A356322 -id:A356322 - cp's OEIS frontend

A369276 Numbers k in A126706 such that either k-1 or k+1 or both are also in A126706.

44, 45, 75, 76, 98, 99, 100, 116, 117, 135, 136, 147, 148, 152, 153, 171, 172, 175, 176, 188, 189, 207, 208, 224, 225, 244, 245, 260, 261, 275, 276, 279, 280, 296, 297, 315, 316, 324, 325, 332, 333, 350, 351, 352, 363, 364, 368, 369, 375, 376, 387, 388, 404, 405

Offset: 1

Michael De Vlieger, Mar 24 2024

A369954 is a proper subset.
Complement of A369516 relative to A126706.
Seen as a table where terms are consecutive, row n contains no primes; corollary: numbers in row n exceed prime(i) but are less than prime(i+1) for some i.
Smallest k such that row n has length m appear in A356322. Rows have length m > 1.

			Seen as a table T(n,j), row n contains the following terms:
      n
      1:     44,     45;
      2:     75,     76;
      3:     98,     99,    100;
      4:    116,    117;
      5:    135,    136;
      6:    147,    148;
      7:    152,    153;
            ...
     59:    844,    845,    846,    847,    848;
            ...
    235:   2888,   2889,   2890,   2891,   2892;
            ...
    255:   3174,   3175,   3176,   3177;
            ...
    293:   3624,   3625,   3626,   3627,   3628;
            ...
   1898:  22020,  22021,  22022,  22023,  22024,  22025;
            ...
  19018: 217070, 217071, 217072, 217073, 217074, 217075, 217076;
            ...

Michael De Vlieger, Table of n, a(n) for n = 1..11053

Cf. A126706, A356322, A369516, A369954.

Select[Select[Range[500], Nor[SquareFreeQ[#], PrimePowerQ[#]] &], AnyTrue[{# - 1, # + 1}, Nor[SquareFreeQ[#], PrimePowerQ[#]] &] &]

A369954 Numbers k that are neither squarefree nor prime powers and also coprime to 6.

Original entry on oeis.org

175, 245, 275, 325, 425, 475, 539, 575, 605, 637, 725, 775, 833, 845, 847, 875, 925, 931, 1025, 1075, 1127, 1175, 1183, 1225, 1325, 1375, 1421, 1445, 1475, 1519, 1525, 1573, 1625, 1675, 1715, 1775, 1805, 1813, 1825, 1859, 1925, 1975, 2009, 2023, 2057, 2075, 2107

Offset: 1

Michael De Vlieger, Mar 24 2024

Define quality Q to signify a number k neither squarefree nor prime power, i.e., k is in A126706. For example, 12 has quality Q but numbers k = 1..11 do not.
Numbers k in this sequence have quality Q and are such that either (k-1) or (k+1) also have quality Q. Hence k also appears in A369276, but not in A369516.
Numbers k such that k mod 12 = 1 or k mod 12 = 5 imply (k-1) in A126706, since 4 divides (k-1).
Numbers k such that k mod 12 = 7 or k mod 12 = 11 imply (k+1) in A126706, since 4 divides (k+1).
Proper subset of A367455.
By definition these odd numbers are such that A053669(k) = 2, therefore A053669(k) < A003557(k), hence this sequence is a proper subset of A360765.

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

Cf. A003557, A007310, A008586, A013929, A024619, A053669, A126706, A356322, A369276, A360765, A367455, A369516.

Select[Flatten[Array[6 # + {1, 5} &, 360]], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]

isok(k) = !issquarefree(k) && !isprimepower(k) && (gcd(k, 6)==1); \\ Michel Marcus, Mar 25 2024

Intersection of A007310 and A126706.

Intersection of A007310, A013929, and A024619.

A358089 First differences of A126706.

Original entry on oeis.org

6, 2, 4, 4, 8, 4, 4, 1, 3, 2, 2, 2, 2, 4, 3, 5, 4, 3, 1, 4, 4, 4, 2, 2, 4, 2, 1, 1, 4, 4, 4, 4, 1, 3, 4, 2, 6, 3, 1, 4, 4, 3, 1, 2, 2, 1, 3, 4, 2, 2, 4, 3, 1, 3, 1, 4, 4, 4, 1, 3, 4, 2, 2, 4, 3, 1, 4, 4, 4, 4, 1, 3, 4, 2, 2, 4, 2, 2, 1, 3, 2, 2, 8, 1, 3, 4, 2

Offset: 1

Michael De Vlieger, Oct 31 2022

A356322 relates to the first instances of exactly k consecutive 1's in this sequence.
a(n) - 1 = number of 0's between 1's in A355447.
For prime p, m such that m mod p^2, unless m = p^e, e > 1, is in A126706, as a consequence of definition of A126706. Therefore m <= 4 is common, m <= 9 much less so. Consequently, the arrangement of A126706 mod M for M in A061742 presents a quasi-modular pattern as seen in the example and raster link at A355447.
a(51265) = 7; m = 9 is not observed in the first 6577230 terms of the sequence, a dataset corresponding to terms k <= 2^24 in A126706.

			The smallest numbers that are neither squarefree nor a prime power are {12, 18, 20, 24, 28 ...}, therefore the first terms of this sequence are {6, 2, 4, 4, ...}.

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

Cf. A126706, A355447, A356322.

k = 0; Rest@ Reap[Do[If[And[#2 > 1, #1 != #2] & @@ {PrimeOmega[n], PrimeNu[n]}, Sow[n - k]; Set[k, n] ],{n, 270}] ][[-1, -1]]
(* Generate 317359 terms of this sequence from the image at A355447: *)
Differences@ Position[Flatten@ ImageData[Import["https://oeis.org/A355447/a355447_1.png", "PNG"]], 0.][[All, -1]]

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A358089(n):
    def f(x): return int(n+sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    r, k = n+1, f(n+1)+1
    while r != k:
        r, k = k, f(k)+1
    return r-m # Chai Wah Wu, Aug 15 2024

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A369276 Numbers k in A126706 such that either k-1 or k+1 or both are also in A126706.

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A369954 Numbers k that are neither squarefree nor prime powers and also coprime to 6.

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A358089 First differences of A126706.

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