A364293 -id:A364293 - cp's OEIS frontend

A364258 a(n) = A163511(n) - n.

1, 1, 2, 0, 4, 4, 0, -2, 8, 18, 8, 14, 0, 2, -4, -8, 16, 64, 36, 106, 16, 54, 28, 26, 0, 20, 4, 8, -8, -8, -16, -20, 32, 210, 128, 590, 72, 338, 212, 304, 32, 184, 108, 202, 56, 102, 52, 74, 0, 86, 40, 124, 8, 52, 16, 22, -16, 6, -16, -4, -32, -28, -40, -50, 64, 664, 420, 3058, 256, 1806, 1180, 2330, 144, 1052, 676

Offset: 0

Antti Karttunen, Jul 25 2023

Compare also to the scatter plot of A364294.

Antti Karttunen, Table of n, a(n) for n = 0..12288

Cf. A007283, A163511, A364255 [= gcd(n,a(n))], A364287 (positions of negative terms), A364292 (of terms <= 0), A364288, A364294 [= -a(A364293(n))].

f[n_] := Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; {1}~Join~Table[-n + Function[t, Prime[t] Product[Prime[m]^(f[n][[m]]), {m, t}] ][DigitCount[n, 2, 1]], {n, 120}] (* Michael De Vlieger, Jul 25 2023 *)

from sympy import nextprime
def A364258(n):
    c, p, k = 1, 1, n
    while k:
        c *= (p:=nextprime(p))**(s:=(~k&k-1).bit_length())
        k >>= s+1
    return c*p-n # Chai Wah Wu, Jul 25 2023

a(n) = A364288(A163511(n)).
For n >= 1, a(2*n) = 2*a(n).
For n >= 0, a(A007283(n)) = 0.

A364292 Numbers k such that A163511(k) <= k.

Original entry on oeis.org

3, 6, 7, 12, 14, 15, 24, 28, 29, 30, 31, 48, 56, 58, 59, 60, 61, 62, 63, 96, 112, 116, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 192, 223, 224, 232, 236, 238, 239, 240, 242, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 383, 384, 446, 447, 448, 464, 472, 476, 478, 479, 480, 484, 488, 490, 492

Offset: 1

Antti Karttunen, Jul 25 2023

nonn

Michael De Vlieger, Fan-style binary tree, for n = 0..2^12-1 and sequence b = A364258, showing b(n) > n in dark blue, b(n) < n in red, and b(n) = n in light blue. Numbers n shown in red appear in A364287, while n shown either in red or light blue appear in this sequence.

Positions of nonpositive terms in A364258.
Cf. A163511.
Subsequences: A007283, A364287, A364293 (odd terms).
Cf. also A364289.

f[n_] := Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; Select[Range[2, 500], Function[t, Prime[t] Product[Prime[m]^(f[#][[m]]), {m, t}]][DigitCount[#, 2, 1]] <= # &] (* Michael De Vlieger, Jul 25 2023 *)

A364294 Difference k - A163511(k) computed for those odd numbers k for which the difference is nonnegative.

Original entry on oeis.org

0, 2, 8, 8, 20, 4, 28, 50, 28, 22, 58, 86, 110, 2, 52, 50, 128, 132, 166, 202, 236, 22, 124, 232, 136, 286, 146, 74, 246, 370, 352, 412, 452, 488, 238, 458, 216, 568, 362, 692, 68, 236, 338, 606, 754, 550, 536, 728, 854, 846, 904, 952, 994, 694, 478, 1124, 744, 1368, 96, 484, 1084, 1490, 10, 236, 812, 746, 1254

Offset: 1

Antti Karttunen, Jul 25 2023

Conjecture: a(1) is the only zero in this sequence, which is equal to the claim that A007283 gives all fixed points of the map n -> A163511(n).

Question: What can be said about the occurrence of small values later in the sequence? Does the sequence admit any lower bound that is monotonic, and keeps on growing, thus never setting to any constant value? See the scatter plot.

Antti Karttunen, Table of n, a(n) for n = 1..15157

Cf. A007283, A163511, A364258, A364293.

f[n_] := Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]]; Subtract @@ # & /@ Select[Array[{2 # - 1, Function[t, Prime[t] Product[Prime[m]^(f[2 # - 1][[m]]), {m, t}]][DigitCount[2 # - 1, 2, 1]]} &, 1024], #1 >= #2 & @@ # &] (* Michael De Vlieger, Jul 25 2023 *)

a(n) = -A364258(A364293(n)).

A364543 Odd numbers k for which A005940(k) <= k.

Original entry on oeis.org

1, 3, 5, 9, 17, 33, 35, 65, 67, 69, 129, 131, 133, 135, 137, 257, 259, 261, 263, 265, 267, 273, 289, 385, 513, 515, 517, 519, 521, 523, 525, 527, 529, 531, 545, 577, 641, 769, 1025, 1027, 1029, 1031, 1033, 1035, 1037, 1039, 1041, 1043, 1045, 1047, 1057, 1059, 1089, 1091, 1153, 1281, 1537, 2049, 2051, 2053, 2055

Offset: 1

Antti Karttunen, Aug 06 2023

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11023; terms less than 2^23

Odd terms of A364541.
Cf. A005940, A364563 [= -A364499(a(n))].
Subsequences: A364547, A364573.
Cf. also A364293.

A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
isA364543(n) = ((n%2)&&(A005940(n)<=n));

cp's OEIS Frontend

A364258 a(n) = A163511(n) - n.

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A364292 Numbers k such that A163511(k) <= k.

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A364294 Difference k - A163511(k) computed for those odd numbers k for which the difference is nonnegative.

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A364543 Odd numbers k for which A005940(k) <= k.

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