A288814 -id:A288814 - cp's OEIS frontend

A293652 a(n) is the smallest prime number whose a056240-type is n (see Comments).

5, 211, 4327, 4547, 25523, 81611, 966109, 1654111, 3851587, 1895479, 66407189, 134965049, 129312889, 425845151, 35914507, 504365461, 2400397969, 8490141637, 8429770031, 20416021309, 23555107819, 23912414437

Offset: 1

David James Sycamore, Feb 06 2018

For a prime p >= 5 whose prime-index is m, the a056240-type of p is defined to be the unique integer k such that A288814(p) = prime(m-k)*A056240(prime(m)-prime(m-k)).
In other words, k is such that prime(n-k) is the greatest prime divisor of the smallest composite number whose sum of prime factors (taken with multiplicity) is prime(n).
The sequence lists the smallest prime of each successive a056240-type.
In the Examples section, the a056240-type k (=a(k)) of a prime p = prime(m) is indicated by p ~ k(g1,g2,...,gk) where gi = prime(m - i + 1) - prime(m - i). See also A295185.
For the values of the a056240-types of the primes 2, 3, 5, 7, ... see A299912. - N. J. A. Sloane, Mar 10 2018
a(20), a(21) > 14 * 10^9. Conjecture: a(k) > 14 * 10^9 for k > 22. - David A. Corneth, Mar 25 2018
a(20), a(21) computed on the basis of the above conjecture. Note that A321983 records the smallest composite number whose sum of prime divisors (with repetition) is a(n). - David James Sycamore, Nov 30 2018
a(23)..a(25) > 45.8 * 10^9. - David A. Corneth, Dec 02 2018

			a(1) = 5 since 6 = 3 * 2, the smallest composite number whose prime divisors add to 5, is a multiple of 3, the greatest prime < 5, so k=1; 5 ~ 1(2).
a(2) = 211 since 6501 = 3 * 11 * 197, the smallest composite whose prime divisors add to 211, and 197 < 199 < 211 is the second prime below 211, so k=2, and 211 ~ 2(12,2), and since no smaller prime has this property, a(2)=211.
a(3) = 4327 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, so k=3, 4327 ~ 3(30,8,6) and since no smaller prime has this property, a(3)=4327. Likewise,
a(4) = 4547 ~ 4(24, 4, 2, 4),
a(5) = 25523 ~ 5(52, 2, 6, 6, 4),
a(6) = 81611 ~ 6(42, 6, 4, 6, 2, 4),
a(7) = 966109 ~ 7(68, 12, 16, 2, 22, 6, 14),
a(8) = 1654111 ~ 8(54, 14, 4, 6, 2, 4, 6, 2),
a(9) = 3851587 ~ 9(128, 16, 12, 2, 6, 10, 14, 10, 2),
a(10) = 1895479 ~ 10(120, 2, 6, 30, 4, 30, 14, 10, 2, 12),
a(11) = 66407189 ~ 11(120, 6, 6, 16, 14, 6, 4, 8, 10, 2, 4),
a(12) = 134965049 ~ 12(138, 10, 2, 22, 18, 20, 6, 12, 18, 16, 8, 10),
a(13) = 129312889 ~ 13(98, 60, 22, 18, 8, 4, 18, 12, 38, 24, 6, 4, 8),
a(14) = 425845151 ~ 14(148, 2, 42, 16, 50, 24, 12, 6, 4, 20, 6, 48, 10, 12),
a(15) = 35914859 ~ 15(126, 82, 8, 4, 18, 12, 8, 4, 14, 6, 16, 8, 6, 30, 10),
a(16) = 504365461 ~ 16(122, 42, 10, 14, 36, 4, 6, 6, 12, 48, 2, 6, 10, 20, 6, 6),
a(17) = 2400397969 ~ 17(122, 58, 8, 4, 18, 36, 2, 4, 6, 32, 10, 2, 16,12,18,32,12),
a(18) = 8490141637 ~ 18(126, 2, 82, 8, 52, 20, 34, 2, 10, 24, 8, 6,34,2,6,28,24,2),
a(19) = 8429770031 ~ 19(148, 26, 16, 18, 12, 2, 18, 18, 10,20,4,2,6,18,6,4,2,18,4),
a(20) = 20416021309 ~ 20(122, 4, 2, 64, 20, 40, 6, 12, 12, 20, 10, 6, 8, 10, 30, 2, 10, 38, 22, 140,
a(21) = 23555107819 ~ 21(192, 20, 156, 30, 18, 10, 2, 12, 58, 12, 12, 26, 28, 32, 4, 6, 12, 2, 6, 22, 2),
a(22) = 23912414437 ~ 22(344, 4, 12, 14, 40, 2, 4, 18, 2, 36, 10, 12, 2, 10, 26, 10, 24, 14, 40, 30, 14, 12).

David A. Corneth, PARI program

Cf. A056240, A288814, A289993, A295185, A299912, A300334, A300359, A321983.

isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]) == n;
snumbr(n) = my(k=2); while(!isok(k, n), k++); k; /* A056240 */
scompo(n) = forcomposite(k=4, , if (isok(k, n), return(k))); /* A288814 */
a(n) = {forprime(p=5,,ip = primepi(p); if (ip > n, x = scompo(p); fmax = vecmax(factor(x)[,1]); ifmax = primepi(fmax); if (ip - ifmax == n, y = fmax*snumbr(p - fmax;); if (y == x, return (p);););););} \\ Michel Marcus, Feb 17 2018

PARI
```
\\ see Corneth link
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a(7)-a(10) from Michel Marcus, Feb 23 2018
Name changed by N. J. A. Sloane, Mar 10 2018
a(11)-a(19) from David A. Corneth, Mar 24 2018, Mar 25 2018
a(20)-a(21) from David James Sycamore, Nov 30 2018
a(22) from David A. Corneth, Dec 02 2018

A038869 Primes p such that both p-2 and 2p-1 are prime.

Original entry on oeis.org

7, 19, 31, 139, 199, 229, 271, 601, 619, 661, 811, 829, 1279, 1429, 1609, 2029, 2089, 2131, 2311, 2551, 2791, 3169, 3331, 3391, 3529, 3769, 4051, 4159, 4231, 4261, 4339, 4639, 4801, 5419, 5479, 5659, 5851, 6271, 6301, 6361, 6691, 6961, 7561, 7951, 8539

Offset: 1

Thomas Kellar

nonn

Primes p such that A(2*p) - 3*A(p) = 3 (7, 31, 661, 811, 2551, ...) and primes p such that 7*A(p) - A(2*p) = 21 (19, 139, 619, 1429, ...), where A=A288814, are both subsequences of A038869. - David James Sycamore, Aug 07 2017

Harvey P. Dale, Table of n, a(n) for n = 1..10000

Cf. A005382, A118071.

[n: n in [0..10000]|IsPrime(n) and IsPrime(n-2) and IsPrime(2*n-1)]; // Vincenzo Librandi, Dec 18 2010

Transpose[Select[Partition[Prime[Range[1200]],2,1],#[[2]]-#[[1]]==2 && PrimeQ[2#[[2]]-1]&]][[2]] (* Harvey P. Dale, Jun 19 2014 *)

is(n)=n%6==1 && isprime(n-2) && isprime(n) && isprime(2*n-1) \\ Charles R Greathouse IV, Aug 09 2017

A288189 a(n) is the smallest composite number whose sum of prime divisors (with multiplicity) is divisible by prime(n).

Original entry on oeis.org

4, 8, 6, 10, 28, 22, 52, 34, 76, 184, 58, 213, 148, 82, 172, 309, 424, 118, 393, 268, 142, 584, 316, 664, 573, 388, 202, 412, 214, 436, 753, 508, 813, 274, 1465, 298, 933, 974, 652, 1336, 1384, 358, 1137, 382, 772, 394, 1257, 1329, 892, 454, 916, 1864, 478, 1497, 1538, 1569

Offset: 1

David James Sycamore, Jul 01 2017

nonn

In most cases a(n) = A288814(prime(n)) but there are exceptions, e.g., a(37)=213, whereas A288814(37)=248. Other exceptions include a(53), a(67), a(127), a(137), etc. These examples occur when there is a number r such that A001414(r*p) is less than A288814(p).

The strictly increasing subsequence of terms (10, 22, 34, 58, 82, 118, 142, 202, 214, 274, 298, ...) where for all m>n, a(m)>a(n) gives the semiprimes with prime sum of prime factors, A108605. The sequence of the indices of this subsequence (5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, ...) gives the greater of twin primes, A006512.

			a(5)=6 because 6 = 2*3 is the smallest number whose sum of prime divisors (2+3 = 5) is divisible by 5.
a(37) = 213 = A288814(74) = A288814(2*37).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000040, A001414, A006512, A056240, A108605, A288814.

With[{s = Array[Boole[CompositeQ@ #] Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #] &, 10^4] /. 0 -> ""}, Table[FirstPosition[s, ?(Mod[#, p] == 0 &)][[1]], {p, Prime@ Range@ 56}]] (* _Michael De Vlieger, Apr 14 2018 *)

sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
a(n) = my(pn=prime(n)); forcomposite(c=pn, , if (sopfr(c) % pn == 0, return(c))); \\ Michel Marcus, Jul 03 2017

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

David James Sycamore, Feb 16 2018

nonn

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.

			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.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A056240, A288814, A293652, A295185.

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

Select[Partition[Prime[Range[2500]],3,1],Differences[#]=={2,12}&][[All,3]] (* Harvey P. Dale, Feb 29 2020 *)

isok(p) =  isprime(p) && (pp=precprime(p-1)) && (p-pp == 12) && (ppp=precprime(pp-1)) && (pp-ppp == 2); \\ Michel Marcus, Feb 16 2018

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)).

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

David James Sycamore, Feb 17 2018

nonn

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

			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.

Robert Israel, Table of n, a(n) for n = 1..2000

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

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

With[{s = Partition[Prime@ Range[10^5], 4, 1]}, Select[s, Differences@ # == {6, 8, 30} &][[All, -1]]] (* Michael De Vlieger, Feb 18 2018 *)

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)).

A289556 Primes p such that both 5p - 4 and 4p - 5 are prime.

Original entry on oeis.org

3, 7, 13, 43, 67, 109, 127, 151, 163, 211, 277, 307, 373, 457, 463, 601, 613, 673, 727, 853, 919, 967, 1021, 1117, 1171, 1231, 1399, 1471, 1483, 1747, 1789, 1933, 2029, 2251, 2311, 2389, 2503, 2521, 2557, 2659, 2851, 2857, 3019, 3067, 3121, 3229, 3583, 3613, 3637, 3691, 3697

Offset: 1

David James Sycamore, Aug 02 2017

nonn

The terms of this sequence belong to two disjoint subsequences, namely those for which |A(5*p) - A(4*p)| = 9; (3,7,13,43,67,127,163,211,277,307,457,...), and those for which 5*A(4*p) - 3*A(5*p) = 3, (109,151,373,673,919,...), where A = A288814.

Note: A288814(n) = A056240(n) for all composite n.

			P=7: 5*7 - 4 = 31, 4*7 - 5 = 23, both prime so 7 is in this sequence, and belongs to the subsequence of terms satisfying A(4*p) - A(3*p) = 9.
P=109: 5*109 - 4 = 541, 4*109 - 5 = 431, both prime so 109 is in this sequence, and belongs to the subsequence of terms satisfying 5*A(4*p) - 3*A(5*p) = 3.

Cf. A259730, A288814, A290163, A290164, A056240.

Intersection of A136051 and A156300. - Michel Marcus, Aug 04 2017

Select[Prime@ Range@ 516, Times @@ Boole@ Map[PrimeQ, {5 # - 4, 4 # - 5}] > 0 &] (* Michael De Vlieger, Aug 02 2017 *)

More terms from Altug Alkan, Aug 02 2017

cp's OEIS Frontend

A293652 a(n) is the smallest prime number whose a056240-type is n (see Comments).

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A038869 Primes p such that both p-2 and 2p-1 are prime.

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A288189 a(n) is the smallest composite number whose sum of prime divisors (with multiplicity) is divisible by prime(n).

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

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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.

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A289556 Primes p such that both 5*p - 4 and 4*p - 5 are prime.

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Mathematica

Extensions

A289556 Primes p such that both 5p - 4 and 4p - 5 are prime.