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.

A264885 Numbers in A007504 such that omega(a(n)) = Omega(a(n)) = 3.

Original entry on oeis.org

238, 874, 2914, 3266, 3638, 4438, 5117, 6601, 7982, 8582, 9854, 10191, 10538, 10887, 11966, 13101, 17283, 19113, 23069, 38238, 40313, 41741, 46191, 53342, 54998, 56690, 68341, 74139, 80189, 84341, 88585, 90763, 95165, 98534, 100838
Offset: 1

Views

Author

Debapriyay Mukhopadhyay, Nov 27 2015

Keywords

Comments

The corresponding numbers of prime summands, k(n), are 13, 23, 39, 41, 43, 47, 50, 56, 61, 63, 67, 68, 69, 70, 73, 76, 86, 90, 98, 123, 126, 128, 134, 143, 145, 147, 160, 166, 172, 176, 180, 182, 186, 189, 191, 196, 197, 200, 215, 220, 222, 225, 229, 238, 241, 251, 252, 265, 266, 267, ....
Intersection of A007504 and A007304 (sphenic numbers). - Michel Marcus, Dec 15 2015

Examples

			For n = 1, k(n) = 13 and a(n) = A007504(13) = 238 = 2*7*17.
For n = 2, k(n) = 23 and a(n) = A007504(23) = 874 = 2*19*23.
For n = 3, k(n) = 39 and a(n) = A007504(39) = 2914 = 2*31*47.
For n = 4, k(n) = 41 and a(n) = A007504(41) = 3266 = 2*23*71.
For n = 5, k(n) = 43 and a(n) = A007504(43) = 3638 = 2*17*107.
For n = 6, k(n) = 47 and a(n) = A007504(47) = 4438 = 2*7*317.
Note that for each of the elements of the sequence, omega(a(n)) = Omega(a(n)) = 3, i.e., the number of prime factors of a(n) = the number of distinct prime factors of a(n) = 3.

Links

Crossrefs

Cf. A001221, A001222, A007304, A007504, A013918, A189072, A264887.

Programs

  • Maple
    N:= 10^4: # to use primes up to N
select(t -> numtheory:-bigomega(t)=3 and numtheory:-issqrfree(t),
ListTools:-PartialSums(select(isprime,[2,seq(i,i=3..N,2)]))); # Robert Israel, Dec 15 2015
  • Mathematica
    t = Accumulate@ Prime@ Range@ 300; Select[Range[2*10^5], And[MemberQ[t, #], PrimeNu@ # == PrimeOmega@ # == 3] &] (* Michael De Vlieger, Nov 27 2015, after Zak Seidov at A007504 *)
  • PARI
    lista(nn) = {my(s = 0); for (n=1, nn, s += prime(n); if ((omega(s) == 3) && (bigomega(s)==3), print1(s, ", ")););} \\ Michel Marcus, Nov 28 2015

A264887 Numbers in A007504 such that omega(a(n)) = Omega(a(n)) = 4.

Original entry on oeis.org

5830, 6870, 13490, 16401, 58406, 60146, 61910, 65534, 75130, 136114, 148827, 153178, 213538, 257358, 269074, 273054, 327198, 354102, 377310, 382038, 403611, 443685, 475323, 488774, 496905, 665130, 684510, 691026, 799846, 817563
Offset: 1

Views

Author

Debapriyay Mukhopadhyay, Nov 27 2015

Keywords

Comments

Omega and omega are given in A001221 and A001222, respectively.
The corresponding numbers of prime summands, k(n), are 53, 57, 77, 84, 149, 151, 153, 157, 167, 219, 228, 231, 269, 293, 299, 301, 327, 339, 349, 351, 360, 376, 388, 393, 396, 453, 459, 461, 493, 498, ...
Intersection of A007504 and A046386 (products of four distinct primes). - Michel Marcus, Dec 15 2015

Examples

			For n = 1, k(n) = 53 and a(n) = A007504(53) = 5830 = 2*5*11*53.
For n = 2, k(n) = 57 and a(n) = A007504(57) = 6870 = 2*3*5*229.
For n = 3, k(n) = 77 and a(n) = A007504(77) = 13490 = 2*5*19*71.
For n = 4, k(n) = 84 and a(n) = A007504(84) = 16401 = 3*7*11*71.
For n = 5, k(n) = 149 and a(n) = A007504(149) = 58406 = 2*19*29*53.
For n = 6, k(n) = 151 and a(n) = A007504(151) = 60146 = 2*17*29*61.
Note that for each of the elements of the sequence, omega(a(n)) = Omega(a(n)) = 4, i.e., the number of prime factors of a(n) = the number of distinct prime factors of a(n) = 4.

Links

Crossrefs

Cf. A001221, A001222, A007504, A013918, A046386, A189072, A264885.

Programs

  • Mathematica
    t = Accumulate@ Prime@ Range@ 600; Select[t, PrimeNu@ # == PrimeOmega@ # == 4 &] (* Michael De Vlieger, Nov 27 2015, after Zak Seidov at A007504 *)
  • PARI
    lista(nn) = {my(s = 0); for (n=1, nn, s += prime(n); if ((omega(s) == 4) && (bigomega(s)==4), print1(s, ", ")););} \\ Michel Marcus, Nov 28 2015

A249679 Terms of A007504 divisible by 3.

Original entry on oeis.org

0, 129, 381, 501, 639, 963, 1161, 1371, 1593, 1851, 2127, 2427, 3087, 3447, 3831, 4227, 5589, 6081, 6870, 10191, 10887, 12339, 13101, 13887, 14697, 15537, 16401, 17283, 18189, 19113, 22548, 23592, 25800, 26940, 28104, 30504, 31734, 35568, 36888, 38238, 39612, 41022, 42468, 46191
Offset: 1

Views

Author

Zak Seidov, Nov 03 2014

Keywords

Comments

Conjecture: a(n) ~ 4.5 n^2 log n. - Charles R Greathouse IV, Nov 03 2014

Examples

			a(2) = 129 = A007504(10), a(3) = 381 = A007504(16).

Links

Crossrefs

Intersection of A007504 and A008585. Cf. A007504, A008585, A024011, A045345, A128165, A189072.

Programs

A274182 Semiprimes that are the sum of the first n odd primes for some n.

Original entry on oeis.org

15, 26, 39, 158, 326, 566, 789, 961, 1159, 1262, 1369, 1478, 1591, 1718, 1849, 2582, 3085, 3829, 4659, 5587, 7697, 8891, 10189, 13885, 14695, 16838, 17281, 18187, 19111, 20057, 22546, 24131, 25798, 26938, 27515, 28102, 35566, 36886, 38919, 41739, 43199, 50885
Offset: 1

Views

Author

K. D. Bajpai, Jun 12 2016

Keywords

Comments

The set of semiprimes in A071148.
Intersection of A001358 and A071148.

Examples

			26 appears in the sequence because 26 = 2*13 that is semiprime. Also, 3+5+7+11 = 26.
158 appears in the sequence because 158 = 2*79 that is semiprime. Also, 3+5+7+11+13+17+19+23+29+31 = 158.

Links

Crossrefs

Cf. A001358, A071148, A189072, A274120.

Programs

  • Maple
    P:= select(isprime,[seq(i,i=3..10^4,2)]):
select(t -> numtheory:-bigomega(t)=2, ListTools:-PartialSums(P)); # Robert Israel, Sep 23 2019
  • Mathematica
    Select[a = 0; Table[a = a + Prime[k], {k, 2, 300}], PrimeOmega[#] == 2 &]
  • PARI
    s = 0; forprime(p=3, 1e4, s += p; if (bigomega(s)==2, print1(s, ", ")))
Showing 1-4 of 4 results.

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