A014076 -id:A014076 - cp's OEIS frontend

A370482 Characteristic function of primes plus characteristic function of even numbers.

1, 0, 2, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1

Offset: 0

Jens Ahlström, Mar 31 2024

There is only one 2 in the sequence, so if the value 2 is blanked out, a riddle is created that demands some out-of-the-box thinking.

			1 is neither prime nor even so a(1) = 0 + 0 = 0.
2 is both a prime and even so a(2) = 1 + 1 = 2.
3 is a prime but odd so a(3) = 1 + 0 = 1.
4 is not a prime but even so a(4) = 0 + 1 = 1.

Antti Karttunen, Table of n, a(n) for n = 0..100000

Cf. A010051, A059841.

If a(2) were 1 instead of 2, then this would the characteristic function of {0} U A106092, whose complement A014076 gives the positions of 0's. - Antti Karttunen, Jan 17 2025

a[n_] := Boole[PrimeQ[n]] + Boole[EvenQ[n]]; Array[a, 100, 0] (* Amiram Eldar, Mar 31 2024 *)

A370482(n) = (!(n%2) + isprime(n)); \\ Antti Karttunen, Jan 17 2025

from sympy import isprime
def A370482(n): return isprime(n)+(n&1^1) # Chai Wah Wu, Apr 25 2024

a(n) = A010051(n) + A059841(n).

A046446 Nonprimes whose prime factors contain only odd digits.

Original entry on oeis.org

1, 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 75, 77, 81, 85, 91, 93, 95, 99, 105, 111, 117, 119, 121, 125, 133, 135, 143, 147, 153, 155, 159, 165, 169, 171, 175, 177, 185, 187, 189, 195, 209, 213, 217, 219, 221, 225, 231, 237, 243, 245, 247, 255

Offset: 1

Patrick De Geest, Jul 15 1998

			247 = 13 * 19; 1, 3 and 9 are odd digits.

Subsequence of A014076.

isA046446 := proc(n)
    if not isprime(n) then
        fset := numtheory[factorset](n) ;
        for f in fset do
            convert(convert(f,base,10),set) ;
            if % intersect {0,2,4,6,8} <> {} then
                return false;
            end if;
        end do:
        true ;
    else
        false;
    end if;
end proc:
for n from 1 to 260 do
    if isA046446(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Oct 02 2018

Select[Range[255], ! PrimeQ[#] && And @@ OddQ[Union[Flatten[IntegerDigits[First /@ FactorInteger[#]]]]] &] (* Jayanta Basu, Jun 24 2013 *)

Definition clarified by Harvey P. Dale, Jul 07 2018

A155474 Odd nonprimes n with distinct smallest digits of n and n-th prime.

Original entry on oeis.org

1, 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 85, 87, 91, 93, 95, 99, 105, 111, 117, 119, 123, 129, 143, 145, 147, 153, 155, 159, 161, 165, 169, 171, 175, 177, 183, 185, 201, 203, 205, 207, 209, 213, 225, 235, 237, 243, 245, 247, 249, 253

Offset: 1

Juri-Stepan Gerasimov, Jan 23 2009

			If n=1=odd nonprime and 2=prime(1), then 1<2 and 1=a(1). If n=9=odd nonprime and 23=prime(9), then 9>2 and 9=a(2). If n=15=odd nonprime and 47=prime(15), then 1<4 and 15=a(3), etc.

Cf. A000027, A000040, A014076.

Corrected (89 removed) by R. J. Mathar, May 10 2010

A156333 Odd nonprimes q=A141468(j) such that largest digit(j)+largest digit(q) is a prime.

Original entry on oeis.org

1, 39, 57, 91, 115, 121, 133, 143, 145, 147, 153, 155, 161, 175, 189, 203, 205, 207, 225, 231, 235, 243, 253, 255, 261, 273, 279, 285, 295, 299, 301, 303, 323, 325, 327, 341, 343, 345, 351, 355, 377, 385, 387, 411, 415, 427, 429, 441, 445, 447, 451, 465, 481, 483

Offset: 1

Juri-Stepan Gerasimov, Feb 08 2009

Subsequence of A014076 - R. J. Mathar, Feb 10 2009

Cf. A000027, A000040, A141468.

Rephrased definition. Inserted 207. - R. J. Mathar, Feb 10 2009

A162832 The odd nonprimes in A162939.

Original entry on oeis.org

1, 35, 65, 119, 125, 155, 161, 209, 245, 287, 299, 335, 341, 377, 395, 407, 425, 497, 539, 551, 575, 629, 665, 695, 731, 737, 749, 755, 785, 845, 899, 965, 989, 1025, 1037, 1079, 1115, 1127, 1235, 1241, 1295, 1325, 1331, 1379, 1421, 1457, 1475, 1529, 1547, 1577

Offset: 1

Juri-Stepan Gerasimov, Jul 14 2009

nonn

Intersection of A014076 and A162939.

Cf. A000040, A005408, A014076, A162939.

Definition simplified, 507 removed, extended by R. J. Mathar, Aug 12 2009

A162887 Odd nonprimes in an alternating 1-based sum up to some odd nonprime.

Original entry on oeis.org

15, 33, 39, 51, 69, 75, 87, 105, 117, 123, 129, 141, 159, 177, 183, 189, 195, 201, 213, 219, 231, 243, 249, 255, 267, 279, 285, 303, 309, 315, 321, 327, 333, 339, 357, 369, 375, 381, 393, 399, 411, 429, 435, 447, 453, 459, 465, 483, 489, 495, 501, 513, 519

Offset: 1

Juri-Stepan Gerasimov, Jul 16 2009

Define the alternating sum S(u) = sum_{k=0..u} (1-(-1)^k*k) = A064455(u+1) as in A162886.

Evaluate the sum with upper limits u= 1,9, 15, 21... from A014076, and add S(u) to this sequence here whenever it is itself a member of A014076.

			The sequence S(u) = 3, 15, 24, 33, 39, 42, 51, 54, 60, 69, ... is generated by u = 1, 9, 15, 21, ...
The first S-term, 3, is prime, and therefore not added to the sequence. The second S-term, 15, is an odd nonprime and added to the sequence. The third, 24, is even and not added to the sequence.

Cf. A014076.

3 removed by R. J. Mathar, Aug 27 2009

A326586 Odd numbers which do not satisfy Korselt's criterion, complement of A324050.

Original entry on oeis.org

9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 129, 133, 135, 141, 143, 145, 147, 153, 155, 159, 161, 165, 169, 171, 175, 177, 183, 185, 187, 189, 195, 201, 203, 205

Offset: 1

Peter Luschny, Jul 21 2019

nonn

The first number k > 1 not in this sequence but in A014076 is 561, because for every p in {3, 11, 17}, which are the prime divisors of 561, p - 1 divides 560.

Cf. A324050, A014076.

Select[Range[3, 205, 2], CompositeQ[#] && Mod[#, CarmichaelLambda[#]] != 1&] (* Jean-François Alcover, Oct 16 2019 *)

isA326586(n) = true if n != 1 and not isEven(n) and not isPrime(n) and not isCarmichael(n).

A335419 Integers m such that every group of order m is not simple.

Original entry on oeis.org

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94

Offset: 1

Bernard Schott, Jul 09 2020

nonn

Officially, the group of order 1 is not considered to be simple; "a group <> 1 is simple if it has no normal subgroups other than G and 1" (See reference for Joseph J. Rotman's definition).
There is no prime term because there exists only one group of order p and this cyclic group Z/pZ is simple.
As a consequence of Feit-Thompson theorem, all odd composites are terms of this sequence.
The first composite even number that is not present in the data is 60 that is the order of simple alternating group Alt(5), the second one that is missing is 168 corresponding to simple Lie group PSL(3,2) [A031963].

			There exist 5 (nonisomorphic) groups of order 8: Z/8Z, Z/2Z × Z/4Z, (Z/2Z)^3, D_4 and H_8; none of these 5 groups is simple, so 8 is a term.
There exist 13 (nonisomorphic) groups of order 60 (see A000001), 12 are not simple but the alternating group Alt(5) is simple, hence 60 is not a term.

Pascal Ortiz, Exercices d'Algèbre, Collection CAPES / Agrégation, Ellipses, Exercice 1.44 p.96.
Joseph J. Rotman, The Theory of Groups: An Introduction, 4th ed., Springer-Verlag, New-York, 1995. Page 39, Definition.

Craig Cato, The orders of the known simple groups as far as one trillion, Math. Comp., 31 (1977), 574-577.
Wikipedia, Feit-Thompson theorem.

Complement of A005180 (except for 1).
Subsequence: A014076 (odd nonprimes).
Cf. A000001, A031963, A051532 (similar for Abelian), A056867 (similar for nilpotent).

A348852 Numbers k such that the number of odd nonprimes <= k is equal to the number of primes <= k.

Original entry on oeis.org

2, 93, 94, 97, 98, 101, 102, 105, 106, 117, 118

Offset: 1

Giorgos Kalogeropoulos, Nov 01 2021

This sequence is finite. For k > 118, there are always more odd nonprimes than primes <= k.

Numbers k such that A000720(k) = A033271(k). - Michel Marcus, Nov 06 2021

			a(4) = 97 is a term because there are 25 odd nonprimes <= 97 and 25 primes <= 97.

Cf. A000040, A014076, A000720, A033271.

Select[Range@1000,(k=#;Length@Select[Range@k,OddQ@#&&!PrimeQ@#&]==PrimePi@k)&]

isok(k) = primepi(k) == #select(x->(!isprime(x) && (x%2)), [1..k]); \\ Michel Marcus, Nov 06 2021

A386963 Gaps between positions of odd terms in A065090.

Original entry on oeis.org

5, 4, 4, 3, 2, 4, 2, 3, 4, 3, 2, 3, 2, 4, 2, 3, 4, 2, 3, 3, 2, 3, 2, 2, 3, 4, 4, 3, 2, 2, 2, 2, 2, 3, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 3, 2, 3, 2, 4, 2, 2, 2, 4, 4, 2, 2, 2, 2, 3, 2, 2, 2, 2, 3, 4, 3, 2, 4, 2, 2, 2, 3, 2, 3, 2, 3, 2, 4, 2, 3, 4, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 4, 3, 2, 2, 2, 2, 2, 3, 2

Offset: 1

Aied Sulaiman, Aug 11 2025

For n >= 2 we have a(n) in {2,3,4}:
  a(n) = 2 if no prime lies between the two successive odd terms,
  a(n) = 3 if a single prime lies between them,
  a(n) = 4 if two primes lie between them.
The initial 5 comes from 3, 5, 7 between 1 and 9.
Conjecture: a(n) tends to 2 in frequency (i.e., {n : a(n) = 2} has natural density 1).
Conjecture is true because the primes have natural density 0. - Robert Israel, Aug 29 2025

			A065090: 1, 2, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, ...
Odd terms occur at positions: 1, 6, 10, 14, 17, 19, 23, 25, ...
Hence a(n): 5, 4, 4, 3, 2, 4, 2, ...

Cf. A065090, A014076.

a065090=Select[Range[335],#==2||!PrimeQ[#]&];l=Length[a065090];p={};Do[If[OddQ[a065090[[i]] ],AppendTo[p,i]],{i,l}];Differences[p] (* James C. McMahon, Aug 29 2025 *)

lista(nn) = my(vio = select(x->(x % 2), select(m->(!isprime(m) || m==2), [1..nn]), 1)); vector(#vio-1, k, vio[k+1] - vio[k]); \\ Michel Marcus, Aug 16 2025

from sympy import isprime
def gaps_generator():
    pos = 0
    last = None
    k = 1
    while True:
        if not (k % 2 == 1 and isprime(k)):  # in A065090
            pos += 1
            if k % 2 == 1:  # odd term (A014076)
                if last is None:
                    last = pos
                else:
                    yield pos - last
                    last = pos
        k += 1
def a(n: int) -> int:
    g = gaps_generator()
    for _ in range(n - 1):
        next(g)
    return next(g)

cp's OEIS Frontend

A370482 Characteristic function of primes plus characteristic function of even numbers.

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A046446 Nonprimes whose prime factors contain only odd digits.

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Maple

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A155474 Odd nonprimes n with distinct smallest digits of n and n-th prime.

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A156333 Odd nonprimes q=A141468(j) such that largest digit(j)+largest digit(q) is a prime.

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A162832 The odd nonprimes in A162939.

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A162887 Odd nonprimes in an alternating 1-based sum up to some odd nonprime.

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A326586 Odd numbers which do not satisfy Korselt's criterion, complement of A324050.

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A335419 Integers m such that every group of order m is not simple.

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A348852 Numbers k such that the number of odd nonprimes <= k is equal to the number of primes <= k.

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Mathematica