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 1-6 of 6 results.

A321983 Let p be A293652(n), a(n) is the smallest composite number whose greatest prime factor is the n-th prime below p and whose prime factors add up to p.

Original entry on oeis.org

6, 6501, 526809, 419709, 5116053, 14923101, 397013259, 441623073, 2276169717, 1290664569, 38449648947, 112155723039, 122976253119, 507181098441, 25104075429, 525044080551, 2801263972359, 11894687774967, 8825968853913, 27500380094379
Offset: 1

Views

Author

Michel Marcus, Nov 23 2018

Keywords

Examples

			a(1) = 6 since 6 = 3 * 2, the smallest composite number whose prime divisors add to 5, is a multiple of 3, the greatest prime < 5, where 5 = A293652(1).
a(2) = 6501 since 6501 = 3 * 11 * 197, the smallest composite whose prime divisors add to 211, and 197 < 199 < 211 is the second prime below 211, where 211 = A293652(2)
a(3) = 526809 since 526809 = 3 * 41 * 4283, the smallest composite whose prime divisors add to 4327, and 4283 < 4289 < 4297 < 4327 is the third prime below 4327, where 4327 = A293652(3).

Crossrefs

Cf. A001414, A056240, A151799, A293652, A295185.

Programs

  • PARI
    sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]); \\ A001414
isok(k, n) = sopfr(k) == n;
a056240(n) = my(k=2); while(!isok(k, n), k++); k;
a(p, n) = {newp = p; for (k=1, n, newp = precprime(newp-1)); newp*a056240(p-newp);}
lista() = {vp = [5, 211, 4327, 4547, ...,  ]; /* A293652 */ for (n=1, #vp, print1(chk(vp[n], n), ", "););}

Formula

a(n) = q*A056240(p-q) where p = A293652(n) and q = A151799^n(p) where A151799^n is A151799(A151799(...)) repeated n times.
a(n) = A295185(A293652(n)).

A299912 a(n) = a056240-type of n-th prime, or -1 if the a056240-type is not defined.

Original entry on oeis.org

-1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

N. J. A. Sloane, Mar 10 2018

Keywords

Comments

For the definition of the a056240-type of a prime, see A293652, which also gives the smallest prime of a056240-type k for k = 1, 2, 3, ....

Links

Crossrefs

Cf. A056240, A293652.
For primes of a056240-types 2 and 3 see A300334 and A300359.

Programs

  • PARI
    \\ See Marcus link.

A299110 Prime(r) for r such that prime(r) - prime(r-1) = 12 and prime(r-1) - prime(r-2) = 2.

Original entry on oeis.org

211, 631, 673, 1801, 3181, 3271, 3343, 3571, 3943, 4561, 4813, 5431, 6673, 6883, 7321, 7573, 7603, 7963, 8443, 8641, 9643, 9733, 9781, 9871, 10513, 10723, 10903, 11083, 11131, 11731, 11953, 12391, 13411, 14401, 14461, 15373, 15661, 15901, 16843, 17203, 17431, 17761, 17851, 17971, 18301, 18553, 20161, 20521, 20563, 20731
Offset: 1

Views

Author

David James Sycamore, Feb 16 2018

Keywords

Comments

These are the primes of a056240-type 2(12,2); k=2 (see definition in A293652). prime(r-2) is the greatest prime factor of the smallest composite number whose prime divisors (with multiplicity) sum to prime(r).
Conjecture: Sequence has infinitely many terms. Note: p~2(12,2) is just one particular form of a prime of A056240-type k=2; there are others, e.g., 2(18,2), 2(18,4), 2(28,12), 2(24,10). All such prime sequences are also conjectured to produce infinitely many terms.

Examples

			a(1)=211=prime(47), the first prime of type k=2. prime(46)=199 and prime(45)=197; 211-199=12 and 199-197=2.

Links

Crossrefs

Cf. A056240, A288814, A293652, A295185.

Programs

  • Maple
    N:=21000:
for X from 2 to N do
if isprime(X) then
A:=prevprime(X);
B:=prevprime(A);
a:=X-A;
b:=A-B;
if a=12 and b=2 then print(X);
end if
end if
end if
end do
# alternative:
P:= select(isprime, {seq(i,i=3..10^6,2)}):
Q:= P intersect map(t -> t-12, P) intersect map(t -> t+2, P):
Q:= remove(t -> ormap(isprime, [seq(t+i,i=2..10,2)]), Q):
map(t -> t+12, Q); # Robert Israel, Feb 16 2018
  • Mathematica
    Select[Partition[Prime[Range[2500]],3,1],Differences[#]=={2,12}&][[All,3]] (* Harvey P. Dale, Feb 29 2020 *)
  • PARI
    isok(p) =  isprime(p) && (pp=precprime(p-1)) && (p-pp == 12) && (ppp=precprime(pp-1)) && (pp-ppp == 2); \\ Michel Marcus, Feb 16 2018

Formula

For every prime(r) in this sequence A288814(prime(r)) = prime(r-2)*A056240(prime(r) - prime(r-2)) = prime(r-2)*A288814(prime(r) - prime(r-2)).

A300334 Primes of a056240-type 2.

Original entry on oeis.org

211, 541, 631, 673, 1693, 1801, 2879, 3181, 3271, 3299, 3343, 3571, 3943, 4177, 4441, 4561, 4751, 4783, 4813, 4861, 5147, 5381, 5431, 5501, 5779, 6029, 6197, 6421, 6469, 6521, 6599, 6637, 6883, 7103, 7321, 7369, 7477, 7573, 7603, 7789, 7901, 7963, 8419, 8443, 8641, 8923, 9091, 9587, 9643, 9733, 9781, 9871, 10513
Offset: 1

Views

Author

David James Sycamore, Mar 03 2018

Keywords

Comments

Prime(r) has a056240-type k if A295185(prime(r))=prime(r-k)*A056240(prime(r)-prime(r-k)).
This sequence lists primes having a056240-type k=2, each having form ~2(g1,g2) where g1 is the first gap below prime(r), and g2 is the second (notation explained in A295185). The majority of primes appear to be of a056240-type 1.

Examples

			211 is included because the smallest composite number whose sum of prime factors (with repetition)=211 is 6501=197*33, a multiple of the second prime below 211, not the first. 211~2(12,2) is the smallest prime to have this property. Likewise 541~2(18,2), 1693~2(24,2), 2879~2(18,4), etc.

Crossrefs

Cf. A056240, A295185, A299110, A293652, A299912, A300359.

Extensions

Edited by N. J. A. Sloane, Mar 10 2018

A300359 Primes of a056240-type 3.

Original entry on oeis.org

4327, 6947, 7283, 7993, 11863, 12143, 14591, 15859, 18583, 18617, 18839, 18947, 19661, 20593, 21059, 21313, 22853, 29567, 29917, 31307, 32531, 32831, 35023, 35491, 35671, 39499, 39659, 40343, 41579, 42821, 43853, 46807, 48221, 49253, 49363, 49597, 50411, 51109, 52667, 54401
Offset: 1

Views

Author

David James Sycamore, Mar 03 2018

Keywords

Examples

			4327 is included because the smallest composite number whose sum of prime factors (with repetition)=4327 is 526809=4283*123, a multiple of the third prime below 4327 (not the first or second). 4327~3(30,8,6) is the smallest prime to have this property. Likewise 6947~3(30,6,4), and 7993~3(30,12,2).

Crossrefs

Cf. A056240, A295185, A299704, A293652, A299912, A300334.

Extensions

Edited by N. J. A. Sloane, Mar 10 2018

A299704 List of primes prime(r) such that prime(r)-prime(r-1)=30, prime(r-1)-prime(r-2)=8 and prime(r-2)-prime(r-3)=6.

Original entry on oeis.org

4327, 91621, 111697, 123001, 190027, 240997, 243517, 244291, 300277, 309667, 315937, 317827, 362137, 393517, 440131, 457087, 467587, 517861, 554167, 567097, 590071, 609571, 617917, 640771, 651727, 653311, 719101, 776551, 788071, 793591, 804157, 809491, 812431, 850177, 861391, 1007857, 1070287
Offset: 1

Views

Author

David James Sycamore, Feb 17 2018

Keywords

Comments

These are the primes of a056240-type 3(30,8,6); k=3 (see definition in A293652).
A prime of a056240-type 3 is a prime, prime(r)>3, such that prime(r-3) is the greatest prime factor of the smallest composite number whose prime divisors (with multiplicity) sum to prime(r).
Conjecture: Sequence has infinitely many terms.
Note: p~3(30,8,6) is one particular form of a prime of a056240-type 3; there are others, e.g., 3(30,12,2), 3(24,6,2), 3(36,6,4), 3(38,10,2), etc. All such prime sequences are also conjectured to produce infinitely many terms.
All terms == 1 (mod 3). - Robert Israel, May 13 2020

Examples

			a(1)=4327=prime(591), the first prime of a056240-type 3. Prime(590)=4297, prime(589)=4289, prime(588)=4283. 4327-4297=30, 4297-4289=8, 4289-4283=6.

Links

Crossrefs

Cf. A056240, A288814, A293652, A295185, A299110.

Programs

  • Maple
    N:=2000000:
for X from 100 to N do
if isprime(X) then
A:=prevprime(X);
B:=prevprime(A);
C:=prevprime(B);
a:=X-A;
b:=A-B;
c:=B-C;
if a=30 and b=8 and c=6 then print(X);
end if
end if
end if
end do
  • Mathematica
    With[{s = Partition[Prime@ Range[10^5], 4, 1]}, Select[s, Differences@ # == {6, 8, 30} &][[All, -1]]] (* Michael De Vlieger, Feb 18 2018 *)

Formula

For every prime(r) in this sequence A288814(prime(r)) = prime(r-3)*A056240(prime(r) - prime(r-3)) = prime(r-3)*A288814(prime(r) - prime(r-3)).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.