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.

Previous Showing 71-77 of 77 results.

A375908 Sphenic numbers that are sandwiched between products of exactly 4 distinct primes (or tetraprimes).

Original entry on oeis.org

18446, 39766, 74306, 83434, 94106, 100346, 107966, 111154, 111814, 113366, 140834, 144754, 145606, 146014, 147406, 149854, 154946, 155702, 156146, 165346, 171786, 189034, 190618, 191806, 197354, 201686, 203314, 206194, 211394, 211946, 219386, 231286, 234394, 253114, 258266, 262294, 263966
Offset: 1

Views

Author

Massimo Kofler, Sep 02 2024

Keywords

Comments

Terms are of the form 4*k+2.

Examples

			18446 = 2 * 23 * 401  (between 18445 = 5*7*17*31 and 18447 = 3*11*13*43).
39766 = 2 * 59 * 337  (between 39765 = 3*5*11*241 and 39767 = 7*13*19*23).
74306 = 2 * 53 * 701  (between 74305 = 5*7*11*193 and 74307 = 3*17*31*47).

Crossrefs

Cf. A007304, A046386. Subsequence of A075819.

Programs

  • Maple
    N:= 5*10^5: # for terms <= N
P:= select(isprime,[seq(i,i=3..N/3,2)]): nP:= nops(P):
R:= NULL:
for i from 1 to nP while 2*P[i]*P[i+1] <= N do
  for j from i+1 to nP do
    x:= 2*P[i]*P[j];
    if x > N then break fi;
    if numtheory:-bigomega(x-1) = 4 and numtheory:-bigomega(x+1) = 4 and
      numtheory:-issqrfree(x-1) and numtheory:-issqrfree(x+1) then
        R:= R,x
    fi
od od:
sort([R]); # Robert Israel, Sep 02 2024
  • Mathematica
    e[n_] := FactorInteger[n][[;; , 2]]; SequencePosition[e /@ Range[300000], {{1, 1, 1, 1}, {1, 1, 1}, {1, 1, 1, 1}}][[;; , 1]] + 1 (* Amiram Eldar, Sep 02 2024 *)

A375964 Semiprime numbers that are sandwiched between products of exactly 4 distinct primes (or tetraprimes).

Original entry on oeis.org

41614, 51358, 55166, 67066, 97054, 97294, 100834, 101914, 109838, 122758, 126634, 130646, 133286, 148394, 154582, 175594, 180214, 180454, 184666, 197086, 212486, 224258, 226654, 227446, 231386, 242906, 258446, 276394, 284866, 285086, 300182, 305066, 308606, 309506, 317054, 344434, 350206, 352834, 360634, 362594
Offset: 1

Views

Author

Massimo Kofler, Sep 04 2024

Keywords

Examples

			41614 = 2 * 20807 (between 41613 = 3*11*13*97 and 41615 = 5*7*29*41).
51358 = 2 * 25679 (between 51357 = 3*17*19*53 and 51359 = 7*11*23*29).
55166 = 2 * 27583 (between 55165 = 5*11*17*59 and 55167 = 3*7*37*71).

Crossrefs

Cf. A006881, A046386.
Subsequence of A100484.

Programs

  • Mathematica
    e[n_] := FactorInteger[n][[;; , 2]]; SequencePosition[e /@ Range[400000], {{1, 1, 1, 1}, {1, 1}, {1, 1, 1, 1}}][[;; , 1]] + 1 (* Amiram Eldar, Sep 04 2024 *)

Formula

a(n) == 2 (mod 4).

A380348 Tetraprimes (or products of exactly four distinct prime numbers) that are the sum of two successive tetraprimes.

Original entry on oeis.org

4785, 11739, 13035, 14685, 17535, 17690, 24115, 24871, 26061, 28203, 33605, 34419, 35061, 37515, 37765, 37851, 38335, 40803, 41205, 48202, 48685, 48895, 49215, 52535, 52955, 55605, 58179, 58245, 59015, 59345, 59595, 62643, 62895, 64785, 66815, 70091, 71205, 71355, 72215
Offset: 1

Views

Author

Massimo Kofler, Jan 22 2025

Keywords

Examples

			4785 = 3*5*11*29 is a member because 4785 = 2370+2415, sum of two successive tetraprime numbers.
11739 = 3*7*13*43 is a member because 11739 = 5865+5874, sum of two successive tetraprime numbers.

Crossrefs

Cf. A001043, A046386, A092192, A380316.

Programs

  • Mathematica
    tetQ[n_] := FactorInteger[n][[;; , 2]] == {1, 1, 1, 1}; Select[MovingMap[Total, Select[Range[40000], tetQ], 1], tetQ] (* Amiram Eldar, Jan 22 2025 *)
  • PARI
    ist4(n) = omega(n)==4 && bigomega(n)==4; \\ A046386
lista(nn) = my(v=select(ist4, [1..nn])); select(ist4, vector(#v-1, k, v[k]+v[k+1])); \\ Michel Marcus, Jan 22 2025

A381919 Pentagonal numbers which are products of four distinct primes.

Original entry on oeis.org

210, 330, 2262, 3290, 4030, 4510, 4845, 5370, 6902, 7315, 8855, 10542, 13490, 15555, 15862, 16485, 18095, 18426, 19437, 21182, 23002, 24130, 28497, 29330, 30602, 31465, 36426, 44290, 46905, 49595, 50142, 54626, 60501, 67310, 67947, 72490, 77862, 79235, 83426, 84135
Offset: 1

Views

Author

Massimo Kofler, Mar 10 2025

Keywords

Examples

			A000326(12) = 210 = 12*(3*12-1)/2 = 2*3*5*7.
A000326(15) = 330 = 15*(3*15-1)/2 = 2*3*5*11.
A000326(57) = 4845 = 57*(3*57-1)/2 = 3*5*17*19.

Crossrefs

Intersection of A000326 and A046386.
Cf. A245365, A381650.

Programs

  • Maple
    N:= 10^5: # for terms <= N
P:= select(isprime,[2,seq(i,i=3..N/30,2)]): R:= {}:
nP:= nops(P):
for i1 from 3 to nP do
   p1:= P[i1];
   for i2 from 1 to i1-1 while p1 * P[i2] <= N/6 do
     p1p2:= p1*P[i2];
   for i3 from 1 to i2-1 while p1p2 * P[i3] <= N/2 do
     p1p2p3:= p1p2 * P[i3];
     m:= ListTools:-BinaryPlace(P[1..i3-1],N/p1p2p3);
     V:=select(ispent, P[1..m] *~ p1p2p3);
     if V <> [] then
        R:= R union convert(V,set);
     fi
od od od:
sort(convert(R,list)); # Robert Israel, Mar 10 2025
  • Mathematica
    Select[Table[n*(3*n-1)/2, {n, 1, 250}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Mar 10 2025 *)

A381920 Hexagonal numbers that are products of exactly four distinct primes.

Original entry on oeis.org

1326, 1770, 2145, 2415, 3003, 3486, 4186, 5565, 6670, 7626, 8385, 8646, 9730, 13695, 17205, 17578, 24531, 25878, 27730, 28203, 35245, 35778, 37401, 42486, 47278, 47895, 51681, 59685, 60378, 63190, 63903, 66795, 72010, 74305, 75855, 81406, 84666, 87153, 91378, 95703
Offset: 1

Views

Author

Massimo Kofler, Mar 10 2025

Keywords

Examples

			1326 = 2*3*13*17 is the product of 4 distinct primes and the 26th hexagonal number  hex(26) = 26(2*26-1).
1770 = 2*3*5*59 is the product of 4 distinct primes and the 30th hexagonal number  hex(30) = 30(2*30-1).
2145 = 3*5*11*13 is the product of 4 distinct primes and the 33rd hexagonal number  hex(33) = 33(2*33-1).

Crossrefs

Intersection of A000384 and A046386.
Cf. A129521, A380007.

Programs

  • Mathematica
    Select[Table[n*(2*n-1), {n, 1, 220}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Mar 10 2025 *)

A382133 Products of 4 distinct primes that are the average of two consecutive primes.

Original entry on oeis.org

462, 570, 714, 858, 870, 1190, 1230, 1254, 1290, 1302, 1482, 1590, 1722, 1785, 1806, 1995, 2046, 2130, 2170, 2210, 2470, 2490, 2870, 3030, 3045, 3255, 3390, 3410, 3705, 3774, 3795, 3885, 3930, 4002, 4218, 4242, 4278, 4422, 4510, 4515, 4641, 4785, 4935, 5010, 5110
Offset: 1

Views

Author

Massimo Kofler, Mar 17 2025

Keywords

Examples

			462 is a term because 462=2*3*7*11 is the product of four distinct primes and 462 = (461+463)/2.
714 is a term because 714=2*3*7*17 is the product of four distinct primes and 714 = (709+719)/2.
210 is not a term because although 210=2*3*5*7 is the product of four distinct primes 210 != (199 + 211)/2.

Crossrefs

Intersection of A024675 and A046386.
Cf. A078443, A130178.

Programs

  • Mathematica
    Select[Range[5200], 2*# == Plus @@ NextPrime[#, {-1, 1}] && FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Mar 17 2025 *)

A382451 Centered pentagonal numbers which are the products of four distinct primes.

Original entry on oeis.org

5406, 12426, 20026, 23766, 40641, 55131, 83266, 115026, 118266, 136306, 142206, 145806, 176226, 184281, 205206, 209526, 245706, 279726, 284766, 315951, 326706, 371526, 387106, 407031, 413106, 419226, 425391, 498406, 505126, 553426, 623751, 638826, 672106, 685131
Offset: 1

Views

Author

Massimo Kofler, Mar 26 2025

Keywords

Examples

			A005891(46) = 5406 = (5*46^2 + 5*46 + 2)/2 = 2*3*17*53.
A005891(70) = 12426 = (5*70^2 + 5*70 + 2)/2 = 2*3*19*109.
A005891(127) = 40641 = (5*127^2 + 5*127 + 2)/2 = 3*19*23*31.

Crossrefs

Intersection of A005891 and A046386.
Cf. A364610.

Programs

  • Mathematica
    Select[Table[5*n*(n+1)/2+1, {n, 0, 600}], FactorInteger[#][[;; , 2]] == {1, 1, 1, 1} &] (* Amiram Eldar, Mar 26 2025 *)
Previous Showing 71-77 of 77 results.

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