A295185 -id:A295185 - cp's OEIS frontend

A299760 Primes p with index k >= 3 such that A288189(k) = A295185(k).

5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 43, 47, 59, 61, 71, 73, 79, 83, 89, 101, 103, 107, 109, 113, 131, 139, 151, 167, 173, 179, 181, 193, 197, 199, 227, 229, 233, 239, 241, 269, 271, 281, 283, 311, 313, 317, 349, 353, 359, 379, 383, 389, 401, 421, 433, 439, 443, 449, 461, 463, 467, 491, 503, 509, 523, 569, 571, 599, 601, 607

Offset: 1

David James Sycamore, Feb 18 2018

nonn

Let A,B,X respectively, represent A288189, A295185, A056240. For prime p with index k >= 3, A(p) = X(t)(rp-t) for some multiple r >= 1 of p, and some integer t such that rp-t is prime; then sopfr(A(p)) = rp. Similarly B(p) = X(g)(p-g) where g = p-q for some prime q < p, where q = p-g is the greatest prime divisor of A295185(p); then sopfr(B(p)) = p. A(p) < B(p) if r and t exist such that (rp-t) is prime, with X(t)(rp-t) < X(g)(p-g), otherwise r = 1, t = g and A(p) = B(p). So A(p) <= B(p) and this sequence lists primes p for which this equality holds. All primes for which g = 2 or 4 are in this sequence, since then both 2(p-2), 4(p-4) are < 3(2p-3), the minimum possible value for any r > 1, t of X(t)(rp-t). Equivocal results are found for g >= 6, though in the great majority of cases (up to k=400), g > 6 ==> A(p) < B(p).

			p=29 is included because 2p-3 and 3p-2 are both composite so A(29) = 8(p-6) = 8(p-6) = 8*23 = 184 = B(29).
p=37 is not included since A(37) = 3(2p-3) = 213 whereas B(37) = X(6)(37-6) = 8*31 = 248, so A(37) < B(37). In both examples g=6.

Cf. A001414, A056240, A288189, A295185, A302720.

sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
ap288189(p) = forcomposite(c=p, , if (!(sopfr(c) % p), return(c)));
ap295185(p) = forcomposite(c=p, , if (sopfr(c) == p, return(c)));
isok(p) = isprime(p) && (ap288189(p)==ap295185(p)); \\ Michel Marcus, Apr 14 2018

A302720 Primes with index k >= 3 such that A288189(prime(k)) < A295185(prime(k)).

Original entry on oeis.org

37, 53, 67, 97, 127, 137, 149, 157, 163, 191, 211, 223, 251, 257, 263, 277, 293, 307, 331, 337, 347, 367, 373, 397, 409, 419, 431, 457, 479, 487, 499, 521, 541, 547, 557, 563, 577, 587, 593, 613, 631, 641, 653, 673, 691, 701, 709, 719, 727, 751, 757, 769, 787, 797, 809, 821, 839, 853, 877, 907, 919, 929, 937, 953, 967, 977

Offset: 1

David James Sycamore, Apr 12 2018

nonn

Let A,B,X represent A288189, A295185, A056240 respectively. A(p) is defined for every prime, B(p) is defined for primes >= 5. For a prime p with index k >= 3, A(p) = X(t)(rp-t) for some multiple r of p, and some integer t such that rp-t is prime. Then Sopfr(A(p)) = Sopfr(X(t))+(rp-t) = t+rp-t = rp. B(p) = X(g)(p-g) where g = p-q for some prime q = p-g < p. q is the greatest prime divisor of A295185(p), so Sopfr(B(p)) = p. A(p) < B(p) if r and t exist such that (rp-t) is prime, with X(t)(rp-t) < X(g)(p-g). A(p) is computed from the list of possible values in the list of inequalities: 3(2p-3) < 2(3p-2) < 5(2p-5) < 2(5p-2) < ... < X(g)(p-g), selecting the first (smallest) value of (rp-t) which is prime. If such a term exists and is < X(p)(p-g), then A(p) < B(p) and p is in this sequence. Otherwise A(p) = B(p) = X(p)(p-g) and p is in A299760.

			k=12, prime(12)=37, A288189(37) = 213 < 248 = A295185(37). 37 is the smallest prime with this property, so a(1)=37.

Cf. A288189, A295185, A299760, A299760.

sopfr(k) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]);
ap288189(p) = forcomposite(c=p, , if (!(sopfr(c) % p), return(c)));
ap295185(p) = forcomposite(c=p, , if (sopfr(c) == p, return(c)));
isokp(p) = (ap288189(p) < ap295185(p));
lista(nn) = forprime(p=5, nn, if (isokp(p), print1(p, ", "))); \\ Michel Marcus, May 13 2018

a(53) corrected by Georg Fischer, Mar 20 2022

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

Original entry on oeis.org

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
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\\ 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

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)).

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

David James Sycamore, Mar 03 2018

nonn

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.

			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.

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

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

David James Sycamore, Mar 03 2018

nonn

			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).

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

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

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)).

A300097 Primes for which A288814 gives a new record.

Original entry on oeis.org

5, 7, 11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127, 191, 211, 223, 307, 331, 479, 521, 541, 809, 877, 907, 1087, 1277, 1361, 1931, 2179, 2203, 2999, 3299, 4201, 4327, 4861, 5779, 7993, 8923, 12889, 14143, 15859, 16411, 16603, 18839, 19661, 24317, 25523, 28277

Offset: 1

Michel Marcus, Feb 24 2018

nonn

These are the primes that require the most effort when searching for the least composite c such that A001414(c) is a given prime, where A001414 is sopfr (sum of prime factors with repetition).
From David James Sycamore, Feb 25 2018: (Start)
Also primes for which A295185 gives a new record.
A006512 gives primes p requiring least effort, since then c=2*(p-2). (End)

Cf. A001414, A006512, A288814, A295185, A300098.

isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]) == n;
scompo(n) = forcomposite(k=4, ,if (isok(k, n), return(k)));
lista(nn) = {my(last = 0); forprime(p=4, nn, my(val = scompo(p)); if (val > last, print1(p, ", "); last = val););}

A300098 Records of A288814.

Original entry on oeis.org

6, 10, 28, 52, 76, 184, 248, 376, 424, 488, 584, 664, 1335, 3729, 3801, 6501, 7385, 9669, 10461, 16345, 17815, 26571, 27895, 28479, 45237, 69485, 81835, 123411, 124345, 140465, 207005, 341665, 361749, 396815, 526809, 592491, 890165, 977727, 1377485, 1992215, 2186585

Offset: 1

Michel Marcus, Feb 24 2018

nonn

Also records of A295185.

Cf. A288814, A295185, A300097.

isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]) == n;
scompo(n) = forcomposite(k=4, ,if (isok(k, n), return(k)));
lista(nn) = {my(last = 0); forprime(p=4, nn, my(val = scompo(p)); if (val > last, print1(val, ", "); last = val););}

a(n) = A288814(A300097(n)).

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

Michel Marcus, Nov 23 2018

nonn

			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).

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

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), ", "););}

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)).

cp's OEIS Frontend

A299760 Primes p with index k >= 3 such that A288189(k) = A295185(k).

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A302720 Primes with index k >= 3 such that A288189(prime(k)) < A295185(prime(k)).

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

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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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A300334 Primes of a056240-type 2.

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A300359 Primes of a056240-type 3.

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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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A300097 Primes for which A288814 gives a new record.

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