A079000 -id:A079000 - cp's OEIS frontend

A080596 a(1)=1; for n >= 2, a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+3.

1, 4, 5, 7, 9, 10, 11, 12, 13, 15, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 35, 37, 39, 41, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102

Offset: 1

N. J. A. Sloane, Feb 23 2003

Hsien-Kuei Hwang, S Janson, TH Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint, 2016; http://140.109.74.92/hk/wp-content/files/2016/12/aat-hhrr-1.pdf. Also Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585

Index entries for sequences of the a(a(n)) = 2n family

Cf. A079000. Apart from initial terms, same as A079945.

a(1) = 1; then a(6*2^k-3+j) = 8*2^k-3+3j/2+|j|/2 for k >= 0, -2^(k+1) <= j < 2^(k+1).

A080712 a(0) = 4; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

Original entry on oeis.org

4, 5, 7, 8, 9, 12, 13, 15, 18, 21, 22, 23, 24, 27, 28, 30, 31, 32, 33, 34, 35, 36, 39, 42, 45, 46, 47, 48, 51, 52, 54, 57, 60, 63, 66, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 84, 87, 90, 91, 92, 93, 96, 97, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111

Offset: 0

N. J. A. Sloane, Mar 05 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)
Index entries for sequences of the a(a(n)) = 2n family

Cf. A079000, A003605, A079253, A080711, A080710.

{a=4; m=[4]; for(n=1,67,print1(a,","); a=a+1; if(m[1]==n, while(a%3>0,a++); m=if(length(m)==1,[],vecextract(m,"2..")),if(a%3==0,a++)); m=concat(m,a))}

a(a(n)) = 3*(n+3).

More terms and PARI code from Klaus Brockhaus, Mar 06 2003

A080731 a(1)=1; a(2)=6; for n > 2, a(n) is taken as the smallest positive integer greater than a(n-1) such that the condition "n is a member of the sequence if and only if a(n) is odd" is satisfied.

Original entry on oeis.org

1, 6, 8, 10, 12, 13, 14, 15, 16, 17, 18, 19, 21, 23, 25, 27, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98

Offset: 1

Matthew Vandermast, Mar 08 2003

Increases according to a pattern analogous to the one for A079000.

			Because a(2)=6, a(6)=13 is the next odd member of the sequence after 1; terms a(3)-a(5) are the smallest even numbers greater than 6, in order.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A080752.

A080752 a(1)=1; a(2)=8; for n > 2, a(n) is smallest integer greater than a(n-1) that satisfies the condition "n is in the sequence if and only if a(n) is odd.".

Original entry on oeis.org

1, 8, 10, 12, 14, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109

Offset: 1

Matthew Vandermast, Mar 08 2003

Increases according to a pattern analogous to the one for A079000.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A080731.

A080753 a(1)=2; for n > 1, a is the second-smallest positive integer greater than a(n-1) such that the condition "n is in the sequence if and only if a(n) is odd" is satisfied.

Original entry on oeis.org

2, 5, 8, 12, 15, 18, 22, 25, 28, 32, 36, 39, 42, 46, 49, 52, 56, 59, 62, 66, 70, 73, 76, 80, 83, 86, 90, 93, 96, 100, 104, 107, 110, 114, 118, 121, 124, 128, 131, 134, 138, 141, 144, 148, 152, 155, 158, 162, 165, 168, 172, 175, 178, 182, 186, 189, 192, 196, 199, 202

Offset: 1

Matthew Vandermast, Mar 08 2003

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000.

A322385 2 and prime numbers whose prime index is a product of at least two not necessarily distinct prime numbers already in the sequence.

Original entry on oeis.org

2, 7, 19, 43, 53, 107, 131, 163, 227, 263, 311, 383, 443, 521, 577, 613, 719, 751, 881, 1021, 1193, 1301, 1307, 1423, 1619, 1667, 1699, 1993, 2003, 2161, 2309, 2311, 2437, 2539, 2693, 2939, 2969, 3167, 3209, 3671, 3767, 3779, 3833, 4423, 4481, 4597, 4871, 5147

Offset: 1

Gus Wiseman, Dec 05 2018

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.

			We have 1993 = prime(301) = prime(7 * 43). Since 7 and 43 already belong to the sequence, so does 1993.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000002, A001462, A007097, A079000, A079254, A277098, A280996, A291636, A304360, A320628, A322386.

ppQ[n_]:=And[PrimeQ[n],!PrimeQ[PrimePi[n]],And@@ppQ/@First/@If[n==2,{},FactorInteger[PrimePi[n]]]];
Select[Range[1000],ppQ]

A322386 Numbers whose prime indices are not prime and already belong to the sequence.

Original entry on oeis.org

1, 2, 4, 7, 8, 14, 16, 19, 28, 32, 38, 43, 49, 53, 56, 64, 76, 86, 98, 106, 107, 112, 128, 131, 133, 152, 163, 172, 196, 212, 214, 224, 227, 256, 262, 263, 266, 301, 304, 311, 326, 343, 344, 361, 371, 383, 392, 424, 428, 443, 448, 454, 512, 521, 524, 526, 532

Offset: 1

Gus Wiseman, Dec 05 2018

nonn

Union of A291636 (Matula-Goebel numbers of series-reduced rooted trees) and A322385.
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.
A multiplicative semigroup: if x and y are in the sequence, then so is x*y. - Robert Israel, Dec 06 2018

			1 has no prime indices, so the definition is satisfied vacuously. - _Robert Israel_, Dec 07 2018
We have 301 = prime(4) * prime(14). Since 4 and 14 already belong to the sequence, so does 301.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000002, A001462, A007097, A079000, A079254, A214577, A276625, A291636, A304360, A320628, A322385.

Res:= 1: S:= {1}:
for n from 2 to 1000 do
  F:= map(numtheory:-pi, numtheory:-factorset(n));
  if F subset S then
    Res:= Res, n;
    if not isprime(n) then S:= S union {n} fi
fi
od:
Res; # Robert Israel, Dec 06 2018

tnpQ[n_]:=With[{m=PrimePi/@First/@If[n==1,{},FactorInteger[n]]},And[!MemberQ[m,_?PrimeQ],And@@tnpQ/@m]]
Select[Range[1000],tnpQ]

A334067 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a term of the sequence if and only if a(n) is prime" where indices start from 0.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 11, 13, 14, 15, 16, 17, 18, 19, 23, 29, 31, 37, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 132, 133, 134, 135, 137, 139, 140, 149

Offset: 0

Adnan Baysal, Apr 13 2020

nonn

a(n) is the minimal sequence for which the sequence generated by the indices of primes in this sequence is equal to itself, where indices start from 0.
So if f is a function on 0-indexed integer sequences with infinitely many primes where f returns the increasing sequence of indices of primes of the input sequence b(n), then a(n) is the lexicographically minimal fixed point of f.
a(n) has almost the same definition as A079254, except that a(n) starts indices from 0 instead of 1. But the resulting sequences do not seem to have any correlation.

			a(0) cannot be 0, since then 0 should be prime, which it is not.
a(0) = 1 is valid hence a(1) must be the next prime, which is a(1) = 2.
Then a(2) should be the next prime, hence a(2) = 3.
a(3) should be prime, hence a(3) = 5.
Since 4 is not in the sequence so far, a(4) must be the next nonprime, which means a(4) = 6.

The same definition as A079254 except here the indices start from 0 instead of 1.

Cf. A079000, A079313.

# is_prime(n) is a Python function which returns True if n is prime, and returns False otherwise. In the form stated below runs with SageMath.
def a_list(length):
    """Returns the list [a(0), ..., a(length-1)]."""
    num = 1
    b = [1]
    for i in range(1, length):
        num += 1
        if i in b:
            while not is_prime(num):
                num += 1
            b.append(num)
        else:
            while is_prime(num):
                num += 1
            b.append(num)
    return b
print(a_list(63))

A081260 a(1)=4; for n>1, a(n) is taken to be the third-smallest integer greater than a(n-1) such that the condition "n is a member of the sequence if and only if a(n) is odd" is satisfied.

Original entry on oeis.org

4, 10, 16, 21, 26, 32, 38, 44, 50, 55, 60, 66, 72, 78, 84, 89, 94, 100, 106, 112, 117, 122, 128, 134, 140, 145, 150, 156, 162, 168, 174, 179, 184, 190, 196, 202, 208, 213, 218, 224, 230, 236, 242, 247, 252, 258, 264, 270, 276, 281, 286, 292, 298, 304, 309, 314

Offset: 1

Matthew Vandermast, Mar 14 2003

			a(1)=4, implying that the fourth term is the first odd member of the sequence; hence a(2) and a(3) are even. The third-smallest even integer greater than 4 is 10; therefore a(2)=10. The third-smallest integers that can satisfy the given condition if taken as a(3) and a(4) are 16 and 21, respectively.

B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence, J. Integer Seqs., Vol. 6 (2003), #03.2.2.
B. Cloitre, N. J. A. Sloane and M. J. Vandermast, Numerical analogues of Aronson's sequence (math.NT/0305308)

Cf. A079000, A080753.

A306719 Lexicographically earliest sequence containing 2 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

Original entry on oeis.org

2, 4, 8, 10, 20, 22, 28, 30, 50, 58, 64, 72, 80, 82, 88, 108, 114, 134, 148, 172, 190, 204, 214, 230, 238, 244, 262, 272, 312, 322, 340, 344, 360, 362, 400, 410, 422, 442, 458, 498, 514, 552, 554, 568, 594, 610, 620, 640, 688, 712, 730, 750, 758, 784, 792, 814

Offset: 1

Gus Wiseman, Mar 11 2019

nonn

A self-describing sequence, similar to A304360.

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.

Prime indices of A324699.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324700, A324701, A324702, A324703, A324704, A324705.

aQ[n_]:=Switch[n,0,False,2,True,,And@@Cases[FactorInteger[n-1],{p,k_}:>aQ[PrimePi[p]]]];
Select[Range[0,100],aQ]

a(n) = A324699(n) + 1.

cp's OEIS Frontend

A080596 a(1)=1; for n >= 2, a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = 2n+3.

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A080712 a(0) = 4; for n>0, a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a member of the sequence if and only if a(n) is a multiple of 3".

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A080731 a(1)=1; a(2)=6; for n > 2, a(n) is taken as the smallest positive integer greater than a(n-1) such that the condition "n is a member of the sequence if and only if a(n) is odd" is satisfied.

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A080752 a(1)=1; a(2)=8; for n > 2, a(n) is smallest integer greater than a(n-1) that satisfies the condition "n is in the sequence if and only if a(n) is odd.".

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A080753 a(1)=2; for n > 1, a is the second-smallest positive integer greater than a(n-1) such that the condition "n is in the sequence if and only if a(n) is odd" is satisfied.

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A322385 2 and prime numbers whose prime index is a product of at least two not necessarily distinct prime numbers already in the sequence.

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Mathematica

A322386 Numbers whose prime indices are not prime and already belong to the sequence.

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Maple

Mathematica

A334067 a(n) is taken to be the smallest positive integer greater than a(n-1) which is consistent with the condition "n is a term of the sequence if and only if a(n) is prime" where indices start from 0.

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Python

A081260 a(1)=4; for n>1, a(n) is taken to be the third-smallest integer greater than a(n-1) such that the condition "n is a member of the sequence if and only if a(n) is odd" is satisfied.

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A306719 Lexicographically earliest sequence containing 2 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

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