cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A364998 Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) <= k but rad(k)*A053669(k) > k.

Original entry on oeis.org

18, 24, 90, 120, 126, 150, 168, 180, 198, 234, 264, 306, 312, 342, 408, 414, 456, 522, 552, 558, 630, 666, 696, 738, 744, 774, 840, 846, 888, 954, 984, 990, 1032, 1050, 1062, 1098, 1128, 1170, 1206, 1260, 1272, 1278, 1314, 1320, 1386, 1416, 1422, 1464, 1470, 1494
Offset: 1

Views

Author

Michael De Vlieger, Aug 16 2023

Keywords

Comments

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) > 0, A360543(k) = 0. There exist nondivisors m < k such that rad(m) = rad(k); however, m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k do not exist.

Examples

			Let b(n) = A126706(n), S = A360768, and T = A363082.
b(1) = 12 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30; both exceed 12, thus 12 is not in S.
b(2) = a(1) = 18 since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, 18 does not exceed 18 and 30 is larger than 18, hence 18 is in both S and T.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36, therefore 36 is in S but not T.
b(7) = 40 is not in the sequence since p*r = 5*10 = 50 and q*r = 3*10 = 30. Though 50 > 40, 30 < 40, thus 40 is neither in S nor T, etc.

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..10000
  • Michael De Vlieger, Annotated plot of b(n) = A126706(n), with n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus showing 400 terms. Terms in this sequence are colored black, those in A364999 in blue, in A364997 in green, and in A361098 in red.
  • Michael De Vlieger, Plot of b(n), with n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus showing 14400 terms, using the same color scheme as described immediately above.
  • Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms in this sequence are colored black, else white. Demonstrates a strong quasiperiodic pattern approximately mod 169.

Crossrefs

Cf. A007947, A053669, A119288, A126706, A355432, A360432, A360768, A361098, A363082, A364997, A364999.

Programs

  • Mathematica
    Select[Select[Range[1500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r <= k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Formula

Intersection of A363082 and A360768.

A364997 Numbers k neither squarefree nor prime power such that rad(k)*A119288(k) > k but rad(k)*A053669(k) < k.

Original entry on oeis.org

40, 45, 56, 63, 88, 99, 104, 117, 136, 152, 153, 171, 175, 176, 184, 207, 208, 232, 248, 261, 272, 275, 279, 280, 296, 297, 304, 315, 325, 328, 333, 344, 351, 368, 369, 376, 387, 423, 424, 425, 440, 459, 464, 472, 475, 477, 488, 495, 496, 513, 520, 531, 536, 539
Offset: 1

Views

Author

Michael De Vlieger, Aug 16 2023

Keywords

Comments

Subset of A126706, numbers that are neither squarefree nor prime powers.
For k in this sequence, let p = A119288(k), q = A053669(k), and r = A007947(k).
A355432(k) = 0, A360543(k) > 0. There exist m < k, gcd(m,k) > 1 such that both omega(k) > omega(m) and rad(m) | k, but nondivisors m < k do not exist such that rad(m) = rad(k).

Examples

			Let b(n) = A126706(n), S = A360767, and T = A360765.
b(1) = 12 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30; both exceed 12, thus 12 is in S but not in T.
b(2) = 18 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6 = 30. Indeed, neither is less than 18, hence 18 is not in S but is in T.
b(6) = 36 is not in the sequence since p*r = 3*6 = 18 and q*r = 5*6, and both do not exceed 36, therefore 36 is not in S but is in T.
b(7) = a(1) = 40 since p*r = 5*10 = 50 and q*r = 3*10 = 30. We have both 50 > 40 and 30 < 40, thus 40 is in both S and T, etc.

Links

  • Michael De Vlieger, Table of n, a(n) for n = 1..10000
  • Michael De Vlieger, Annotated plot of b(n) = A126706(n), n = 20*(y-1) + x at (x, -y), for x = 1..20 and y = 1..20, thus for n = 1..400. Terms in this sequence are colored black, those in A364999 in blue, in A364998 in gold, and in A361098 in red.
  • Michael De Vlieger, Plot of b(n), n = 120*(y-1) + x at (x, -y), for x = 1..120 and y = 1..120, thus for n = 1..14400 using the same color scheme as immediately above.
  • Michael De Vlieger, Plot of b(n), with n = 1016*(y-1) + x at (x, -y), for x = 1..1016 and y = 1..1016, thus showing 1032256 terms. Terms b(n) in this sequence are colored black, else white.

Crossrefs

Cf. A007947, A053669, A119288, A126706, A355432, A360432, A360765, A360767, A361098, A364998, A364999.

Programs

  • Mathematica
    Select[Select[Range[500], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, And[p r > k, q r < k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

Formula

Intersection of A360765 and A360767.

A364996 Union of A360767 and A363082.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 52, 56, 60, 63, 68, 76, 84, 88, 90, 92, 99, 104, 116, 117, 120, 124, 126, 132, 136, 140, 148, 150, 152, 153, 156, 164, 168, 171, 172, 175, 176, 180, 184, 188, 198, 204, 207, 208, 212, 220, 228, 232, 234, 236, 244, 248, 260, 261
Offset: 1

Views

Author

Michael De Vlieger, Aug 26 2023

Keywords

Examples

			This sequence is A126706 \ A361098.
Union of A364997, A364998, A364999.

Links

Crossrefs

Cf. A013929, A024619, A126706, A360767, A361098, A363082, A364997, A364998, A364999.

Programs

  • Mathematica
    Select[Select[Range[261], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], Function[{k, f}, Function[{p, q, r}, Or[p r > k, q r > k]] @@ {f[[2, 1]], SelectFirst[Prime@ Range[PrimePi[f[[-1, 1]]] + 1], ! Divisible[k, #] &], Times @@ f[[All, 1]]}] @@ {#, FactorInteger[#]} &]

A366460 Odd terms in A366825.

Original entry on oeis.org

45, 63, 99, 117, 153, 171, 175, 207, 261, 275, 279, 315, 325, 333, 369, 387, 423, 425, 475, 477, 495, 531, 539, 549, 575, 585, 603, 637, 639, 657, 693, 711, 725, 747, 765, 775, 801, 819, 833, 855, 873, 909, 925, 927, 931, 963, 981, 1017, 1025, 1035, 1071, 1075
Offset: 1

Views

Author

Michael De Vlieger, Jan 05 2024

Keywords

Comments

Proper subset of A364997, in turn a proper subset of A364996, which is a proper subset of A126706.
Prime signature of a(n) is 2 followed by at least one 1.
Numbers of the form A065642(k) where k is an odd term in A120944.
Numbers of the form p^2 * m, squarefree m > 1, odd prime p < lpf(m), where lpf(m) = A020639(m).
The asymptotic density of this sequence is (2/(3*Pi^2)) * Sum_{p odd prime} ((1/p^2) * (Product_{odd primes q <= p} (q/(q+1)))) = 0.0537475047... . - Amiram Eldar, Jan 08 2024

Examples

			a(1) = 45 = 9*5 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(5), i.e., 3 < 5.
a(2) = 63 = 9*7 = p^2 * m, squarefree m > 1; sqrt(9) < lpf(7), i.e., 3 < 7.
Prime powers p^k, k > 2, are not in the sequence since m = p^(k-2) is not squarefree and p = lpf(m).

Links

Crossrefs

Cf. A007947, A020639, A065642, A120944, A126706, A364996, A364997, A364999.

Programs

  • Mathematica
    Select[Select[Range[1, 1100, 2], PrimeOmega[#] > PrimeNu[#] > 1 &], And[OddQ[#1], #1/(Times @@ #2) == #2[[1]]] & @@ {#, FactorInteger[#][[All, 1]]} &]
  • PARI
    is(n) = {my(e); n%2 && e = factor(n)[, 2]; #e > 1 && e[1] == 2 && vecmax(e[2..#e]) == 1; } \\ Amiram Eldar, Jan 08 2024

Formula

{a(n)} = {A366825 \ A364999}.
Showing 1-4 of 4 results.

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