A001747 -id:A001747 - cp's OEIS frontend

A071681 Number of ways to represent the n-th prime as arithmetic mean of two other primes.

0, 0, 1, 1, 2, 2, 3, 1, 3, 3, 2, 4, 4, 4, 4, 5, 5, 3, 5, 7, 5, 4, 5, 6, 6, 8, 6, 7, 6, 6, 8, 8, 10, 6, 10, 8, 8, 6, 10, 8, 9, 7, 9, 11, 10, 6, 10, 11, 11, 8, 12, 10, 10, 14, 13, 14, 13, 9, 10, 13, 12, 12, 14, 16, 11, 13, 13, 14, 18, 13, 18, 14, 14, 17, 14, 16, 14, 16, 15, 16, 16, 17, 16, 16

Offset: 1

Reinhard Zumkeller, May 31 2002

Conjecture: a(n)>0 for n>2.
a(A137700(n))=n and a(m)<>n for m < A137700(n), A000040(A137700(n))=A126204(n). - Reinhard Zumkeller, Feb 07 2008
The conjecture follows from a slightly strengthened version of Goldbach's conjecture: that every even number > 6 is the sum of two distinct primes. - T. D. Noe, Jan 10 2011 [Corrected by Barry Cherkas and Robert Israel, May 21 2015]
a(n) = A116619(n) + 1. - Reinhard Zumkeller, Mar 27 2015
Number of primes q < prime(n), such that 2*prime(n) - q is prime. - Dmitry Kamenetsky, May 27 2023

			a(7)=3 as prime(7) = 17 = (3+31)/2 = (5+29)/2 = (11+23)/2 and 2*17-p is not prime for the other primes p < 17: {2,7,13}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mladen Vassilev-Missana, Goldbach's n-perfect numbers as a key for proving the Goldbach's Conjecture, Notes on Number Theory and Discrete Mathematics (2005) Vol. 11, No. 1, 20-22.

Cf. A071680, A000040, A129363, A178609, A001358, A100484, A001747, A010051, A116619, A253138.

a071681 n = sum $ map a010051' $
   takeWhile (> 0) $ map (2 * a000040 n -) $ drop n a000040_list
-- Reinhard Zumkeller, Mar 27 2015

f[n_] := Block[{c = 0, k = PrimePi@n - 1}, While[k > 0, If[ PrimeQ[2n - Prime@k], c++ ]; k-- ]; c]; Table[ f@ Prime@n, {n, 84}] (* Robert G. Wilson v, Mar 22 2007 *)

A071681(n)={s=2*prime(n);a=0;for(i=1,n-1,a=a+isprime(s-prime(i)));a}

A073703 Smallest prime p such that also p+prime(n)*2 is a prime.

Original entry on oeis.org

3, 5, 3, 3, 7, 3, 3, 3, 7, 3, 5, 5, 7, 3, 3, 3, 13, 5, 3, 7, 3, 5, 7, 3, 3, 31, 5, 13, 5, 3, 3, 7, 3, 3, 13, 5, 3, 5, 3, 3, 31, 5, 7, 3, 3, 3, 11, 3, 3, 3, 13, 13, 5, 7, 7, 31, 3, 5, 3, 7, 3, 7, 3, 19, 5, 7, 11, 3, 7, 3, 3, 43, 5, 5, 3, 3, 19, 3, 7, 3, 19, 11, 19, 11, 3, 43, 13, 5, 7, 3, 3, 13, 3

Offset: 1

Reinhard Zumkeller, Aug 04 2002

nonn

If Polignac's conjecture (1849) is correct, the sequence is defined for all n (as is A020483).

Also: least k-prime(n) such that k-prime(n) and k+prime(n) are both primes. - Pierre CAMI, Aug 27 2004

			n=5: prime(5)=11; 2+11*2=24, 3+11*2=25 and 5+11*2=27 are not prime, but 7+11*2=29 is prime, therefore a(5)=7.

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

Cf. A073704, A001747, A000040, A020483.

Cf. A010051.

a073703 n = head [p | p <- a000040_list, a010051 (p + 2 * a000040 n) == 1]
-- Reinhard Zumkeller, Oct 29 2013

f[n_] := Block[{k = Prime[n], p = Prime[n]}, While[ !PrimeQ[k - p] || !PrimeQ[k + p], k++ ]; k - p]; Table[ f[n], {n, 95}] (* Robert G. Wilson v, Aug 28 2004 *)

forprime(q=2,500,forprime(p=2,default(primelimit),if(isprime(2*q+p),print1(p", ");next(2)));error("Not enough precomputed primes")) \\ Charles R Greathouse IV, Aug 21 2011

Merged with Pierre CAMI's submission of Aug 2004 - R. J. Mathar, Jul 29 2008

A143201 Product of distances between prime factors in factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 6, 3, 1, 1, 2, 1, 4, 5, 10, 1, 2, 1, 12, 1, 6, 1, 6, 1, 1, 9, 16, 3, 2, 1, 18, 11, 4, 1, 10, 1, 10, 3, 22, 1, 2, 1, 4, 15, 12, 1, 2, 7, 6, 17, 28, 1, 6, 1, 30, 5, 1, 9, 18, 1, 16, 21, 12, 1, 2, 1, 36, 3, 18, 5, 22, 1, 4, 1, 40, 1, 10, 13, 42, 27, 10, 1, 6, 7, 22

Offset: 1

Reinhard Zumkeller, Aug 12 2008

nonn

a(n) is the product of the sum of 1 and first differences of prime factors of n with multiplicity, with a(n) = 1 for n = 1 or prime n. - Michael De Vlieger, Nov 12 2023.
a(A007947(n)) = a(n);
A006093 and A001747 give record values and where they occur:
A006093(n)=a(A001747(n+1)) for n>1.
a(n) = 1 iff n is a prime power: a(A000961(n))=1;
a(n) = 2 iff n has exactly 2 and 3 as prime factors:
a(A033845(n))=2;
a(n) = 3 iff n is in A143202;
a(n) = 4 iff n has exactly 2 and 5 as prime factors:
a(A033846(n))=4;
a(n) = 5 iff n is in A143203;
a(n) = 6 iff n is in A143204;
a(n) = 7 iff n is in A143205;
a(n) <> A006512(k)+1 for k>1.
a(A033849(n))=3; a(A033851(n))=3; a(A033850(n))=5; a(A033847(n))=6; a(A033848(n))=10. [Reinhard Zumkeller, Sep 19 2011]

			a(86) = a(43*2) = 43-2+1 = 42;
a(138) = a(23*3*2) = (23-3+1)*(3-2+1) = 42;
a(172) = a(43*2*2) = (43-2+1)*(2-2+1) = 42;
a(182) = a(13*7*2) = (13-7+1)*(7-2+1) = 42;
a(276) = a(23*3*2*2) = (23-3+1)*(3-2+1)*(2-2+1) = 42;
a(330) = a(11*5*3*2) = (11-5+1)*(5-3+1)*(3-2+1) = 42.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for primes, gaps between

Cf. A000961, A001747, A006093, A020639, A033845, A033846.
Cf. A033847, A033848, A033849, A033850.
Cf. A143203, A143204, A143205.
Cf. A027746.

a143201 1 = 1
a143201 n = product $ map (+ 1) $ zipWith (-) (tail pfs) pfs
   where pfs = a027748_row n
-- Reinhard Zumkeller, Sep 13 2011

Table[Times@@(Differences[Flatten[Table[First[#],{Last[#]}]&/@ FactorInteger[ n]]]+1),{n,100}] (* Harvey P. Dale, Dec 07 2011 *)

a(n) = f(n,1,1) where f(n,q,y) = if n=1 then y else if q=1 then f(n/p,p,1)) else f(n/p,p,y*(p-q+1)) with p = A020639(n) = smallest prime factor of n.

A360678 Sum of the left half (inclusive) of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 1, 4, 2, 2, 1, 5, 2, 6, 1, 2, 2, 7, 3, 8, 2, 2, 1, 9, 2, 3, 1, 4, 2, 10, 3, 11, 3, 2, 1, 3, 2, 12, 1, 2, 2, 13, 3, 14, 2, 4, 1, 15, 3, 4, 4, 2, 2, 16, 3, 3, 2, 2, 1, 17, 2, 18, 1, 4, 3, 3, 3, 19, 2, 2, 4, 20, 3, 21, 1, 5, 2, 4, 3, 22, 3, 4, 1

Offset: 1

Gus Wiseman, Mar 05 2023

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 810 are {1,2,2,2,2,3}, with left half (inclusive) {1,2,2}, so a(810) = 5.
The prime indices of 3675 are {2,3,3,4,4}, with left half (inclusive) {2,3,3}, so a(3675) = 8.

Positions of first appearances are 1 and A001248.
Positions of 1's are A001747.
These partitions are counted by A360675 with rows reversed.
The exclusive version is A360676.
The right version is A360679.
A112798 lists prime indices, length A001222, sum A056239, median* A360005.
A360616 gives half of bigomega (exclusive), inclusive A360617.
A360673 counts multisets by right sum (exclusive), inclusive A360671.
First for prime indices, second for partitions, third for prime factors:
- A360676 gives left sum (exclusive), counted by A360672, product A361200.
- A360677 gives right sum (exclusive), counted by A360675, product A361201.
- A360678 gives left sum (inclusive), counted by A360675, product A347043.
- A360679 gives right sum (inclusive), counted by A360672, product A347044.
Cf. A026424, A280076, A359912, A360006, A360457.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[Take[prix[n],Ceiling[Length[prix[n]]/2]]],{n,100}]

A360676(n) + A360679(n) = A001222(n).

A360677(n) + A360678(n) = A001222(n).

A346703 Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 6, 13, 2, 3, 4, 17, 6, 19, 10, 3, 2, 23, 4, 5, 2, 9, 14, 29, 10, 31, 8, 3, 2, 5, 6, 37, 2, 3, 4, 41, 14, 43, 22, 15, 2, 47, 12, 7, 10, 3, 26, 53, 6, 5, 4, 3, 2, 59, 6, 61, 2, 21, 8, 5, 22, 67, 34, 3, 14, 71, 12, 73, 2, 15, 38

Offset: 1

Gus Wiseman, Aug 08 2021

nonn

			The prime factors of 108 are (2,2,3,3,3), with odd bisection (2,3,3), with product 18, so a(108) = 18.
The prime factors of 720 are (2,2,2,2,3,3,5), with odd bisection (2,2,3,5), with product 60, so a(720) = 60.

Positions of 2's are A001747.
Positions of primes are A037143 (complement: A033942).
The even reverse version appears to be A329888.
Positions of first appearances are A342768.
The sum of prime indices of a(n) is A346697(n), reverse: A346699.
The reverse version is A346701.
The even version is A346704.
A001221 counts distinct prime factors.
A001222 counts all prime factors.
A056239 adds up prime indices, row sums of A112798.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A209281 (shifted) adds up the odd bisection of standard compositions.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A335433/A335448 rank separable/inseparable partitions.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346633 adds up the even bisection of standard compositions.
A346698 gives the sum of the even bisection of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf. A025047, A027187, A027193, A053738, A097805, A106356, A341446, A344653, A345957, A345958, A345959.

Table[Times@@First/@Partition[Append[Flatten[Apply[ConstantArray,FactorInteger[n],{1}]],0],2],{n,100}]

a(n) * A346704(n) = n.

A056239(a(n)) = A346697(n).

A283312 a(n) = smallest missing positive number, unless a(n-1) was a prime in which case a(n) = 2*a(n-1).

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 7, 14, 8, 9, 11, 22, 12, 13, 26, 15, 16, 17, 34, 18, 19, 38, 20, 21, 23, 46, 24, 25, 27, 28, 29, 58, 30, 31, 62, 32, 33, 35, 36, 37, 74, 39, 40, 41, 82, 42, 43, 86, 44, 45, 47, 94, 48, 49, 50, 51, 52, 53, 106, 54, 55, 56, 57, 59, 118, 60, 61, 122, 63, 64, 65, 66, 67, 134, 68, 69

Offset: 1

N. J. A. Sloane, Mar 08 2017

nonn

Comments from N. J. A. Sloane, Nov 02 2020: (Start)
Alternatively, this is the lexicographically earliest infinite sequence of distinct positive numbers such that every prime is followed by its double.
Theorem: This is a permutation of the positive integers.
Proof. Sequence is clearly infinite, so for any k there is a number N_0(k) such that n >= N_0(k) implies a(n) > k.
Suppose m is missing. Consider a(n) for n = N_0(m). Then a(n) must be a prime p (otherwise it would have been m, which is missing), a(n+1) = 2*p, and a(n+2) = m, a contradiction. QED.
(End)
A toy model of A280864, A280985, and A127202.
Alternative definition: a(1,2) = 1,2. Let P(k) = rad(a(1)*a(2)*...*a(k)), then for n > 2, a(n) = P(n)/P(n-1), where rad is A007947. - David James Sycamore, Jan 27 2024

			The offset is 1. What is a(1)? It is the smallest missing positive number, which is 1. Similarly, a(2)=2.
What is a(3)? Since the previous term was the prime 2, a(3) = 4.
And so on.

N. J. A. Sloane, Table of n, a(n) for n = 1..75000

Cf. A127202, A280864, A280985.
See A283313 for smallest missing number, A338362 for inverse, A338361 for indices of primes, A338357 for first differences.
For records see A338356 and A001747.

a:=[1];
H:=Array(1..1000,0); MMM:=1000;
H[1]:=1; smn:=2; t:=2;
for n from 2 to 100 do
if t=smn then a:=[op(a),t]; H[t]:=1;
   if isprime(t) then a:=[op(a),2*t]; H[2*t]:=1; fi;
   t:=t+1;
# update smallest missing number smn
   for i from smn+1 to MMM do if H[i]=0 then smn:=i; break; fi; od;
else t:=t+1;
fi;
od:
a;

Module[{nmax = 100, smn = 1}, Nest[Append[#, If[PrimeQ[Last[#]], 2*Last[#], While[MemberQ[#, ++smn]]; smn]]&, {1}, nmax-1]] (* Paolo Xausa, Feb 12 2024 *)

There is an explicit formula for the n-th term of the inverse permutation: see A338362.
The graph: Numbers appear in the sequence in their natural order, except when interrupted by the appearance of primes. Suppose a(n)=x, where x is neither a prime nor twice a prime. Then if p is a prime in the range x/2 < p < x, 2p appears in the sequence between p and p+1. Therefore we have the identity
n = x + pi(x) - pi(x/2). ... (1)
If a(n) = x = a prime, then (1) is replaced by
n = x + pi(x) - pi(x/2) - 1. ... (2)
If a(n) = x = twice a prime then
n = x/2 + pi(x/2) - pi(x/4). ... (3)
These equations imply that the lower line in the graph of the sequence is
x approx= n(1 - 1/(2*log n)) ... (4)
while the upper line is
x approx= 2n(1 - 1/(2*log n)). ... (5)
a(2*n-1 + A369610(n)) = prime(n). - David James Sycamore, Jan 27 2024

Entry revised by N. J. A. Sloane, Nov 03 2020

A096014 a(n) = (smallest prime factor of n) * (least prime that is not a factor of n), with a(1)=2.

Original entry on oeis.org

2, 6, 6, 6, 10, 10, 14, 6, 6, 6, 22, 10, 26, 6, 6, 6, 34, 10, 38, 6, 6, 6, 46, 10, 10, 6, 6, 6, 58, 14, 62, 6, 6, 6, 10, 10, 74, 6, 6, 6, 82, 10, 86, 6, 6, 6, 94, 10, 14, 6, 6, 6, 106, 10, 10, 6, 6, 6, 118, 14, 122, 6, 6, 6, 10, 10, 134, 6, 6, 6, 142, 10, 146, 6, 6, 6, 14, 10, 158, 6, 6, 6

Offset: 1

Reinhard Zumkeller, Jun 15 2004

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

Cf. A001747, A020639, A053669, A073582, A096015.

f:= proc(n) local p;
p:= 3;
if n::even then
  while type(n/p,integer) do p:= nextprime(p) od;
else
  while not type(n/p,integer) do p:= nextprime(p) od:
fi;
2*p;
end proc:
f(1):= 2:
map(f, [$1..100]); # Robert Israel, Jun 22 2018

PrimeFactors[n_] := Flatten[ Table[ #[[1]], {1} ] & /@ FactorInteger[n]]; f[1] = 2; f[n_] := Block[ {k = 1}, While[ Mod[ n, Prime[k]] == 0, k++ ]; Prime[k]PrimeFactors[n][[1]]]; Table[ f[n], {n, 83}] (* Robert G. Wilson v, Jun 15 2004 *)
spfn[n_]:=Module[{fi=FactorInteger[n][[;;,1]],k=2},While[MemberQ[fi,k],k=NextPrime[k]];fi[[1]]*k]; Array[spfn,90] (* Harvey P. Dale, Sep 22 2024 *)

dnd(n) = forprime(p=2, , if (n % p, return(p)));
lpf(n) = if (n==1, 1, forprime(p=2, , if (!(n % p), return(p))));
a(n) = dnd(n)*lpf(n); \\ Michel Marcus, Jun 22 2018

a(n) = A020639(n)*A053669(n);
A096015(n) = a(n)/2.
If n (mod 6) = 2, 3 or 4, then a(n) = 6. If n (mod 6) = 0, 1 or 5, then a(n) belongs to A001747 less the first three terms or belongs to A073582 less the first two terms. - Robert G. Wilson v, Jun 15 2004
From Bill McEachen, Jul 26 2024: (Start)
a(n) <= 2*n, except when n = 2.
a(n) = 2*n for n an odd prime. (End)

A116366 Triangle read by rows: even numbers as sums of two odd primes.

Original entry on oeis.org

6, 8, 10, 10, 12, 14, 14, 16, 18, 22, 16, 18, 20, 24, 26, 20, 22, 24, 28, 30, 34, 22, 24, 26, 30, 32, 36, 38, 26, 28, 30, 34, 36, 40, 42, 46, 32, 34, 36, 40, 42, 46, 48, 52, 58, 34, 36, 38, 42, 44, 48, 50, 54, 60, 62, 40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74, 44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82

Offset: 1

Reinhard Zumkeller, Feb 06 2006

T(n,k) = 2*A065305(n,k) = A065342(n+1,k+1);
Row sums give A116367; central terms give A116368;
T(n,1) = A113935(n+1);
T(n,n-2) = A048448(n) for n>2;
T(n,n-1) = A001043(n) for n>1;
T(n,n) = A001747(n+2) = A100484(n+1).

			Triangle begins:
  6;
  8,  10;
  10, 12, 14;
  14, 16, 18, 22;
  16, 18, 20, 24, 26;
  20, 22, 24, 28, 30, 34;
  22, 24, 26, 30, 32, 36, 38;
  26, 28, 30, 34, 36, 40, 42, 46;
  32, 34, 36, 40, 42, 46, 48, 52, 58;
  34, 36, 38, 42, 44, 48, 50, 54, 60, 62;
  40, 42, 44, 48, 50, 54, 56, 60, 66, 68, 74;
  44, 46, 48, 52, 54, 58, 60, 64, 70, 72, 78, 82; etc. - _Bruno Berselli_, Aug 16 2013

G. C. Greubel, Rows n = 1..100 of triangle, flattened
Index entries for sequences related to Goldbach conjecture

Cf. A001043, A001747, A048448, A065305, A065342, A100484, A113935, A116367, A116368.

[NthPrime(n+1)+NthPrime(k+1): k in [1..n], n in [1..15]]; // Bruno Berselli, Aug 16 2013

Table[Prime[n+1] + Prime[k+1], {n,1,12}, {k,1,n}]//Flatten (* G. C. Greubel, May 12 2019 *)

{T(n,k) = prime(n+1) + prime(k+1)}; \\ G. C. Greubel, May 12 2019

[[nth_prime(n+1) + nth_prime(k+1) for k in (1..n)] for n in (1..12)] # G. C. Greubel, May 12 2019

T(n,k) = prime(n+1) + prime(k+1), 1 <= k <= n.

A347045 Smallest divisor of n with exactly half as many prime factors (counting multiplicity) as n, or 1 if there are none.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 3, 4, 1, 1, 1, 1, 3, 2, 1, 4, 5, 2, 1, 1, 1, 1, 1, 1, 3, 2, 5, 4, 1, 2, 3, 4, 1, 1, 1, 1, 1, 2, 1, 1, 7, 1, 3, 1, 1, 6, 5, 4, 3, 2, 1, 4, 1, 2, 1, 8, 5, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 7, 1, 1, 1, 9, 2, 1, 4, 5, 2, 3

Offset: 1

Gus Wiseman, Aug 16 2021

nonn

			The divisors of 90 with half bigomega are: 6, 9, 10, 15, so a(90) = 6.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The smallest divisor without the condition is A020639 (greatest: A006530).
Positions of 1's are A026424.
Positions of even terms are A063745 = 2*A026424.
The case of powers of 2 is A072345.
Positions of 2's are A100484.
Divisors of this type are counted by A345957 (rounded: A096825).
The rounded version is A347043.
The greatest divisor of this type is A347046 (rounded: A347044).
A000005 counts divisors.
A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A056239 adds up prime indices, row sums of A112798.
A207375 lists central divisors (min: A033676, max: A033677).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Cf. A000720, A001747, A027746, A038548, A056924, A063962, A106529, A140271, A244991, A324522, A335433, A335448.

Table[If[#=={},1,Min[#]]&@Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&],{n,100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]], np}, np = Length[p]; If[OddQ[np], 1, Times @@ p[[1 ;; np/2]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)

from sympy import divisors, factorint
def a(n):
    npf = len(factorint(n, multiple=True))
    for d in divisors(n)[1:-1]:
        if 2*len(factorint(d, multiple=True)) == npf: return d
    return 1
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Aug 18 2021

from math import prod
from sympy import factorint
def A347045(n):
    fs = factorint(n,multiple=True)
    q, r = divmod(len(fs),2)
    return 1 if r else prod(fs[:q]) # Chai Wah Wu, Aug 20 2021

a(n) = Product_{k=1..A001222(n)/2} A027746(n,k) if A001222(n) is even, and 1 otherwise. - Amiram Eldar, Nov 02 2024

A036438 Integers which can be written as m*tau(m) for some m, where tau = A000005.

Original entry on oeis.org

1, 4, 6, 10, 12, 14, 22, 24, 26, 27, 32, 34, 38, 40, 46, 56, 58, 60, 62, 72, 74, 75, 80, 82, 84, 86, 88, 94, 104, 106, 108, 118, 120, 122, 132, 134, 136, 140, 142, 146, 147, 152, 156, 158, 166, 168, 178, 184, 192, 194, 202, 204, 206, 214, 218, 220, 226, 228, 232

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Invented by the HR concept formation program.

			10 = 5 * tau(5).

Antti Karttunen, Table of n, a(n) for n = 1..11329
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics, 1998-1999. [Wayback Machine link]

Cf. A000005, A001747, A036434.

Range of A038040.

q[k_] := AnyTrue[Divisors[k], # * DivisorSigma[0, #] == k &]; Select[Range[250], q] (* Amiram Eldar, Feb 01 2025 *)

isok(n) = {for (k=1, n, if (k*numdiv(k) == n, return (1));); return (0);} \\ Michel Marcus, Dec 09 2014

up_to = 65536;
A036438list(up_to) = { my(v=vector(up_to), m = Map()); for(n=1,#v,mapput(m,n*numdiv(n),n)); my(k=0,u=0); while((k<#v)&&(u<#v), u++; if(mapisdefined(m,u), k++; v[k] = u)); vector(k,i,v[i]); };
v036438 = A036438list(up_to);
A036438(n) = v036438[n]; \\ Antti Karttunen, Jul 18 2020

cp's OEIS Frontend

A071681 Number of ways to represent the n-th prime as arithmetic mean of two other primes.

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A073703 Smallest prime p such that also p+prime(n)*2 is a prime.

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A143201 Product of distances between prime factors in factorization of n.

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A360678 Sum of the left half (inclusive) of the prime indices of n.

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A346703 Product of primes at odd positions in the weakly increasing list (with multiplicity) of prime factors of n.

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A283312 a(n) = smallest missing positive number, unless a(n-1) was a prime in which case a(n) = 2*a(n-1).

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A096014 a(n) = (smallest prime factor of n) * (least prime that is not a factor of n), with a(1)=2.

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