 70, 72, 70, 78, 90, 92, 92, 92, 100, 102, 110, 106, 108, 112, 118, 120, 124, 124, 128

Offset: 1

Zak Seidov, Apr 01 2007

For small n's, a(n) is negative, while for large n's, a(n) is positive.

Cf. A065091, A071904, A129131, A129145.

from sympy import primepi, prime
def A129146(n):
    if n == 1: return -6
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return prime(n+1)-m # Chai Wah Wu, Aug 01 2024

a(n) = A065091(n) - A071904(n).

A160522 The n-th odd composite number minus the n-th even composite number.

Original entry on oeis.org

5, 9, 13, 15, 15, 19, 19, 21, 25, 27, 27, 29, 29, 33, 33, 35, 39, 39, 41, 43, 43, 45, 45, 45, 47, 51, 55, 57, 57, 57, 57, 57, 57, 59, 61, 61, 65, 65, 65, 65, 69, 69, 71, 71, 73, 75, 75, 77, 77, 81, 81, 81, 81, 85, 89, 89, 89, 89, 89, 91, 91, 91, 91, 91, 93, 97, 99, 99, 103, 103

Offset: 1

Kyle Stern, May 16 2009

K. Stern, Table of n, a(n) for n = 1..1000

Cf. A002808, A005843, A071904, A047846.

composite function [a] = A160522(k) j = 1; n = 1; even = 4; while j < k n = n + 1; if isprime(n) == 1 else if mod(n,2) == 0 else a(j,1) = n - even; even = even + 2; j = j + 1; end end end

Last[t = GatherBy[Select[Range[4, 245], ! PrimeQ[#] &], OddQ]] - Take[First[t], Length[Last[t]]] (* Jayanta Basu, Aug 11 2013 *)

m=70; v=vector(m); k=4; n=0; while(n0&&!isprime(k), n++; v[n]=k-2*(n+1)); k++); v \\ Klaus Brockhaus, May 22 2009

from sympy import primepi
def A160522(n):
    if n == 1: return 5
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return m-(n+1<<1) # Chai Wah Wu, Aug 01 2024

a(n) = A071904(n) - A005843(n+1).

Extended and formula edited by Klaus Brockhaus, May 22 2009

A226025 Odd composite numbers that are not squares of primes.

Original entry on oeis.org

15, 21, 27, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 123, 125, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 165, 171, 175, 177, 183, 185, 187, 189, 195, 201, 203, 205, 207, 209, 213, 215, 217

Offset: 1

Arkadiusz Wesolowski, Jun 07 2013

nonn

Numbers that are in A071904 (odd composite numbers) but not in A001248 (squares of primes).
First differs from its subsequence A082686 in a(16)=81 which is not in A082686. More precisely, A226025 \ A082686 = A062532 \ {1} = A014076^2 \ {1}. - M. F. Hasler, Oct 20 2013
Odd numbers that are greater than the square of their least prime factor - Odimar Fabeny, Sep 08 2014

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10000

Subsequence of A071904. Cf. A226603.

a226025 n = a226025_list !! (n-1)
a226025_list = filter ((/= 2) . a100995) a071904_list
-- Reinhard Zumkeller, Jun 15 2013

[n: n in [3..217 by 2] | not IsPrime(n) and not IsSquare(n) or IsSquare(n) and not IsPrime(Floor(n^(1/2)))];

select(n -> not(isprime(n)) and (not(issqr(n)) or not(isprime(sqrt(n)))), [seq(2*i+1,i=1..1000)]); # Robert Israel, Sep 08 2014

Select[Range[3, 217, 2], ! PrimeQ[#] && ! PrimeQ@Sqrt[#] &]
r = Prime@Range[2, 6]^2; Complement[Select[Range[3, Last[r] - 2, 2], ! PrimeQ[#] &], Most[r]]
Select[Range[3,251,2],NoneTrue[{#,Sqrt[#]},PrimeQ]&] (* Harvey P. Dale, Sep 06 2021 *)

is_A226025(n)={bittest(n,0)&&!isprime(n,0)&&!(issquare(n)&&isprime(sqrtint(n)))&&n>1} \\ - M. F. Hasler, Oct 20 2013

A226025 = { odd x>1 | A100995(x) = 0 or A100995(x) > 2 }. - M. F. Hasler, Oct 20 2013

A262044 Partial sum of the first n odd composite numbers.

Original entry on oeis.org

9, 24, 45, 70, 97, 130, 165, 204, 249, 298, 349, 404, 461, 524, 589, 658, 733, 810, 891, 976, 1063, 1154, 1247, 1342, 1441, 1546, 1657, 1772, 1889, 2008, 2129, 2252, 2377, 2506, 2639, 2774, 2915, 3058, 3203, 3350, 3503, 3658, 3817, 3978, 4143, 4312

Offset: 1

R. J. Mathar, Sep 09 2015

Intersection with A002113 gives A058850.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A058850.

A262044 := proc(n)
    add(A071904(i),i=1..n) ;
end proc:

Accumulate[Select[Range[9,191,2],CompositeQ]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 09 2017 *)

a(n) = Sum_{i=1..n} A071904(i).

A266419 Odd nonludic numbers.

Original entry on oeis.org

9, 15, 19, 21, 27, 31, 33, 35, 39, 45, 49, 51, 55, 57, 59, 63, 65, 69, 73, 75, 79, 81, 85, 87, 93, 95, 99, 101, 103, 105, 109, 111, 113, 117, 123, 125, 129, 133, 135, 137, 139, 141, 145, 147, 151, 153, 155, 159, 163, 165, 167, 169, 171, 177, 183, 185, 187, 189, 191, 195, 197, 199, 201, 203, 205, 207, 213, 215

Offset: 1

Antti Karttunen, Jan 28 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10220

Intersection of A005408 and A192607.
Cf. A192490, A266410, A237126.
Cf. also A071904, A266420.

Other identities. For all n >= 1:

a(n) = 1 + 2*A266410(n).

A322568 Integers k such that the least prime factor of 2^k - 1 is not in A122094.

Original entry on oeis.org

169, 221, 323, 611, 779, 793, 923, 1121, 1159, 1271, 1273, 1349, 1513, 1717, 1829, 1919, 2033, 2077, 2197, 2201, 2413, 2533, 2603, 2759, 2873, 2951, 3097, 3131, 3173, 3193, 3211, 3281, 3379, 3599, 3721, 3757, 3791, 3937, 3953, 4043, 4199, 4223, 4309, 4331

Offset: 1

Jeppe Stig Nielsen, Aug 29 2019

nonn

Clearly, the terms are odd and composite (A071904).

The first term which is itself of form 2^j - 1 is 34359738367 = 2^35 - 1. The least prime factor of 2^34359738367 - 1 is 136463, and the multiplicative order of 2 modulo 136463 is 2201 = 31*71. In A309130, it is asked if a member of A322568 can be of form 2^p - 1 with p prime.

			169 is included because the least prime factor of 2^169-1 is 4057, and the multiplicative order of 2 modulo 4057 is 169 which is not prime. The divisor 4057 is less than the "algebraic" divisor 2^13-1 = 8192 (Mersenne prime).
4199 (= 13*17*19) is included because the least prime factor of 2^4199-1 is 647, and the multiplicative order of 2 modulo 647 is 323 (= 17*19) which is not prime. The divisor 647 is less than the smallest "algebraic" divisor which is 2^13-1 = 8192 (Mersenne prime).
289 is NOT included; its least prime factor is 2^17 - 1.
1073 (= 29*37) is NOT included; its least prime factor is 223, but 223 is a divisor of one of the "algebraic" factors, namely 223 is a divisor of composite Mersenne number 2^37 - 1.

Cf. A000225, A020639, A122094, A309130.

for(k=2,+oo,isprime(k)&&next();forprime(p=3,,if(Mod(2,p)^k-1==0,!isprime(znorder(Mod(2,p)))&&print1(k,", ");next(2))))

A325573 Odd numbers n that have divisor d > 1 such that A048720(A065621(d),n/d) = n.

Original entry on oeis.org

9, 21, 33, 35, 45, 49, 65, 75, 93, 105, 129, 133, 135, 153, 155, 161, 165, 189, 195, 217, 225, 259, 273, 279, 297, 309, 315, 341, 345, 381, 385, 403, 441, 465, 513, 525, 527, 561, 567, 585, 589, 597, 611, 621, 635, 645, 651, 681, 693, 705, 713, 729, 765, 775, 793, 819, 837, 889, 899, 945, 961, 1025, 1029, 1035, 1057, 1065

Offset: 1

Antti Karttunen, May 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A048720, A065621, A115872, A325566, A325567, A325571.

Subsequence of A071904 and of A325572.

A048720(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A065621(n) = bitxor(n-1,n+n-1);
isA325573(n) = ((n%2)&&fordiv(n,d,if(A048720(A065621(n/d),d)==n,return(d

A339519 Odd composite integers m such that A087130(2m-J(m,29)) == 5J(m,29) (mod m), where J(m,29) is the Jacobi symbol.

Original entry on oeis.org

9, 15, 27, 39, 45, 91, 117, 121, 135, 143, 195, 287, 351, 507, 585, 741, 1521, 1547, 1573, 1755, 1935, 2015, 2067, 2535, 2601, 3157, 3227, 3445, 3505, 3519, 3731, 4563, 4879, 4921, 6201, 6273, 6543, 6591, 6721, 7605, 7803, 8099, 10335, 10377, 10403, 10515

Offset: 1

Ovidiu Bagdasar, Dec 07 2020

nonn

The generalized Pell-Lucas sequences of integer parameters (a,b) defined by V(m+2)=a*V(m+1)-b*V(m) and V(0)=2, V(1)=a, satisfy V(k*p-J(p,D)) == V(k-1)*J(p,D) (mod p) whenever p is prime, k is a positive integer, b=-1 and D=a^2+4.
The composite integers m with the property V(k*m-J(m,D)) == V(k-1)*J(m,D) (mod m) are called generalized Pell-Lucas pseudoprimes of level k- and parameter a.
Here b=-1, a=5, D=29 and k=2, while V(m) recovers A087130(m).

D. Andrica, O. Bagdasar, Recurrent Sequences: Key Results, Applications and Problems. Springer, 2020.
D. Andrica, O. Bagdasar, On some new arithmetic properties of the generalized Lucas sequences, Mediterr. J. Math. (to appear, 2021).
D. Andrica, O. Bagdasar, On generalized pseudoprimality of level k (submitted).

Dorin Andrica, Vlad Crişan, and Fawzi Al-Thukair, On Fibonacci and Lucas sequences modulo a prime and primality testing, Arab Journal of Mathematical Sciences, 24(1), 9-15 (2018).

Cf. A087130, A071904, A339127 (a=5, b=-1, k=1).

Cf. A339517 (a=1, b=-1), A339518 (a=3, b=-1), A339520 (a=7, b=-1).

Select[Range[3, 20000, 2],  CoprimeQ[#, 29] && CompositeQ[#] && Divisible[LucasL[2*# - JacobiSymbol[#, 29], 5] - 5*JacobiSymbol[#, 29], #] &]

cp's OEIS Frontend

A046344 Sum of the prime factors of the odd composite numbers (counted with multiplicity).

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A073763 Least number of unrelated set belonging to these numbers is odd.

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A129146 a(n) = n-th odd prime minus n-th odd composite number.

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A160522 The n-th odd composite number minus the n-th even composite number.

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A226025 Odd composite numbers that are not squares of primes.

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A262044 Partial sum of the first n odd composite numbers.

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A266419 Odd nonludic numbers.

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A322568 Integers k such that the least prime factor of 2^k - 1 is not in A122094.

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A339519 Odd composite integers m such that A087130(2m-J(m,29)) == 5J(m,29) (mod m), where J(m,29) is the Jacobi symbol.