A288814 -id:A288814 - 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

A290163 Primes p such that A288814(4p) - A288814(3p) = 7.

Original entry on oeis.org

2, 19, 29, 59, 79, 89, 131, 149, 151, 389, 479, 499, 521, 571, 631, 659, 701, 739, 919, 941, 971, 1069, 1279, 1289, 1361, 1381, 1451, 1471, 1489, 1669, 1949, 2069, 2089, 2131, 2549, 2609, 2749, 2791, 3011, 3109, 3181, 3251, 3361, 3389, 3539, 3581, 3659, 4049, 4091, 4139

Offset: 1

David James Sycamore, Jul 22 2017

nonn

Proper subset of A290164.

Terms of A290164 not in this sequence include 5, 11, 61, 191, 431, 541, 1181, 3571, ... corresponding to primes p such that A(4*p) - A(3*p) = A(3*p) - 1, where A=A288814. Examples: A(4*5) - A(3*5) = 51 - 26 = 25; A(4*541) - A(3*541) = 6483 - 3242 = 3241.

			A288814(4*2) - A288814(3*2) = 15 - 8 = 7, therefore prime 2 is in the sequence;
A288814(4*19) - A288814(3*19) = 219 - 212 = 7, therefore prime 19 is a term.

Cf. A259730, A288814, A290164.

With[{s = Array[Boole[CompositeQ@ #] Total@ Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger[#]] &, 10^5]}, Select[Prime@ Range[600], Function[p, FirstPosition[s, ?(# == 4 p &)][[1]] - FirstPosition[s, ?(# == 3 p &)][[1]] == 7]]] (* Michael De Vlieger, Jul 23 2017 *)

More terms from Altug Alkan, Jul 23 2017

Edited by Robert Israel, Jul 24 2017

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)).

A301592 Numbers k at which the ratio A288814(k) / k reaches a record high.

Original entry on oeis.org

4, 5, 6, 7, 8, 10, 11, 12, 17, 18, 24, 28, 29, 37, 53, 59, 67, 79, 89, 95, 97, 121, 123, 125, 223, 305, 329, 479, 521, 539, 541, 905, 1087, 1147, 1277, 1345, 1351, 1355, 1357, 5779, 8923, 10003, 11773, 12883, 19371, 19651, 19657, 28277, 31445

Offset: 1

David A. Corneth, Mar 24 2018

nonn

			A288814(10) / 10 = 21 / 10 and for n < 10 all ratios are less than 21 / 10, so 10 is in the sequence.

Cf. A288814.

Block[{s = Array[If[PrimeQ@ #, 0, Total@ Flatten[ConstantArray[#1, #2] & @@@ FactorInteger@ #]] &, 10^5, 2], t}, t = Table[(1 + FirstPosition[s, k][[1]])/k, {k, 4, LengthWhile[Differences@ Rest@ Union@ s, # == 1 &]}]; Map[3 + FirstPosition[t, #][[1]] &, Union@ FoldList[Max, t]]] (* Michael De Vlieger, Mar 26 2018 *)

isok(k, n) = my(f=factor(k)); sum(j=1, #f~, f[j, 1]*f[j, 2]) == n;
f(n) = forcomposite(k=1, , if (isok(k, n), return(k))); /* A288814 */
lista(nn) = {maxr = 0; for (n=4, nn, if ((newr=f(n)/n) > maxr, print1(n, ", "); maxr = newr););} \\ Michel Marcus, Mar 26 2018

A000792 a(n) = max{(n - i)*a(i) : i < n}; a(0) = 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 243, 324, 486, 729, 972, 1458, 2187, 2916, 4374, 6561, 8748, 13122, 19683, 26244, 39366, 59049, 78732, 118098, 177147, 236196, 354294, 531441, 708588, 1062882, 1594323, 2125764, 3188646, 4782969, 6377292

Offset: 0

N. J. A. Sloane

Numbers of the form 3^k, 2*3^k, 4*3^k with a(0) = 1 prepended.
If a set of positive numbers has sum n, this is the largest value of their product.
In other words, maximum of products of partitions of n: maximal value of Product k_i for any way of writing n = Sum k_i. To find the answer, take as many of the k_i's as possible to be 3 and then use one or two 2's (see formula lines below).
a(n) is also the maximal size of an Abelian subgroup of the symmetric group S_n. For example, when n = 6, one of the Abelian subgroups with maximal size is the subgroup generated by (123) and (456), which has order 9. [Bercov and Moser] - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 19 2001
Also the maximum number of maximal cliques possible in a graph with n vertices (cf. Capobianco and Molluzzo). - Felix Goldberg (felixg(AT)tx.technion.ac.il), Jul 15 2001 [Corrected by Jim Nastos and Tanya Khovanova, Mar 11 2009]
Every triple of alternate terms {3*k, 3*k+2, 3*k+4} in the sequence forms a geometric progression with first term 3^k and common ratio 2. - Lekraj Beedassy, Mar 28 2002
For n > 4, a(n) is the least multiple m of 3 not divisible by 8 for which omega(m) <= 2 and sopfr(m) = n. - Lekraj Beedassy, Apr 24 2003
Maximal number of divisors that are possible among numbers m such that A080256(m) = n. - Lekraj Beedassy, Oct 13 2003
Or, numbers of the form 2^p*3^q with p <= 2, q >= 0 and 2p + 3q = n. Largest number obtained using only the operations +,* and () on the parts 1 and 2 of any partition of n into these two summands where the former exceeds the latter. - Lekraj Beedassy, Jan 07 2005
a(n) is the largest number of complexity n in the sense of A005520 (A005245). - David W. Wilson, Oct 03 2005
a(n) corresponds also to the ultimate occurrence of n in A001414 and thus stands for the highest number m such that sopfr(m) = n, for n >= 2. - Lekraj Beedassy, Apr 29 2002
a(n) for n >= 1 is a paradigm shift sequence with procedural length p = 0, in the sense of A193455. - Jonathan T. Rowell, Jul 26 2011
a(n) = largest term of n-th row in A212721. - Reinhard Zumkeller, Jun 14 2012
For n >= 2, a(n) is the largest number whose prime divisors (with multiplicity) add to n, whereas the smallest such number (resp. smallest composite number) is A056240(n) (resp. A288814(n)). - David James Sycamore, Nov 23 2017
For n >= 3, a(n+1) = a(n)*(1 + 1/s), where s is the smallest prime factor of a(n). - David James Sycamore, Apr 10 2018

			a{8} = 18 because we have 18 = (8-5)*a(5) = 3*6 and one can verify that this is the maximum.
a(5) = 6: the 7 partitions of 5 are (5), (4, 1), (3, 2), (3, 1, 1), (2, 2, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1) and the corresponding products are 5, 4, 6, 3, 4, 2 and 1; 6 is the largest.
G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 18*x^8 + ...

B. R. Barwell, Cutting String and Arranging Counters, J. Rec. Math., 4 (1971), 164-168.
B. R. Barwell, Journal of Recreational Mathematics, "Maximum Product": Solution to Prob. 2004;25(4) 1993, Baywood, NY.
M. Capobianco and J. C. Molluzzo, Examples and Counterexamples in Graph Theory, p. 207. North-Holland: 1978.
S. L. Greitzer, International Mathematical Olympiads 1959-1977, Prob. 1976/4 pp. 18;182-3 NML vol. 27 MAA 1978
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 396.
P. R. Halmos, Problems for Mathematicians Young and Old, Math. Assoc. Amer., 1991, pp. 30-31 and 188.
L. C. Larson, Problem-Solving Through Problems. Problem 1.1.4 pp. 7. Springer-Verlag 1983.
D. J. Newman, A Problem Seminar. Problem 15 pp. 5;15. Springer-Verlag 1982.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..500
R. Bercov and L. Moser, On Abelian permutation groups, Canad. Math. Bull. 8 (1965) 627-630.
Walter Bridges and William Craig, On the distribution of the norm of partitions, arXiv:2308.00123 [math.CO], 2023.
J. Arias de Reyna and J. van de Lune, The question "How many 1's are needed?" revisited, arXiv preprint arXiv:1404.1850 [math.NT], 2014. See M_n.
J. Arias de Reyna and J. van de Lune, Algorithms for determining integer complexity, arXiv preprint arXiv:1404.2183 [math.NT], 2014.
Nigel Derby, 96.21 The MaxProduct partition, The Mathematical Gazette 96:535 (2012), pp. 148-151.
Tomislav Doslic, Maximum Product Over Partitions Into Distinct Parts, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.8.
Hans Havermann, Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums).
J. Iraids, K. Balodis, J. Cernenoks, M. Opmanis, R. Opmanis and K. Podnieks, Integer Complexity: Experimental and Analytical Results, arXiv preprint arXiv:1203.6462 [math.NT], 2012.
Andrew Kenney and Caroline Shapcott, Maximum Part-Products of Odd Palindromic Compositions, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.6.
E. F. Krause, Maximizing The Product of Summands, Mathematics Magazine, MAA Oct 1996, Vol. 69, no. 5 pp. 270-271.
MathPro, 20000 Problems Under the Sea, Problem 14856.Putnam 1979/A1.
J. W. Moon and L. Moser, On cliques in graphs, Israel J. Math. 3 (1965), 23-28.
Natasha Morrison and Alex Scott, Maximizing the number of induced cycles in a graph, Preprint, 2016. See f_2(n).
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
F. Pluvinage, Developing problem solving experiences in practical action projects, The Mathematics Enthusiast, ISSN 1551-3440, Vol. 10, nos. 1 & 2, pp. 219-244.
D. A. Rawsthorne, How many 1's are needed?, Fib. Quart. 27 (1989), 14-17.
J. T. Rowell, Solution Sequences for the Keyboard Problem and its Generalizations, Journal of Integer Sequences, 18 (2015), #15.10.7.
Robert Schneider and Andrew V. Sills, The Product of Parts or "Norm" of a Partition, Integers (2020) Vol. 20A, Article #A13.
J. Scholes, 40th Putnam 1979 Problem A1.
J. Scholes, 18th IMO 1976 Problem 4.
Andrew V. Sills and Robert Schneider, The product of parts or "norm" of a partition, arXiv:1904.08004 [math.NT], 2019.
V. Vatter, Maximal independent sets and separating covers, Amer. Math. Monthly, 118 (2011), 418-423.
Robert G. Wilson v, Letter to N. J. Sloane, circa 1991.
A. C.-C. Yao, On a problem of Katona on minimal separating systems, Discrete Math., 15 (1976), 193-199.
Index to sequences related to the complexity of n
Index entries for linear recurrences with constant coefficients, signature (0,0,3).

See A007600 for a left inverse.
Cf. A000793, A009490, A034891, A062943, A007601, A062723, A069188, A087902.
Cf. array A064364, rightmost (nonvanishing) numbers in row n >= 2.
See A056240 and A288814 for the minimal numbers whose prime factors sums up to n.
A000792, A178715, A193286, A193455, A193456, and A193457 are closely related as paradigm shift sequences for (p = 0, ..., 5 respectively).
Cf. A202337 (subsequence).
Cf. A038754, A005245, A005520.

a000792 n = a000792_list !! n
a000792_list = 1 : f [1] where
   f xs = y : f (y:xs) where y = maximum $ zipWith (*) [1..] xs
-- Reinhard Zumkeller, Dec 17 2011

I:=[1,1,2,3,4]; [n le 5 select I[n] else 3*Self(n-3): n in [1..45]]; // Vincenzo Librandi, Apr 14 2015

A000792 := proc(n)
    m := floor(n/3) ;
    if n mod 3 = 0 then
        3^m ;
    elif n mod 3 = 1 then
        4*3^(m-1) ;
    else
        2*3^m ;
    end if;
    floor(%) ;
end proc: # R. J. Mathar, May 26 2013

a[1] = 1; a[n_] := 4* 3^(1/3 *(n - 1) - 1) /; (Mod[n, 3] == 1 && n > 1); a[n_] := 2*3^(1/3*(n - 2)) /; Mod[n, 3] == 2; a[n_] := 3^(n/3) /; Mod[n, 3] == 0; Table[a[n], {n, 0, 40}]
CoefficientList[Series[(1 + x + 2x^2 + x^4)/(1 - 3x^3), {x, 0, 50}], x] (* Harvey P. Dale, May 01 2011 *)
f[n_] := Max[ Times @@@ IntegerPartitions[n, All, Prime@ Range@ PrimePi@ n]]; f[1] = 1; Array[f, 43, 0] (* Robert G. Wilson v, Jul 31 2012 *)
a[ n_] := If[ n < 2, Boole[ n > -1], 2^Mod[-n, 3] 3^(Quotient[ n - 1, 3] + Mod[n - 1, 3] - 1)]; (* Michael Somos, Jan 23 2014 *)
Join[{1, 1}, LinearRecurrence[{0, 0, 3}, {2, 3, 4}, 50]] (* Jean-François Alcover, Jan 08 2019 *)
Join[{1,1},NestList[#+Divisors[#][[-2]]&,2,41]] (* James C. McMahon, Aug 09 2024 *)

{a(n) = floor( 3^(n - 4 - (n - 4) \ 3 * 2) * 2^( -n%3))}; /* Michael Somos, Jul 23 2002 */

lista(nn) = {print1("1, 1, "); print1(a=2, ", "); for (n=1, nn, a += a/divisors(a)[2]; print1(a, ", "););} \\ Michel Marcus, Apr 14 2015

A000792(n)=if(n>1,3^((n-2)\3)*(2+(n-2)%3),1) \\ M. F. Hasler, Jan 19 2019

G.f.: (1 + x + 2*x^2 + x^4)/(1 - 3*x^3). - Simon Plouffe in his 1992 dissertation.
a(3n) = 3^n; a(3*n+1) = 4*3^(n-1) for n > 0; a(3*n+2) = 2*3^n.
a(n) = 3*a(n-3) if n > 4. - Henry Bottomley, Nov 29 2001
a(n) = n if n <= 2, otherwise a(n-1) + Max{gcd(a(i), a(j)) | 0 < i < j < n}. - Reinhard Zumkeller, Feb 08 2002
A007600(a(n)) = n; Andrew Chi-Chih Yao attributes this observation to D. E. Muller. - Vincent Vatter, Apr 24 2006
a(n) = 3^(n - 2 - 2*floor((n - 1)/3))*2^(2 - (n - 1) mod 3) for n > 1. - Hieronymus Fischer, Nov 11 2007
From Kiyoshi Akima (k_akima(AT)hotmail.com), Aug 31 2009: (Start)
a(n) = 3^floor(n/3)/(1 - (n mod 3)/4), n > 1.
a(n) = 3^(floor((n - 2)/3))*(2 + ((n - 2) mod 3)), n > 1. (End)
a(n) = (2^b)*3^(C - (b + d))*(4^d), n > 1, where C = floor((n + 1)/3), b = max(0, ((n + 1) mod 3) - 1), d = max(0, 1 - ((n + 1) mod 3)). - Jonathan T. Rowell, Jul 26 2011
G.f.: 1 / (1 - x / (1 - x / (1 + x / (1 - x / (1 + x / (1 + x^2 / (1 + x))))))). - Michael Somos, May 12 2012
3*a(n) = 2*a(n+1) if n > 1 and n is not divisible by 3. - Michael Somos, Jan 23 2014
a(n) = a(n-1) + largest proper divisor of a(n-1), n > 2. - Ivan Neretin, Apr 13 2015
a(n) = max{a(i)*a(n-i) : 0 < i < n} for n >= 4. - Jianing Song, Feb 15 2020
a(n+1) = a(n) + A038754(floor( (2*(n-1) + 1)/3 )), for n > 1. - Thomas Scheuerle, Oct 27 2022

More terms and better description from Therese Biedl (biedl(AT)uwaterloo.ca), Jan 19 2000

A108605 Semiprimes with prime sum of factors: twice the lesser of the twin prime pairs.

Original entry on oeis.org

6, 10, 22, 34, 58, 82, 118, 142, 202, 214, 274, 298, 358, 382, 394, 454, 478, 538, 562, 622, 694, 838, 862, 922, 1042, 1138, 1198, 1234, 1282, 1318, 1618, 1642, 1654, 1714, 1762, 2038, 2062, 2098, 2122, 2182, 2302, 2458, 2554, 2578, 2602, 2638, 2854, 2902

Offset: 1

Zak Seidov, Jun 12 2005

All terms are even. (Cf. formula.)
The definition implies that the sum of factors is the sum over the prime factors with multiplicity, as in A001414. - R. J. Mathar, Nov 28 2008
The sum of factors of a semiprime pq is p+q, which can only be prime if {p, q} = {2, odd prime}. Requiring the sum to be prime then implies that the semiprime is twice the lesser of a twin prime pair. - M. F. Hasler, Apr 07 2015
Subsequence of A288814, each term being of the form A288814(p), where p is the greatest of a pair of twin primes. - David James Sycamore, Aug 29 2017

			58=2*29 and 2+29 is prime.

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

Cf. A001358 semiprimes, A001359 lesser of twin primes, A101605 3-almost primes, A108606 semiprimes with prime sum of digits, A108607 intersection of A108605 and A108606.

Select[Range[2, 3000, 2], !IntegerQ[Sqrt[ # ]]&&Plus@@(Transpose[FactorInteger[ # ]])[[2]]==2&&PrimeQ[Plus@@(Transpose[FactorInteger[ # ]])[[1]]]&]
Select[Range[2,3000,2],PrimeOmega[#]==PrimeNu[#]==2&&PrimeQ[Total[ FactorInteger[ #][[;;,1]]]]&] (* Harvey P. Dale, Apr 10 2023 *)

list(lim)=my(v=List(),p=2); forprime(q=3,lim\2+1, if(q-p==2, listput(v,2*p)); p=q); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017

a(n)=2*p, with p and 2+p twin primes: a(n)=2*A001359(n).

Changed division by 2 to multiplication by 2 in formula related to A001359. - R. J. Mathar, Nov 28 2008

A295185 a(n) is the smallest composite number whose prime divisors (with multiplicity) sum to prime(n); n >= 3.

Original entry on oeis.org

6, 10, 28, 22, 52, 34, 76, 184, 58, 248, 148, 82, 172, 376, 424, 118, 488, 268, 142, 584, 316, 664, 1335, 388, 202, 412, 214, 436, 3729, 508, 1048, 274, 2919, 298, 1208, 1256, 652, 1336, 1384, 358, 3801, 382, 772, 394, 6501, 7385, 892, 454, 916, 1864, 478, 5061, 2008, 2056, 2104, 538, 2168, 1108, 562, 5943, 9669

Offset: 3

David James Sycamore, Nov 16 2017

nonn

Sequence is undefined for n=1,2 since no composites exist whose prime divisors sum to 2, 3. For n >= 3, a(n) = A288814(prime(n)) = prime(n-k)*B(prime(n) - prime(n-k)) where B=A056240, and k >= 1 is the "type" of prime(n), indicated as prime(n)~k(g1,g2,...,gk) where gi = prime(n-(i-1)) - prime(n-i); 1 <= i <= k. Thus: 5~1(2), 211~2(12,2), 4327~3(30,8,6) etc. The sequence relates to gaps between odd primes, and in particular to the sequence of k prime gaps below prime(n). The even-indexed terms of B are relevant, as are those of subsequences:
  C=A288313, 2,4 plus terms B(n) where n-3 is prime (A298252),
  D=A297150, terms B(n) where n-5 is prime and n-3 is composite (A297925) and
  E=A298615, terms B(n) where both n-3 and n-5 are composite (A298366).
  The above sequences of indices 2m form a partition of the even numbers and the corresponding terms B(2m) form a partition of the even-indexed terms of A056240. The union of D and E is the sequence A292081 = B-C.
Let g(n,t) = prime(n) - prime(n-t), t < n, and h(n,t) = g(n,t) - g(n,1), 1 < t < n. If g1=g(n,1) is a term in A298252 (g1-3 is prime), then B(g1) is a term in C, so k=1. If g1 belongs to A297925 or A298366 then B(g1) is a term in D or E and the value of k depends on subsequent gaps below prime(n), within a range dependent on g1.
Let range R1(g1) = u - g(n,1) where u is the index in B of the greatest term in C such that C(u) < B(g1). Let range R2(g1) = v-g(n,1) where v is the index in B of the greatest term in D such that D(v) <= B(g1). For all n, R2 < R1, and if g1 is a term in D then R2(g1)=0. Examples: R1(12)=2, R2(12)=0, R1(30)=26, R2(30)=6.
k >= 1 is the smallest integer such that B(g(n,k)) <= B(g(n,t)) for all t satisfying g1 <= g(n,t) <= g1 + R1(g1). For g1-3 prime, k=1. If g1-3 is composite, let z be least integer > 1 such that g(n,z)-3 is prime, and let w be least integer >= 1 such that g(n,w)-5 is prime. Then z "complies" if h(n,z) <= R1, and w "complies" if h(n,w) <= R2. If g1-5 is prime then R2=w=0 and only z is relevant.
B(g1) must belong to C,D or E. If in C (g1-3 is prime) then k=1. If in D (g1-5 is prime), k=z if z complies, otherwise k=1. If B(g1) is in E and z complies but not w then k=z, or if w complies but not z then k=w. If B(g1) is in E and z,w both comply then k=z if 3*(g(n,z)-3) < 5*(g(n,w)-5), otherwise k=w. If neither z nor w comply, then k=1.
Conjecture: For all n >= 3, a(n) >= A288189(n).

			5=prime(3), g(3,1)=5-3=2, a term in C; k=1, and a(3)=3*B(5-3)=3*2=6; 5~1(2).
17=prime(7), g(7,1)=17-13=4, a term in C; k=1, a(7)=13*B(17-13)=13*4=52; 17~1(4).
211=prime(47); g(47,1)=12, a term in D, R1=2, R2=0, k=z=2, a(47)=197*b(211-197)=197*33=6501; 211~2(12,2), and 211 is first prime of type k=2.
8923=prime(1109); g(1109,1)=30, a term in E. R1=26, R2=6, z=3 and w=2 both comply  but 3*(g(n,3)-3)=159 > 5*(g(n,2)-5)=155, so k=w=2. Therefore a(1109)=8887*b(8923-8887)=8887*b(36)=8887*155=1377485; 8923~2(30,6).
40343=prime(4232); g(4232,1)=54, a term in E. R1=58, R2=12,z=6 and w=3, both comply, 3*(g(n,z)-3)=309 and 5*(g(n,w)-5)=305 therefore k=w=3 and a(4232) = 40277*b(40343-40277)=40277*b(66)=40277*305=12284485; 40343~3(54,6,6).
81611=prime(7981); g(81611,1)=42, a term in D, R1=22, R2=0; z complies, k=z=6, a(7981)=81547*b(81611-81547)=81546*b(64)=81546*183=14923101; 81611~6(42,6,4,6,2,4) and is the first prime of type k=6.
If p is the greater of twin/cousin primes then p~1(2), p~1(4), respectively.

Giovanni Resta, Table of n, a(n) for n = 3..10000

Cf. A000040, A056240, A288814, A292081, A289993, A288313, A297150, A298615, A298252, A297925, A298366, A288189.

b[n_] := b[n] = Total[Times @@@ FactorInteger[n]];
a[n_] := For[k = 2, True, k++, If[CompositeQ[k], If[b[k] == Prime[n], Return[k]]]];
Table[a[n], {n, 3, 63}] (* Jean-François Alcover, Feb 23 2018 *)

a(n) = { my(p=prime(n)); forcomposite(x=6, , my(f=factor(x)); if(f[, 1]~*f[, 2]==p, return(x))); } \\ Iain Fox, Dec 08 2017

a(n) = A288814(prime(n)) = prime(n-k)*A056240(prime(n) - prime(n-k)) for some k >= 1 and prime(n-k) = gpf(A288814(prime(n)).

a(n) >= A288189(n).

A259730 Primes p such that both 2p - 3 and 3p - 2 are prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 23, 37, 43, 53, 67, 71, 113, 127, 137, 167, 181, 191, 193, 211, 251, 263, 331, 347, 373, 431, 433, 443, 461, 487, 587, 727, 751, 757, 907, 991, 1021, 1091, 1103, 1171, 1187, 1213, 1231, 1297, 1367, 1453, 1483, 1597, 1637, 1663, 1667, 1733

Offset: 1

Reinhard Zumkeller, Jul 05 2015

nonn

A010051(2*a(n) - 3) * A010051(3*a(n) - 2) = 1;
A259758(n) = (2*a(n) - 3) * (3*a(n) - 2).
Except for a(1)=3 this is the same sequence as primes p such that A288814(3*p) - A288814(2*p) = 5. - David James Sycamore, Jul 22 2017
Furthermore, (A288814(3*p)*A288814(2*p))/6 belongs to A259758. - David James Sycamore, Jul 23 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Intersection of A063908 and A088878; A172287, A259758.

Cf. A288814, A259758.

import Data.List.Ordered (isect)
a259730 n = a259730_list !! (n-1)
a259730_list = a063908_list `isect` a088878_list

Select[Prime@ Range@ 270, Times @@ Boole@ Map[PrimeQ, {2 # - 3, 3 # - 2}] > 0 &] (* Michael De Vlieger, Jul 22 2017 *)
Select[Prime[Range[300]],AllTrue[{2#-3,3#-2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 08 2020 *)

lista(nn) = forprime(p=3, nn, if(isprime(2*p-3) && isprime(3*p-2), print1(p, ", "))); \\ Altug Alkan, Jul 22 2017

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
```
\\ see Corneth link
```

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

cp's OEIS Frontend

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

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A300098 Records of A288814.

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A301592 Numbers k at which the ratio A288814(k) / k reaches a record high.

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A000792 a(n) = max{(n - i)*a(i) : i < n}; a(0) = 1.

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A108605 Semiprimes with prime sum of factors: twice the lesser of the twin prime pairs.

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A290163 Primes p such that A288814(4p) - A288814(3p) = 7.

A259730 Primes p such that both 2p - 3 and 3p - 2 are prime.