A056240 -id:A056240 - cp's OEIS frontend

A289993 Primes p such that gpf(A288814(p)) < q, where q is greatest prime < p.

211, 541, 631, 673, 1693, 1801, 2879, 3181, 3271, 3299, 3343, 3571, 3943, 4177, 4327, 4441, 4547, 4561, 4751, 4783, 4813, 4861, 5147, 5261, 5381, 5431, 5501, 5779, 6029, 6197, 6421, 6469, 6521, 6599, 6673, 6883, 6947, 7103, 7283, 7321, 7369, 7477, 7573, 7603, 7789, 7901, 7963, 7993, 8419, 8443

Offset: 1

David James Sycamore, Sep 13 2017

For prime p in this sequence, b(p) = r*b(p-r) where b(m) = A288814(m), and r = gpf(b(p)) is some prime < q. We can say that prime p_n (n > 2) is of type k if gpf(b(p_n)) = p_(n-k).

Prime gap p-q, and pattern of gaps p-r determines if p is in the sequence or not. Prime p is of type k > 2 only if p-q is one of the even indices of A056240 on which A292081 is defined (12,18,24,28,30,36,...), and if there is a prime r < q < p such that b(p-r) < b(p-q).

			p=211 is a candidate for inclusion because p-q = 211-199 = 12, and b(12)=35 is a term in A292081. Since r=197 is the next prime below q, p-r = 14 and b(14) = 33 < 35, 211 is in the sequence, of type 2.
Conversely, p=809, which also has gap p-q = 12, is not in the sequence because the only number n > 12 for which b(n) < b(12)=35 is n=14, and p-14 = 795 is not prime. Therefore b(809) = 797*b(12) = 27895, and 809 is of type 1.

David A. Corneth, Table of n, a(n) for n = 1..31763 (terms up to 5*10^6; first 544 terms from Robert Israel)
David A. Corneth, PARI program.

Cf. A000040, A056240, A288814, A288313, A292081, A001223.

N:= 10^7: # to get terms before the first prime p>3 such that A288814(p) > N
Res:= NULL:
for x from 4 to N do
  if isprime(x) then next fi;
  F:= ifactors(x)[2];
  p:= add(t[1]*t[2],t=F);
  if not isprime(p) then next fi;
  if not assigned(A288814[p]) then
    A288814[p]:= x;
    w:= max(seq(t[1],t=F));
    if w < prevprime(p) then
      Res:= Res, p
    fi
  fi
od:
pmax:= Res[-1]:
Primes:= select(isprime, [seq(i,i=5..pmax,2)]):
B:= remove(p -> assigned(A288814[p]), Primes):
sort(select(`<`,[Res], min(B))); # Robert Israel, Oct 19 2017

\\ See PARI link. - David A. Corneth, Mar 23 2018

a(30)-a(50) from Robert Israel, Oct 02 2017

Edited by Michel Marcus, Nov 15 2023

A297150 Let b(k) denote A292081(k); the sequence lists numbers b(2n) where for all m > n, b(2m) > b(2n).

Original entry on oeis.org

35, 65, 95, 115, 155, 185, 215, 235, 265, 305, 335, 365, 395, 415, 445, 485, 515, 545, 565, 635, 655, 695, 755, 785, 815, 835, 865, 905, 965, 995, 1055, 1115, 1145, 1165, 1205, 1255, 1285, 1315, 1355, 1385, 1415, 1465, 1535, 1565, 1585, 1655, 1685, 1745, 1765, 1795, 1835, 1865, 1895, 1915, 1945, 1985

Offset: 1

David James Sycamore, Dec 26 2017

nonn

This is also an ascending subsequence of the even-indexed terms of A056240(2n) (of which A292081 is a subsequence). For n >= 1, a(n) is a semiprime of the form a(n)=5*A049591(n), and the index m in A056240 of any term in this sequence belongs to the sequence of even numbers m such that m-5 is prime and m-3 is not prime (A297925). See A297925 for explanation.

			a(1)=5*A049591(1)=5*7=35. Also A056240(A297925(1))=A056240(12)=35.
a(17)=5*A049591(17)=5*103=515. Also A056240(A297925(17))=A056240(108)=515.

Vincenzo Librandi, Table of n, a(n) for n = 1..1690

Similar to A288313.

Cf. A292081, A288313, A056240, A049591, A297925.

[5*p: p in PrimesInInterval(3, 500) | not IsPrime(p + 2)]; // Vincenzo Librandi, Nov 12 2018

5 Select[Prime[Range[3, 100]], ! PrimeQ[(# + 2)] &] (* Vincenzo Librandi, Nov 12 2018 *)

a(n) = 5*A049591(n) = A056240(A297925(n)).

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)).

A330821 a(1) = 1, a(2) = 2. Thereafter a(n+1) is the smallest number k such that A001414(k) = a(n) + a(n-1).

Original entry on oeis.org

1, 2, 3, 5, 15, 51, 305, 1059, 4083, 117737, 3775459, 19465955, 952896293, 13613071346, 14565967639, 112716155924, 11073544747197, 637616871476643, 11027737075804991, 3056322734187753262, 542921033413649799807, 8189660342217563295915, 515222301162241572644117

Offset: 1

David James Sycamore, Jan 02 2020

nonn

			sopfr(3) = 3 = 2 + 1, sopfr(15) = 8 = 3 + 5, etc.

Jinyuan Wang, Table of n, a(n) for n = 1..200

Cf. A000045, A001414, A056240.

s[1] = 0; s[n_] := Plus @@ Times @@@ FactorInteger@n; a[1] = 1; a[2] = 2; a[n_] := a[n] = Module[{k = 1, sum = a[n - 1] + a[n - 2]}, While[s[k] != sum, k++]; k]; Array[a, 10] (* Amiram Eldar, Jan 02 2020 *)

a(n+1) = A056240(a(n) + a(n-1)).

Data corrected by and more terms from Amiram Eldar and David A. Corneth, Jan 02 2020

Data corrected by Jinyuan Wang, Mar 08 2020

A267000 a(n) is the smallest m such that A001414(m)=n and ((m=0) mod n) and m/n is both squarefree and prime to n, or 0 if no such m exists.

Original entry on oeis.org

2, 3, 4, 5, 0, 7, 0, 0, 30, 11, 60, 13, 70, 105, 240, 17, 0, 19, 220, 231, 0, 23, 0, 650, 286, 1755, 476, 29, 2730, 31, 1824, 627, 3570, 805, 4788, 37, 646, 897, 1160, 41, 8778, 43, 1276, 11385, 8970, 47, 1776, 36309, 10850, 1581, 41860, 53, 2322, 4070, 2408, 45885, 16530, 59

Offset: 2

Michel Marcus, Jan 08 2016

nonn

The offset is 2 like A056240 since there is no number m with A001414(m) = 1
Alladi and Erdős state that there is only a finite number of zeros in this sequence.
When a(n) is not zero, A056240(n) <= a(n); a(n) <= A000792(n).

			a(10) = 30 since A001414(30)=10 and 30 is divisible by 10, and 30/10=3 is squarefree and prime to 10.

Amiram Eldar, Table of n, a(n) for n = 2..3000
K. Alladi and P. Erdős, On an additive arithmetic function, Pacific J. Math., Volume 71, Number 2 (1977), 275-294.

Cf. A000792, A001414, A036844, A056240, A064364.

sopfr(n) = {my(f=factor(n)); sum(k=1, #f~, f[k, 1]*f[k, 2]); }
first(n) = {my(k=1); while (sopfr(k) != n, k++); k;}
last(n) = polcoeff((1+x+2*x^2+x^4)/(1-3*x^3) + O(x^(n + 3)), n);
a(n) = {na = first(n); nb = last(n); for (m=na, nb, if ((sopfr(m) == n) && (! (m % n)) && issquarefree(m/n) && (gcd(m/n, n) == 1), return(m)););}

a(p) = p, for p prime.

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

Original entry on oeis.org

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

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)).

A330988 a(1)=2, a(2)=3; then a(n+1) = smallest k such that S(k) = S(a(n)) + S(a(n-1)), (n>=2), where S is sopfr (A001414).

Original entry on oeis.org

2, 3, 5, 15, 13, 38, 93, 106, 89, 695, 233, 1492, 1821, 3932, 1597, 12895, 16708, 13526, 76573, 70828, 28657, 787967, 1125255, 4005507, 6087997, 10487301, 514229, 30784111, 68658699, 150301527, 38770237, 290846217, 525964251, 164233751, 193262488, 1368085495, 1075181473, 8903068701, 10762707995, 4554542743, 433494437

Offset: 1

David James Sycamore, Jan 05 2020

nonn

The subsequence of prime terms is A005478; a term is prime if and only if it is a Fibonacci prime (proved by Giovanni Resta).

			a(3)=5 since 5 is the smallest number whose sum of prime divisors is soprf(2) + sopfr(3) = 2 + 3 = 5. a(4)=15 since sopfr(3)+sopfr(5)=8, and 15 is the smallest number whose sum of prime divisors is 8.

Giovanni Resta, Table of n, a(n) for n = 1..51

Cf. A000045, A001414, A005478, A056240, A330821.

sopfr[n_] := Plus @@ Times @@@ FactorInteger[n]; a[1] = 2; a[2] = 3; a[n_] := a[n] = Block[{t = sopfr@ a[n-1] + sopfr@ a[n-2], k=3}, While[ sopfr[k] != t, k++]; k]; Array[a, 21] (* Giovanni Resta, Jan 07 2020 *)

a(n+1) = A056240(A001414(a(n)) + A001414(a(n-1))).

a(n) = A056240(A000045(n+2)). - Giovanni Resta, Jan 07 2020

cp's OEIS Frontend

A289993 Primes p such that gpf(A288814(p)) < q, where q is greatest prime < p.

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A297150 Let b(k) denote A292081(k); the sequence lists numbers b(2n) where for all m > n, b(2m) > b(2n).

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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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A330821 a(1) = 1, a(2) = 2. Thereafter a(n+1) is the smallest number k such that A001414(k) = a(n) + a(n-1).

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A267000 a(n) is the smallest m such that A001414(m)=n and ((m=0) mod n) and m/n is both squarefree and prime to n, or 0 if no such m exists.

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

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