A352931 -id:A352931 - cp's OEIS frontend

A352932 Where the parity of A352931 changes.

1, 2, 11, 18, 79, 1788, 15023, 38124, 45053, 1036044, 100280245077

Offset: 1

N. J. A. Sloane, May 04 2022

The terms 15023, 38124, 45053, 1036044 are based on the comments from Scott R. Shannon and Michael De Vlieger in A352588.

In fact a(n) = A351498(n) + 1, although this requires proof (see A351498).

Rémy Sigrist, C program

Cf. A093714, A351498, A352588, A352931.

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a(9) corrected and a(11) added by Rémy Sigrist, May 06 2022

A093714 a(n) = smallest number coprime to a(n-1), not equal to a(n-1)+1, and not occurring earlier; a(1)=1.

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 6, 11, 8, 13, 9, 14, 17, 10, 19, 12, 23, 15, 22, 21, 16, 25, 18, 29, 20, 27, 26, 31, 24, 35, 32, 37, 28, 33, 38, 41, 30, 43, 34, 39, 44, 47, 36, 49, 40, 51, 46, 45, 52, 55, 42, 53, 48, 59, 50, 57, 56, 61, 54, 65, 58, 63, 62, 67, 60, 71, 64, 69, 68, 73, 66, 79

Offset: 1

Reinhard Zumkeller, Apr 12 2004

nonn

Lexicographically earliest infinite sequence of distinct positive numbers such that gcd(a(n-1), a(n)) = 1, a(n) != a(n-1) + 1. - N. J. A. Sloane, May 02 2022
Permutation of natural numbers with inverse A093715: a(A093715(n))=A093715(a(n))=n.
Comments from N. J. A. Sloane, May 02 2022: (Start)
Proof that this is a permutation of the natural numbers.
1. As usual for a "lexicographically earliest sequence" of this class, there is a function B(k) such that a(n) > k for all n > B(k).
2. For any prime p, p divides a(n) for some n. [Suppose not. Using 1, find n_0 such that a(n) > p^2 for all n >= n_0. But if a(n) > p^2, then p is a smaller choice for a(n+1), contradiction.]
3. For any prime p, p divides infinitely many terms. [Suppose not. Let p^i be greater than any multiple of p in the sequence. Go out a long way, and find a term greater than p^i. Then p^i is a smaller candidate for the next term. Contradiction.]
4. Every prime p appears naked. [If not, using 3, find a large multiple of p, G*p, say. But then p would have been a smaller candidate than G*p. Contradiction.]
5. The next term after a prime p is the smallest number not yet in the sequence which is relatively prime to p. Suppose k is missing from the sequence, and find a large prime p that does not divide k. Then the term after p will be k.  So every number appears.
This completes the proof.
Conjecture 1: If p is a prime >= 3, p-1 appears after p.
Conjecture 2: If p is a prime, the first term divisible by p is p itself.
Conjecture 3: If a(n) = p is a prime >= 5, then n < p.
(End)
Coincides with A352588 for n >= 17. - Scott R. Shannon, May 02 2022

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A085229, A347113, A352588, A352928 (smallest missing number).
A352929 gives indices of prime terms, A352930 = first differences, A352931 = a(n)-n. See also A352932.
See Comments in A109812 for a set-theory analog.

nn = 120; c[] = 0; a[1] = c[1] = 1; u = 2; Do[k = u; While[Nand[c[k] == 0, CoprimeQ[#, k], k != # + 1], k++] &@ a[i - 1]; Set[{a[i], c[k]}, {k, i}]; If[a[i] == u, While[c[u] > 0, u++]], {i, 2, nn}]; Array[a, nn] (* _Michael De Vlieger, May 02 2022 *)

from math import gcd
from itertools import islice
def agen(): # generator of terms
    an, aset, mink = 1, {1}, 2
    while True:
        yield an
        k = mink
        while k in aset or gcd(an, k) != 1 or k-an == 1: k += 1
        an = k
        aset.add(an)
        while mink in aset: mink += 1
print(list(islice(agen(), 72))) # Michael S. Branicky, May 02 2022

A352096 Fixed points in A097465.

Original entry on oeis.org

1, 21, 27, 33, 39, 297, 429, 478, 495, 501, 2138, 3854, 4712, 6428, 8144, 9711, 21872, 37791, 57052, 80216, 88179, 89654, 94802, 109388, 113373, 163761, 171166, 182318, 188955, 196904, 203768, 211490, 214149, 239343, 1805247, 1820092, 1849821, 1983543, 2072691

Offset: 1

Michael De Vlieger, May 05 2022

Position of zeros in A351646.

Analogous to A352931 regarding A093714.

Cf. A093714, A097465, A351646, A352931, A353069.

nn = 2^20; c = {1}; j = 1; u = 2; {1}~Join~Reap[Do[k = u; While[Nand[FreeQ[c, k], CoprimeQ[j, k], k != j + 1, k != j - 1], k++]; j = k; AppendTo[c, k]; If[k == i, Sow[k]]; If[k == u, While[MemberQ[c, u], u++]; c = DeleteCases[c, _?(# < u &)]], {i, 2, nn}]][[-1, -1]]

A351624 Fixed points of A093714.

Original entry on oeis.org

1, 91, 93, 117, 147, 164, 172, 189, 224, 231, 254, 273, 284, 327, 344, 357, 374, 382, 399, 411, 434, 464, 483, 494, 543, 561, 567, 584, 592, 609, 644, 674, 693, 704, 747, 777, 794, 819, 854, 884, 891, 903, 914, 932, 939, 957, 987, 1001, 1012, 1029, 1064, 1094

Offset: 1

Rémy Sigrist, May 04 2022

nonn

From Michael De Vlieger, May 04 2022: (Start)
Also positions of 0's in A352931.
Fixed points of A352588 are these terms together with 2 and 8. (End)

			A093714(91) = 91 so 91 belongs to this sequence.
A093714(92) = 88 so neither 88 nor 92 belongs to this sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic scatterplot of the first differences of the first 400000 terms
Rémy Sigrist, C program

Cf. A093714, A352588, A352931.

C
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See Links section.
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{1}~Join~Reap[Do[k = u; While[Nand[FreeQ[c, k], CoprimeQ[j, k], k != j + 1], k++]; j = k; AppendTo[c, k]; If[k == i, Sow[k]]; If[k == u, While[MemberQ[c, u], u++]; c = DeleteCases[c, ?(# < u &)]], {i, 2, nn}]][[-1, -1]] (* _Michael De Vlieger, May 04 2022 *)

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A352932 Where the parity of A352931 changes.

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A093714 a(n) = smallest number coprime to a(n-1), not equal to a(n-1)+1, and not occurring earlier; a(1)=1.

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A352096 Fixed points in A097465.

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A351624 Fixed points of A093714.

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