A046317 -id:A046317 - cp's OEIS frontend

A360110 Nonmultiples of 4 whose arithmetic derivative is a multiple of 4.

1, 15, 35, 39, 51, 55, 81, 87, 91, 95, 111, 115, 119, 123, 143, 155, 159, 183, 187, 189, 203, 215, 219, 225, 235, 247, 259, 267, 287, 291, 295, 297, 299, 303, 319, 323, 327, 335, 339, 355, 371, 391, 395, 403, 407, 411, 415, 427, 441, 447, 451, 471, 511, 513, 515, 519, 525, 527, 535, 543, 551, 559, 579

Offset: 1

Antti Karttunen, Jan 31 2023

nonn

A multiplicative semigroup; if m and n are in the sequence then so is m*n.
Terms > 1 do not form a subsequence of A327934: Here 189 = 3^3 * 7 is present, although it is missing from A327934.
This is a subsequence of A046337, numbers with an even number of odd prime factors (with multiplicity). The semiprimes that occur here are all of the type (4m-1)*(4n+1), i.e., in A080774. A product of four odd primes (A046317) occurs here if either all of the primes have the same remainder modulo 4 (i.e., either all are of the type 4m-1 or all are of the type 4m+1), or two are of the other type, and two are of the other type. This follows because A003415(p*q*r*s) = (pqr + pqs + prs + qrs), while the product of four odd primes with just one prime of the different type are all located in A327862. - Antti Karttunen, Feb 05 2024

			189 = 3^3 * 7 has arithmetic derivative 189' = A003415(189) = 216 = 2^3 * 3^3. Because 189 is not a multiple of 4, but 216 is, 189 is included in this sequence.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Intersection of A327864 with A042968, or equally, with A046337.
Setwise difference A046337 \ A327862.
After 1, a subsequence of A327929 and of A099309.
Subsequence of A235992, but not of A359829.
Cf. A003415, A046317, A327934, A360109 (characteristic function).
Cf. A080774 (subsequence).

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[600], ! Divisible[#, 4] && Divisible[d[#], 4] &] (* Amiram Eldar, Jan 31 2023 *)

PARI
```
isA360110(n) = A360109(n);
```

A046390 Squarefree odd numbers with exactly 4 distinct prime factors.

Original entry on oeis.org

1155, 1365, 1785, 1995, 2145, 2415, 2805, 3003, 3045, 3135, 3255, 3315, 3705, 3795, 3885, 3927, 4305, 4389, 4485, 4515, 4641, 4785, 4845, 4935, 5005, 5115, 5187, 5313, 5565, 5655, 5865, 6045, 6105, 6195, 6279, 6405, 6545, 6555, 6699, 6765, 6783, 7035

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A046317, A046406.

f[n_]:=Last/@FactorInteger[n]=={1,1,1,1}&&FactorInteger[n][[1,1]]>2; lst={};Do[If[f[n],AppendTo[lst,n]],{n,8!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
Select[Range[1,8001,2],PrimeNu[#]==PrimeOmega[#]==4&] (* Harvey P. Dale, Mar 28 2013 *)

list(lim)=my(v=List()); forprime(p=3,sqrtnint(lim\=1,4), forprime(q=p+2,sqrtnint(lim\p,3), forprime(r=q+2,sqrtint(lim\p\q), my(t=p*q*r); forprime(s=r+2,lim\t, listput(v,t*s))))); Set(v) \\ Charles R Greathouse IV, Dec 06 2024

A046330 Palindromes with exactly 4 prime factors (counted with multiplicity).

Original entry on oeis.org

88, 232, 414, 424, 444, 484, 525, 585, 636, 666, 676, 686, 808, 858, 868, 999, 1881, 2002, 2332, 2442, 2662, 3003, 3663, 3773, 3993, 4114, 4444, 4774, 5005, 5115, 5225, 6116, 6556, 6666, 7007, 7227, 8668, 9999, 10101, 10701, 11011, 12321, 13431

Offset: 1

Patrick De Geest, Jun 15 1998

John Cerkan, Table of n, a(n) for n = 1..10000

Cf. A014613, A046317, A046410.

Select[Range[14000],PalindromeQ[#]&&PrimeOmega[#]==4&] (* Harvey P. Dale, Jul 07 2024 *)

A046374 Odd numbers with exactly 4 palindromic prime factors (counted with multiplicity).

Original entry on oeis.org

81, 135, 189, 225, 297, 315, 375, 441, 495, 525, 625, 693, 735, 825, 875, 1029, 1089, 1155, 1225, 1375, 1617, 1715, 1815, 1925, 2401, 2541, 2695, 2727, 3025, 3537, 3773, 3993, 4077, 4235, 4545, 4887, 5157, 5895, 5929, 6363, 6655, 6795, 7575, 8145, 8253

Offset: 1

Patrick De Geest, Jun 15 1998

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Intersection of A033620 and A046317.

Cf. A046406.

Offset changed by Andrew Howroyd, Aug 14 2024

A358778 Positions of positive terms in A358777, which is the Dirichlet inverse of A353557.

Original entry on oeis.org

1, 135, 189, 225, 297, 315, 351, 375, 441, 459, 495, 513, 525, 585, 621, 693, 735, 765, 783, 819, 825, 837, 855, 875, 975, 999, 1029, 1035, 1071, 1089, 1107, 1155, 1161, 1197, 1225, 1269, 1275, 1287, 1305, 1365, 1375, 1395, 1425, 1431, 1449, 1521, 1593, 1617, 1625, 1647, 1665, 1683, 1715, 1725, 1785

Offset: 1

Antti Karttunen, Dec 23 2022

nonn

Question: Is A046317 \ (A030514\{16}) a subsequence of this sequence?

30375 = 3^5 * 5^3 is the first term with more than 4 prime factors (when counted with multiplicity).

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A046317, A353557, A358777.

PARI
```
isA358778(n) = (A358777(n)>0);
```

cp's OEIS Frontend

A360110 Nonmultiples of 4 whose arithmetic derivative is a multiple of 4.

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A046390 Squarefree odd numbers with exactly 4 distinct prime factors.

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A046330 Palindromes with exactly 4 prime factors (counted with multiplicity).

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A046374 Odd numbers with exactly 4 palindromic prime factors (counted with multiplicity).

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A358778 Positions of positive terms in A358777, which is the Dirichlet inverse of A353557.

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