A351498 -id:A351498 - cp's OEIS frontend

A352931 a(n) = A093714(n) - n.

0, 1, -1, 1, -1, 1, -1, 3, -1, 3, -2, 2, 4, -4, 4, -4, 6, -3, 3, 1, -5, 3, -5, 5, -5, 1, -1, 3, -5, 5, 1, 5, -5, -1, 3, 5, -7, 5, -5, -1, 3, 5, -7, 5, -5, 5, -1, -3, 3, 5, -9, 1, -5, 5, -5, 1, -1, 3, -5, 5, -3, 1, -1, 3, -5, 5, -3, 1, -1, 3, -5, 7, -3, 7, -1, 1, -5, 5, -4, 2, 4, -6, 4, -4, 4, -8, 8, -4, 8, -4, 0, -4, 0, -2, 4, -2, 4, -8, 4, -4, 6, -4

Offset: 1

N. J. A. Sloane, May 04 2022

sign

From Michael De Vlieger, May 04 2022: (Start)
Zeros in this sequence correspond to fixed points in A093714.
Sequence exhibits intervals where terms have same parity. Parity changes when a run of odd terms in A093714 with even run length occurs. These runs begin with A093714(A351498(k)). Conjecture: the last parity change begins with a(1036044) = -9. (End)
Rival conjecture: The parity changes infinitely often. This would imply that A352932 contains infinitely many terms. - N. J. A. Sloane, May 04 2022

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Scatterplot of a(n), n = 1..2^20, logarithmic along the x-axis, showing even terms in red and odd in blue, accentuating zeros that are confined to even-term intervals, annotating positive and negative records, and labeling the last term a(k) in parity intervals, with k in A351498.

Cf. A093714, A351498, A352932.

nn = 120; c = {1}; s = 0; a[1] = j = 1; u = 2; Do[k = u; While[Nand[FreeQ[c, k], CoprimeQ[j, k], k != j + 1], k++]; j = k; AppendTo[c, k]; a[i] = k - i; If[k == u, While[MemberQ[c, u], u++]; c = DeleteCases[c, ?(# < u &)]], {i, 2, nn}], Array[a, nn] (* _Michael De Vlieger, May 04 2022 *)

A352932 Where the parity of A352931 changes.

Original entry on oeis.org

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.

C
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See Links section.
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a(9) corrected and a(11) added by Rémy Sigrist, May 06 2022

A351499 Odd m in A352928.

Original entry on oeis.org

1, 9, 15, 75, 105, 315, 525, 735, 945, 1155, 1365, 1575, 1995, 3465, 4305, 5775, 6615, 7035, 8085, 8925, 9765, 10395, 11235, 12495, 12705, 13545, 15015, 19635, 26565, 28875, 31185, 33495, 38115, 45045, 255255, 405405, 525525, 765765, 975975, 1036035, 1786785, 2297295

Offset: 1

Michael De Vlieger, May 03 2022

nonn

Cf. A093714, A351498, A352928.

nn = 2^20; c = {1}; j = 1; s = 0; u = 1; {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 == u, If[OddQ[u], Sow[u]]; While[MemberQ[c, u], u++]; c = DeleteCases[c, _?(# < u &)]], {i, 2, nn}]][[-1, -1]]

from math import gcd
from itertools import islice
def agen(): # generator of terms
    an, aset, mink, seen = 1, {1}, 2, {1}
    yield 1
    while True:
        if mink%2 and mink not in seen: yield mink; seen.add(mink)
        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(), 42))) # Michael S. Branicky, May 03 2022

cp's OEIS Frontend

A352931 a(n) = A093714(n) - n.

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A352932 Where the parity of A352931 changes.

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A351499 Odd m in A352928.

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