A381750 -id:A381750 - cp's OEIS frontend

A381864 Numbers k in A024619 such that p^(m+1) == r (mod k) where r is also in A024619 for all p | n.

15, 33, 35, 44, 45, 51, 63, 65, 66, 69, 70, 75, 76, 77, 80, 85, 87, 88, 90, 91, 92, 95, 99, 102, 104, 105, 115, 119, 123, 130, 133, 135, 138, 140, 141, 143, 144, 145, 152, 153, 154, 159, 160, 161, 170, 172, 174, 175, 176, 177, 180, 184, 185, 187, 188, 189, 190

Offset: 1

Michael De Vlieger, Apr 06 2025

nonn

This sequence intersects neither A381750 nor A382120.

			Table of a(n) for n = 1..12, showing prime decomposition (facs(a(n))), p_x^(m+1) mod n, where x = 1 denotes the smallest prime factor, x = 2, the second smallest prime factor, etc. Brackets appear around residues that are not prime powers.
                       p_x^(m+1) mod n
 n  a(n)  facs(a(n))   p_1   p_2   p_3
-----------------------------------------
 1   15   3 * 5        12    10
 2   33   3 * 11       15    22
 3   35   5 * 7        20    14
 4   44   2^2 * 11     20    33
 5   45   3^2 * 5      36    35
 6   51   3 * 17       30    34
 7   63   3^2 * 7      18    28
 8   65   5 * 13       60    39
 9   66   2 * 3 * 11   62    15    55
10   69   3 * 23       12    46
11   70   2 * 5 * 7    58    55    63
12   75   3 * 5^2       6    50

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

Cf. A000961, A024619, A381750, A382120.

nn = 190, Reap[Do[If[! PrimePowerQ[n], If[NoneTrue[Map[PowerMod[#, 1 + Floor@ Log[#, n], n] &, FactorInteger[n][[All, 1]] ], PrimePowerQ], Sow[n]]], {n, 2, nn}] ][[-1, 1]]

A382120 Numbers k in A024619 such that there exists a prime p | k for which p^(m+1) == r (mod k), where r is also in A024619, and a prime q | k for which q^(m+1) == r (mod k), where r is a prime power.

Original entry on oeis.org

10, 18, 20, 21, 22, 26, 28, 30, 34, 36, 38, 40, 42, 46, 48, 50, 52, 54, 55, 57, 58, 60, 68, 72, 74, 78, 82, 84, 86, 93, 94, 96, 98, 100, 106, 108, 110, 111, 114, 116, 117, 118, 122, 124, 126, 129, 132, 134, 136, 142, 146, 147, 148, 150, 156, 158, 162, 164, 165

Offset: 1

Michael De Vlieger, Apr 06 2025

nonn

This sequence intersects neither A381750 nor A381864.

			Table of a(n) for select n, showing prime decomposition (facs(a(n))), p_x^(m+1) mod n, where x = 1 denotes the smallest prime factor, x = 2, the second smallest prime factor, etc. Brackets appear around residues that are not prime powers.
                          p_x^(m+1) mod n
 n  a(n)  facs(a(n))      p_1   p_2   p_3
-----------------------------------------
 1   10   2 * 5           [6]    5
 2   18   2 * 3^2        [14]    9
 3   20   2^2 * 5        [12]    5
 4   21   3 * 7           [6]    7
 5   22   2 * 11         [10]   11
 6   26   2 * 13          [6]   13
 7   28   2^2 * 7          4   [21]
 8   30   2 * 3 * 5        2   [21]    5
 9   34   2 * 17         [30]   17
10   36   2^2 * 3^2      [28]    9
11   38   2 * 19         [26]   19
22   60   2^2 * 3 * 5      4   [21]    5

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

Cf. A000961, A024619, A381750, A381864.

nn = 165, Reap[Do[If[! PrimePowerQ[n], If[CountDistinct@ Map[Boole@ PrimePowerQ@ PowerMod[#, 1 + Floor@ Log[#, n], n] &, FactorInteger[n][[All, 1]] ] == 2, Sow[n]]], {n, 2, nn}] ][[-1, 1]]

A380870 a(n) = A381798(n) - A361373(n) - 1.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 0, 0, 2, 0, 1, 4, 6, 0, 0, 0, 8, 0, 1, 0, 1, 0, 0, 3, 3, 7, 2, 0, 13, 0, 1, 0, 4, 0, 7, 6, 6, 0, 1, 0, 15, 14, 8, 0, 13, 3, 0, 15, 23, 0, 1, 0, 0, 5, 0, 5, 7, 0, 3, 9, 12, 0, 2, 0, 30, 18, 14, 10, 6, 0, 3, 0, 14, 0

Offset: 1

Michael De Vlieger, Apr 08 2025

nonn

a(n) = cardinality of the intersection of A024619 and row n of A381799.
Let S(n,p) = {p^m : p | n, m = 1..floor(log_p n)}. Therefore S(10,2) = {1,2,4,8} and S(30,3) = {1,3,9,27}. Then U({S(n,p) : p|n}) = row n of A377485.
Let T(n,p) = {p^m (mod n) : p | n} the set of prime divisor power residues r (mod n) == p^m, p | n. Thus T(10,2) = {1,2,4,8,6} and T(30,3) = {1,3,9,27,21}. Then U({T(n,p) : p|n}) = row n of A381799.

			Table of n, a(n), and H(n) = intersection of row n of A381799 with A024619.
 n   facs(n)   a(n)  H(n)
--------------------------------------------
 6   2 * 3       0   -
10   2 * 5       1   {6}
12   2^2 * 3     0   -
14   2 * 7       0   -
15   3 * 5       3   {6, 10, 12}
18   2 * 3^2     2   {10, 14}
20   2^2 * 5     1   {12}
21   3 * 7       4   {6, 12, 15, 18}
22   2 * 11      6   {6, 10, 12, 14, 18, 20}
24   2^3 * 3     0   -
30   2 * 3 * 5   1   {21}
.
a(6) = 0 since Q(6) = R(6) = {1,2,3,4}, i.e., all terms in row 6 of A381799 are powers of primes.
a(10) = 1 since Q(10) = {1,2,4,5,8} but R(10) = {1,2,4,5,6,8}; the latter set contains 1 term (i.e., 6) that is not a member of the former set.
a(14) = 0 since R(14) = {1,2,4,7,8} are all powers of primes.
a(15) = 3 since R(15) = {1,3,5,6,9,10,12} has 3 terms {6,10,12} that are not powers of primes.
a(18) = 2 since R(18) = {1,2,3,4,8,9,10,14,16} has 2 terms {6,10} that are not powers of primes, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..16384

Cf. A000961, A024619, A361373, A377485, A381750, A381798, A381799.

f[x_, p_] := Block[{m = 2, r, c},
  Which[
    PrimePowerQ[x],
    Join[{0}, #1^Range[0, #2 - 1]] & @@ FactorInteger[x][[1]],
    PowerMod[p, m, x] == p, {1, p},
    True, c[_] := False;
    c[1] = c[p] = True; {1, p}~Join~
      Reap[While[r = PowerMod[p, m, x]; ! c[r], Sow[r];
        c[r] = True; m++] ][[-1, 1]] ] ]
Table[Count[Union@ Flatten@ Map[f[n, #] &, FactorInteger[n][[All, 1]] ], _?(And[# > 1, ! PrimePowerQ[#]] &)], {n, 120}]

Let Q(n) = {1} joined to row n > 1 of A377485 and let R(n) = row n of A381799.
a(n) = card(U(Q(n) \ R(n))).
a(p^m) = 0 for prime power p^m, m >= 0.
a(n) = 0 for n in A381750.

A382438 Numbers k in A024619 such that all residues r (mod k) in row k of A381801 are such that rad(r) divides k, where rad = A007947.

Original entry on oeis.org

6, 12, 14, 24, 39, 62, 155, 254, 3279, 5219, 16382, 19607, 70643, 97655, 208919, 262142, 363023, 402233, 712979, 1040603, 1048574, 1508597, 2265383, 2391483, 4685519, 5207819, 6728903, 21243689, 25239899, 56328959, 61035155, 67977559, 150508643

Offset: 1

Michael De Vlieger, Mar 27 2025

nonn

Numbers k in A024619 such that A381804(k) = 0.
Let S(n,p) be the set of distinct power residues r (mod n) beginning with empty product and recursively multiplying by prime p | n. For example, S(10,2) = {1,2,4,8,6}.
This sequence builds on A381750, taking the tensor product T(k) (mod k) of S(k,p), p | k. If all products r (mod k) are such that rad(r) | k, then k is in this sequence. Distinct residues r (mod k) in T(k) are listed in row k of A381801.
Proper subset of A381750.
A139257 is a proper subset since 2^m is congruent to 2 (mod 2^m-2).
Conjecture: 12 and 24 are the only nonsquarefree numbers in this sequence, i.e., in A126706.

			Table of a(n) for n = 1..10, showing prime decomposition (facs(a(n))), row a(n) of A381801:
                        Row a(n) of A381801
 n    a(n)  facs(a(n))  k (mod a(n)) such that rad(k) | a(n).
-------------------------------------------------------------
 1      6   2 * 3       {0, 1, 2, 3, 4}
 2     12   2^2 * 3     {0, 1, 2, 3, 4, 6, 8, 9}
 3     14   2 * 7       {0, 1, 2, 4, 7, 8}
 4     24   2^3 * 3     {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
 5     39   3 * 13      {0, 1, 3, 9, 13, 27}
 6     62   2 * 31      {0, 1, 2, 4, 8, 16, 31, 32}
 7    155   5 * 31      {0, 1, 5, 25, 31, 125}
 8    254   2 * 127     {0, 1, 2, 4, 8, 16, 32, 64, 127, 128}
 9   3279   3 * 1093    {0, 1, 3, 9, 27, 81, 243, 729, 1093, 2187}
10   5219   17 * 307    {0, 1, 17, 289, 307, 4913}
Let b = A381750.
a(1) = 6 since T(6) (mod 6) = {1,2,4} X {1,3} = {{1,2,4},{3,0,0}}; all residues r (mod 6) in T(6) (i.e., in row 6 of A381801) are such that rad(r) | 6.
a(2) = 12 since T(12) (mod 12) = {1,2,4,8} X {1,3,9} = {{1,2,4,8},{3,6,0,0},{9,6,0,0}}; all residues r (mod 12) in T(12) are such that rad(r) | 12.
a(3) = 14 since T(14) (mod 14) = {1,2,4,8} X {1,7} = {{1,2,4,8},{7,0,0,0}}; all residues r (mod 14) in T(14) are such that rad(r) | 14.
a(4) = 24 since T(24) (mod 24) = {1,2,4,8,16} X {1,3,9} = {{1,2,4,8,16},{3,6,12,0,0},{9,18,0,0,0}}; all residues r (mod 24) in T(24) are such that rad(r) | 24.
b(6) = 56 is not in the sequence since 49*2 = 98 = 42 (mod 56), rad(42) does not divide 56.
b(8) = 112 is not in the sequence since 49*4 = 196 = 84 (mod 112), rad(84) does not divide 112, etc.

Michael De Vlieger, Efficient Mathematica code for this sequence (2025).

Cf. A007947, A024619, A381750, A381801.

cp's OEIS Frontend

A381864 Numbers k in A024619 such that p^(m+1) == r (mod k) where r is also in A024619 for all p | n.

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A382120 Numbers k in A024619 such that there exists a prime p | k for which p^(m+1) == r (mod k), where r is also in A024619, and a prime q | k for which q^(m+1) == r (mod k), where r is a prime power.

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A380870 a(n) = A381798(n) - A361373(n) - 1.

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A382438 Numbers k in A024619 such that all residues r (mod k) in row k of A381801 are such that rad(r) divides k, where rad = A007947.

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