A360519 -id:A360519 - cp's OEIS frontend

A361106 Numbers k such that w(k), w(k+1), and w(k+2) are all odd, where w is A360519.

12, 4565, 6402, 12255, 20112, 21421, 24818, 28859, 28924, 29257, 31026, 31207, 34856, 36933, 43614, 49287, 51164, 51869, 59526, 60503, 62984, 65273, 70478, 75659, 76632, 78501, 84754, 86195, 90824, 92301, 95598, 103451, 114460, 115025, 115890, 116995, 117608, 118021, 119994, 121439, 123892

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 02 2023

nonn

Scott R. Shannon, Table of n, a(n) for n = 1..545. These are the k values for the first 1 million terms of A360519.

Cf. A360519, A336957, A337644.

A361108 Indices of records in A360519.

Original entry on oeis.org

1, 2, 3, 4, 8, 12, 13, 17, 29, 74, 85, 97, 105, 110, 145, 149, 186, 230, 369, 401, 442, 521, 689, 741, 745, 989, 993, 1062, 1129, 1153, 1274, 1493, 1937, 2722, 2818, 2842, 3237, 4097, 4301, 5939, 6006, 7516, 7560, 9439, 12984, 14141, 14748, 16480, 21610, 21818, 22226, 23110, 23778, 24210, 27607, 29330, 31392, 35201, 43306, 44199, 47795

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2023

nonn

Scott R. Shannon, Table of n, a(n) for n = 1..116. The indices for the record values for the first 1 million terms of A360519.

Cf. A360519, A361107.

A361112 Numbers that begin a run of 3 consecutive odd valued terms in A360519.

Original entry on oeis.org

77, 5775, 7917, 14745, 23925, 25425, 29435, 34035, 34125, 34485, 36495, 36705, 40803, 43275, 50925, 57375, 59565, 60345, 68859, 70035, 72825, 75525, 81435, 87405, 141495, 90705, 97695, 99267, 104355, 106035, 109935, 118755, 143769, 131745, 132765, 134055, 134805, 135225, 138525, 139065, 141945

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2023

nonn

These are the w(k) values in A361106.

Scott R. Shannon, Table of n, a(n) for n = 1..545. These are the values for the first 1 million terms of A360519.

Cf. A361106, A360519, A336957, A337644.

A361117 a(n) is the least k such that A360519(k) is divisible by the n-th prime number.

Original entry on oeis.org

2, 2, 3, 4, 8, 17, 24, 32, 40, 48, 50, 54, 58, 69, 73, 104, 120, 122, 126, 137, 141, 160, 164, 176, 200, 202, 206, 208, 210, 229, 252, 260, 276, 280, 304, 308, 312, 332, 336, 344, 361, 376, 388, 392, 400, 404, 428, 452, 468, 472, 480, 496, 500, 508, 520, 532

Offset: 1

Scott R. Shannon, Rémy Sigrist and N. J. A. Sloane, Mar 03 2023

nonn

			The first terms of A360519 alongside their prime factors and the corresponding terms of this sequence are:
  n  A360519(n)  Primes  Terms
  -  ----------  ------  --------------
  1           1  None
  2           6  2, 3    a(1)=2, a(2)=2
  3          10  2, 5    a(3)=3
  4          35  5, 7    a(4)=4
  5          21  3, 7
  6          12  2, 3
  7          20  2, 5
  8          55  5, 11   a(5)=8

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program

Cf. A360519.

PARI
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a(n) <= a(n+1).

A361127 Let p = n-th odd prime; a(n) = index where 2p appears in A360519, or -1 if 2p never appears.

Original entry on oeis.org

2, 3, 11, 16, 28, 24, 32, 40, 48, 51, 55, 59, 84, 96, 104, 120, 123, 127, 144, 148, 160, 164, 176, 200, 203, 207, 208, 211, 236, 252, 260, 276, 280, 304, 308, 312, 332, 336, 344, 368, 376, 388, 392, 400, 404, 428, 452, 468, 472, 480, 496, 500, 508, 520, 532, 556, 560

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 08 2023

nonn

It is conjectured that every 2*prime(n) (n>1) appears in A360519. A proof of this would be a big step towards proving that every term of C appears in A360519.

			p = 11 is the 4th odd prime, and A360519(16) = 2*11 = 22, so a(4) = 16.

N. J. A. Sloane, Table of n, a(n) for n = 1..5062

Cf. A360519, A360103.

A361105 Fixed points in A360519.

Original entry on oeis.org

1, 88, 92, 112, 116, 172, 268, 272, 324, 17242, 18650, 43208, 55828, 192434, 1497756

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 02 2023

It is unknown if the sequence is infinite.

Cf. A360519, A336957, A338050.

a(15) from Rémy Sigrist, Mar 04 2023

A361102 1 together with numbers having at least two distinct prime factors.

Original entry on oeis.org

1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111, 112

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 02 2023

nonn

This is the union of 1 and A024619. It is the sequence C used in the definition of A360519. Since C is central to the analysis of A360519 it deserves its own entry.

This has the same relationship to A024619 as A000469 does to A120944 for squarefree numbers.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000469, A024619, A120944, A126706, A246655, A360519.

isa := n -> is(irem(ilcm(seq(1..n-1)), n) = 0):
aList := upto -> select(isa, [seq(1..upto)]):
aList(112); # Peter Luschny, May 17 2023

Select[Range[120], Not@*PrimePowerQ] (* Michael De Vlieger, May 17 2023 *)

from sympy import primepi, integer_nthroot
def A361102(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

def A361102List(upto: int) -> list[int]:
    return sorted(Set(1..upto).difference(prime_powers(upto)))
print(A361102List(112))  # Peter Luschny, May 17 2023

From Peter Luschny and Michael De Vlieger, May 17 2023: (Start)
The sequence is the complement of the prime powers in the positive integers, a = A000027 \ A246655.
k is in this sequence <=> k divides lcm(1, 2, ..., k-1). (End)
This sequence is {1} U { A120944 U A126706 } = {1} U A024619. - Michael De Vlieger, May 17 2023

Offset set to 1 by Peter Luschny, May 17 2023

A361321 Lexicographically earliest infinite sequence of distinct elements of A000469 such that, for n > 2, a(n) has a common factor with a(n-1) but not with a(n-2).

Original entry on oeis.org

1, 6, 10, 35, 21, 33, 22, 14, 91, 39, 15, 55, 77, 42, 26, 65, 85, 34, 38, 57, 51, 119, 70, 30, 69, 161, 133, 95, 110, 46, 299, 143, 66, 58, 145, 105, 78, 62, 155, 115, 138, 74, 185, 165, 87, 203, 154, 82, 123, 93, 217, 182, 86, 129, 111, 259, 238, 94, 141, 159

Offset: 1

Scott R. Shannon, Rémy Sigrist and N. J. A. Sloane, Mar 09 2023

nonn

This sequence is a variant of A360519 where we only consider nonprime squarefree numbers (A000469).
Theorem: a(1) = 1, a(2) = 6; thereafter, a(n) is the smallest nonprime squarefree number m not yet in the sequence such that
(i) gcd(m, a(n-1)) > 1,
(ii) gcd(m, a(n-2)) = 1, and
(iii) m does not divide a(n-1).
Conjecture: The sequence is a permutation of A000469.

Scott R. Shannon, Table of n, a(n) for n = 1..100000
Scott R. Shannon, White on black graph of first 50000 terms [The green line is x = y]
Scott R. Shannon, Image of the first 500000 terms in color. The terms with a lowest prime factor of 2,3,5,7,9,11,13,17,19,>=23 are colored white, red, orange, yellow, green, blue, indigo, violet, gray respectively.
Rémy Sigrist, PARI program

Cf. A000469, A336957, A360519.

PARI
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See Links section.
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A361110 a(n) indicates the index of A361109 in C (A361102).

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 18, 18, 18, 18, 22, 22, 22, 22, 24, 24, 24, 24, 28, 28, 28, 28, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 29, 38, 38

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2023

nonn

			After we have calculated A360519(4) = 35, the smallest term of C that is missing from A360519 is 12 = C(3) = A361102(3), so a(4) = 3.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
N. J. A. Sloane, Table showing A360519(1)-A360519(13), also the smallest missing number (smn, A361109 and A361110), binary vectors showing which terms are divisible by the primes 2, 3, 5, 7, 11; and phi, a decimal representation of those binary vectors (A361111). This sequence forms the fourth row of the table.

Cf. A360519, A361102, A361109.

PARI
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See Links section.
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More terms from Rémy Sigrist, Mar 03 2023

A363576 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared such that a(n) has a common factor with a(n-1), has no common factor with a(n-2), while the difference |a(n) - a(n-1)| is distinct from all previous differences |a(i) - a(i-1)|, i=2..n-1.

Original entry on oeis.org

1, 6, 10, 35, 21, 12, 20, 55, 33, 18, 28, 77, 143, 26, 14, 105, 51, 34, 40, 95, 57, 24, 22, 187, 85, 15, 36, 52, 65, 45, 42, 68, 221, 39, 63, 56, 38, 171, 75, 115, 46, 74, 111, 69, 92, 44, 165, 87, 58, 88, 99, 135, 50, 82, 123, 183, 122, 70, 133, 209, 66, 93, 155, 80, 114, 153, 391, 161, 84

Offset: 1

Scott R. Shannon, Jun 10 2023

nonn

This is a variation of A360519 where the difference between consecutive terms is distinct. See A360519 for further details.

In the first 100000 terms the only fixed points are 1, 585 and 3619, although it is possible more exist.

			a(11) = 28 as 28 shares 2 as a common factor with a(10) = 18 while sharing no common factor with a(9) = 33. Also the difference |28 - 18| = 10 is distinct from all previous differences. This is the first term to differ from A360519.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A360519, A361102, A098550, A336957.

cp's OEIS Frontend

A361106 Numbers k such that w(k), w(k+1), and w(k+2) are all odd, where w is A360519.

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A361108 Indices of records in A360519.

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A361112 Numbers that begin a run of 3 consecutive odd valued terms in A360519.

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A361117 a(n) is the least k such that A360519(k) is divisible by the n-th prime number.

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A361127 Let p = n-th odd prime; a(n) = index where 2*p appears in A360519, or -1 if 2*p never appears.

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A361105 Fixed points in A360519.

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A361102 1 together with numbers having at least two distinct prime factors.

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A361321 Lexicographically earliest infinite sequence of distinct elements of A000469 such that, for n > 2, a(n) has a common factor with a(n-1) but not with a(n-2).

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A361110 a(n) indicates the index of A361109 in C (A361102).

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PARI

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A363576 a(1) = 1, a(2) = 6; for n > 2, a(n) is the smallest positive number that has not yet appeared such that a(n) has a common factor with a(n-1), has no common factor with a(n-2), while the difference |a(n) - a(n-1)| is distinct from all previous differences |a(i) - a(i-1)|, i=2..n-1.

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A361127 Let p = n-th odd prime; a(n) = index where 2p appears in A360519, or -1 if 2p never appears.