A086486 -id:A086486 - cp's OEIS frontend

A063908 Numbers k such that k and 2*k-3 are primes.

3, 5, 7, 11, 13, 17, 23, 31, 37, 41, 43, 53, 67, 71, 83, 97, 101, 107, 113, 127, 137, 157, 167, 181, 191, 193, 211, 223, 233, 241, 251, 263, 283, 311, 317, 331, 347, 373, 421, 431, 433, 443, 457, 461, 487, 521, 547, 563, 577, 587, 613, 617, 631, 641, 643, 647

Offset: 1

N. J. A. Sloane, Aug 31 2001

nonn

If p is in this sequence then the products of positive powers of 3, p and 2p-3 are entries in A086486. - Victoria A Sapko (vsapko(AT)canes.gsw.edu), Sep 23 2003
Median prime of AP3's starting at 3, i.e., triples of primes (3,p,q) in arithmetic progression. - M. F. Hasler, Sep 24 2009
a(n) = sum of the coprimes(p) mod (p+1). - J. M. Bergot, Nov 13 2014
A010051(2*a(n)-3) = 1. - Reinhard Zumkeller, Jul 02 2015
A098090 INTERSECT A000040. - R. J. Mathar, Mar 23 2017

			From _K. D. Bajpai_, Nov 29 2019: (Start)
a(5) = 13 is prime and 2*13 - 3 = 23 is also prime.
a(6) = 17 is prime and 2*17 - 3 = 31 is also prime.
(End)

Harry J. Smith and K. D. Bajpai, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Carlos Rivera, Puzzle 34.- Prime Triplets in arithmetic progression, The Prime Puzzles & Problems Connection. [From _M. F. Hasler_, Sep 24 2009]

Cf. A005382.
Cf. A000040, A010051, A088878, A172287.
Cf. A259730.

a063908 n = a063908_list !! (n-1)
a063908_list = filter
   ((== 1) . a010051' . (subtract 3) . (* 2)) a000040_list
-- Reinhard Zumkeller, Jul 02 2015

[n : n in [0..700] | IsPrime(n) and IsPrime(2*n-3)]; // Vincenzo Librandi, Nov 14 2014

select(k -> andmap(isprime, [k, 2*k-3]), [seq(k, k=1.. 10^4)]); # K. D. Bajpai, Nov 29 2019

Select[Prime[Range[6! ]],PrimeQ[2*#-3]&] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2009 *)

{ n=0; p=1; for (m=1, 10^9, p=nextprime(p+1); if (isprime(2*p - 3), write("b063908.txt", n++, " ", p); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 02 2009

forprime( p=1,default(primelimit), isprime(2*p-3) && print1(p",")) \\ M. F. Hasler, Sep 24 2009

a(n) = A241817(n)/2. - Wesley Ivan Hurt, Apr 08 2018

A326847 Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197

Offset: 1

Gus Wiseman, Jul 26 2019

nonn

First differs from A071139, A089352 and A086486 in lacking 60. First differs from A326837 in lacking 268.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The enumeration of these partitions by sum is given by A326842.

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  30: {1,2,3}
  31: {11}
  32: {1,1,1,1,1}
  37: {12}

R. J. Mathar, Table of n, a(n) for n = 1..489

Intersection of A326841 and A316413.

Cf. A001222, A018818, A056239, A067538, A112798, A316413, A326836, A326842.

isA326847 := proc(n)
    psigsu := A056239(n) ;
    for ifs in ifactors(n)[2] do
        p := op(1,ifs) ;
        psig := numtheory[pi](p) ;
        if modp(psigsu,psig) <> 0 then
            return false;
        end if;
    end do:
    psigle := numtheory[bigomega](n) ;
    if modp(psigsu,psigle) = 0 then
        true;
    else
        false;
    end if;
end proc:
n := 1:
for i from 2 to 3000 do
    if isA326847(i) then
        printf("%d %d\n",n,i);
        n := n+1 ;
    end if;
end do: # R. J. Mathar, Aug 09 2019

Select[Range[2,100],With[{y=Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]},Divisible[Total[y],Length[y]]&&And@@IntegerQ/@(Total[y]/y)]&]

Intersection of A326841 and A316413.

A144100 Numbers k such that k is strictly greater than f(k), where f(k) = 1 if k is prime, 2 * rad(k) if 4 divides k and rad(k) otherwise.

Original entry on oeis.org

2, 3, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 24, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 45, 47, 48, 49, 50, 53, 54, 56, 59, 61, 63, 64, 67, 71, 72, 73, 75, 79, 80, 81, 83, 88, 89, 90, 96, 97, 98, 99, 100, 101, 103, 104, 107, 108, 109, 112, 113, 117, 120, 121, 125, 126

Offset: 1

Reikku Kulon, Sep 10 2008

This is the set of all integers k such that there exists a full period linear congruential pseudorandom number generator x -> bx + c (mod k), where b is not a multiple of k, b - 1 is a multiple of f(k) and c is a positive integer relatively prime to k.
4 is the only prime power not a member of the set: f(4) = 2 * rad(4) = 4.
This sequence consists of the primes and 2*A013929. - Charlie Neder, Jan 28 2019

			2 is a member: f(2) = 1 and the sequence (0, 1, 0, ...) given by x -> x + 1 (mod 2) has period 2.
8 is a member: f(8) = 4 and the sequence (0, 1, 6, 7, 4, 5, 2, 3, 0, ...) given by x -> 5x + 1 (mod 8) has period 8.
18 is a member: f(18) = 6 and the sequence (0, 1, 14, 3, 4, 17, 6, 7, 2, 9, 10, 5, 12, 13, 8, 15, 16, 11, 0, ...) given by x -> 13x + 1 (mod 18) has period 18.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000040, A007947, A071139, A082377, A086486, A089352, A133810, A133811.

a144100 n = a144100_list !! (n-1)
a144100_list = filter (\x -> a144907 x < x) [1..]
-- Reinhard Zumkeller, Mar 12 2014

rad(n) = local(p); p=factor(n)[, 1]; prod(i=1, length(p), p[i]) ;
f(n) = if (isprime(n), 1, if ((n % 4)==0 , 2*rad(n), rad(n))); isok(n) = n > f(n); \\ Michel Marcus, Aug 09 2013

A144907(a(n)) < a(n). - Reinhard Zumkeller, Mar 12 2014

A086487 Smallest number with n prime divisors such that the sum of the prime divisors is also a divisor, or 0 if no such number exists.

Original entry on oeis.org

2, 0, 30, 1122, 2730, 72930, 870870, 9699690, 340510170, 9592993410, 265257422430, 8624101075590, 304250263527210, 14299762385778870, 1164365758518632670, 43657174563782890110, 1987938667108592728530

Offset: 1

Amarnath Murthy, Jul 28 2003

nonn

First member of A086486 with n distinct prime factors. A001221(a(n)) = n; A008472(a(n)) divides a(n). - David Wasserman, Mar 09 2005

			a(3) = 30. The prime divisors of 30 are 2,3 and 5 and 2+3+5 = 10, divides 30.

Cf. A086486, A086488.

Cf. A001221, A008472, A104465, A104466.

More terms from David Wasserman, Mar 09 2005

A131647 Composite numbers that are products of distinct primes and divisible by the sum of those primes.

Original entry on oeis.org

30, 70, 105, 231, 286, 627, 646, 805, 897, 1122, 1581, 1798, 2730, 2958, 2967, 3055, 3526, 3570, 4070, 4543, 5487, 5658, 6461, 6745, 7198, 7881, 8778, 8970, 9222, 9282, 9717, 10366, 10370, 10626, 10707, 11130, 14231, 15015, 16377, 16530, 19866

Offset: 1

Tanya Khovanova, Sep 08 2007

nonn

			1122 = 2*3*11*17 and 1122 is divisible by 2+3+11+17 = 33.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Harvey P. Dale)

Sequence union A000040 = A005117 intersect A086486. - Ray Chandler, Nov 29 2011

with(numtheory): P:=proc(q) local a,k,n; for n from 2 to q do
if issqrfree(n) and not isprime(n) then a:=ifactors(n)[2];
if type(n/add(a[k][1],k=1..nops(a)),integer) then print(n); fi;
fi; od; end: P(10^9); # Paolo P. Lava, Sep 19 2014

Select[Range[2, 20000], PrimeQ[ # ] == False && Union[Transpose[FactorInteger[ # ]][[2]]] == {1} && Mod[ #, Plus @@ Transpose[FactorInteger[ # ]][[1]]] == 0 &]
pdpQ[n_]:=Module[{fi=Transpose[FactorInteger[n]]},!PrimeQ[n]&&Max[fi[[2]]] == 1&&Divisible[n,Total[fi[[1]]]]]; Select[Range[2,50000],pdpQ] (* Harvey P. Dale, Oct 16 2013 *)

lista(nn) = {forcomposite(n=1, nn, f = factor(n); nbp = #f~; if ((vecmax(f[,2]) == 1) && !(n % sum(i=1, nbp, f[i, 1])), print1(n, ", ")););} \\ Michel Marcus, Sep 19 2014

A380487 Numbers such that the sum of prime factors without repetition divides the product of prime factors without repetition and each division yields a greater quotient.

Original entry on oeis.org

2, 30, 70, 105, 231, 627, 805, 1122, 2730, 3570, 8778, 9282, 10626, 15015, 24738, 24882, 31746, 33495, 33915, 44330, 45885, 49335, 51051, 62985, 72930, 95095, 106590, 132990, 145145, 156009, 170170, 222870, 230945, 274505, 290598, 329406, 335478, 418285, 449995

Offset: 1

Torlach Rush, Jan 24 2025

nonn

Conjecture: There exists m, the smallest positive integer such that a(k)|a(k+m), for all k.

			2 is a term because sopf(2)|rad(2) = 2|2.
30 is a term because sopf(30)|rad(30) = 10|30.
70 is a term because sopf(70)|rad(70) = 14|70.

Robert Israel, Table of n, a(n) for n = 1..168

Cf. A007947, A008472.

Subsequence of A086486.

f:= proc(n,m) local q,S;
   S:= numtheory:-factorset(n);
   q:= convert(S,`*`)/convert(S,`+`);
   if q::integer and q > m  then q else 0 fi;
end proc:
m:= 0: R:= NULL: count:= 0:
for n from 2 while count < 50 do
  v:= f(n,m);
  if v > 0 then
    m:= v;  R:= R, n; count:= count+1;
  fi;
od:
R; # Robert Israel, Apr 30 2025

s={};q=0;Do[pf=Times@@First/@FactorInteger[n];sf=Total[First/@FactorInteger[n]];If[Divisible[pf,sf]&&pf/sf>q,AppendTo[s,n];q=pf/sf],{n,2,449995}];s (* James C. McMahon, Apr 03 2025 *)

lista(nn) = my(m=0, list=List()); for (n=2, nn, my(f=factor(n)[,1], q=factorback(f)/vecsum(f)); if ((denominator(q) == 1) && (q>m), listput(list, n); m=q);); Vec(list); \\ Michel Marcus, Mar 29 2025

def sopf(n): return sum(set(prime_factors(n)))
def rad(n):
    rad = 1
    for p in set(prime_factors(n)): rad *= p
    return rad
def output(limit=39):
    results = []
    n = 2
    result = 0
    while len(results) < limit:
        sopf_n = sopf(n)
        rad_n = rad(n)
        if rad_n % sopf_n == 0 and result < rad_n / sopf_n:
            results.append(n)
            result = rad_n / sopf_n
            print(n, end=', ')
        n += 1
    return results
output()

A086488 A086487(n)/S where S is the sum of the prime divisors.

Original entry on oeis.org

1, 0, 3, 34, 91, 1430, 12441, 125970, 3095547, 67083870, 1560337779, 42483256530, 1278362451795, 50174604862382, 3146934482482791, 109416477603465890, 4497598794363331965, 327313882010942897070, 14916466618879283766165

Offset: 1

Amarnath Murthy, Jul 28 2003

nonn

			a(3) = 3. A086487(3) = 30 /(2+3+5) = 3.

Cf. A086486, A086487.

a(n) = A086487(n)/A008472(A086487(n)). - David Wasserman, Mar 09 2005

More terms from David Wasserman, Mar 09 2005

cp's OEIS Frontend

A063908 Numbers k such that k and 2*k-3 are primes.

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A326847 Heinz numbers of integer partitions of m >= 0 using divisors of m whose length also divides m.

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A144100 Numbers k such that k is strictly greater than f(k), where f(k) = 1 if k is prime, 2 * rad(k) if 4 divides k and rad(k) otherwise.

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A086487 Smallest number with n prime divisors such that the sum of the prime divisors is also a divisor, or 0 if no such number exists.

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A131647 Composite numbers that are products of distinct primes and divisible by the sum of those primes.

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A380487 Numbers such that the sum of prime factors without repetition divides the product of prime factors without repetition and each division yields a greater quotient.

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A086488 A086487(n)/S where S is the sum of the prime divisors.

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