A000040 -id:A000040 - cp's OEIS frontend

A058698 a(n) = p(P(n)), P = primes (A000040), p = partition numbers (A000041).

2, 3, 7, 15, 56, 101, 297, 490, 1255, 4565, 6842, 21637, 44583, 63261, 124754, 329931, 831820, 1121505, 2679689, 4697205, 6185689, 13848650, 23338469, 49995925, 133230930, 214481126, 271248950, 431149389, 541946240, 851376628, 3913864295, 5964539504, 11097645016

Offset: 1

N. J. A. Sloane, Dec 31 2000

nonn

Number of partitions of n-th prime. - Omar E. Pol, Aug 05 2011

			a(2) = 3 because the second prime is 3 and there are three partitions of 3: {1, 1, 1}, {1, 2}, {3}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..500

Cf. A058697.

Cf. A000040, A000041, A260798.

import Data.MemoCombinators (memo2, integral)
a058698 n = a058698_list !! (n-1)
a058698_list = map (pMemo 1) a000040_list where
   pMemo = memo2 integral integral p
   p _ 0 = 1
   p k m | m < k     = 0
         | otherwise = pMemo k (m - k) + pMemo (k + 1) m
-- Reinhard Zumkeller, Aug 09 2015

Table[PartitionsP[Prime[n]], {n, 30}] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)

a(n) = A000041(A000040(n)). - Omar E. Pol, Aug 05 2011

A246281 Numbers k for which A003961(k) < 2k; Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then product_{k >= 1} (p_{k+1})^(c_k) < 2n, where p_k indicates the k-th prime, A000040(k).

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 17, 19, 22, 23, 25, 26, 29, 31, 33, 34, 37, 38, 41, 43, 46, 47, 51, 53, 55, 58, 59, 61, 62, 65, 67, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 121, 122, 123, 127, 129, 131, 133, 134, 137, 139

Offset: 1

Antti Karttunen, Aug 24 2014

nonn

Numbers n such that A003961(n) < 2*n.
Numbers n such that A048673(n) <= n.
All primes (A000040) are members. (Cf. Bertrand's postulate).
All terms are deficient (in A005100). See A286385. - Antti Karttunen, Aug 27 2020

			1 is present, as 1 = empty product and 1 < 2.
2 = p_1 is in the sequence, as p_2 = 3 and 3/2 < 2.
4 = p_1 * p_1 is not a member, as p_2 * p_2 = 3*3 = 9, and 9/4 > 2.
22 = 2*11 = p_1 * p_5 is a member, as p_2 * p_6 = 39, and 39/22 < 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Wikipedia, Bertrand's postulate
Index entries for sequences computed from indices in prime factorization

Complement: A246282.
Union of A246351 and A048674.
Subsequence: A000040.
Subsequence of A005100.
Positions of zeros in A252742, in A336836, and in A337345.
Positions of negative terms in A252748.
Cf. A003961, A048673, A246361, A286385.

A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i,1] = nextprime(f[i,1]+1)); factorback(f);
isA246281(n) = (A003961(n) < (n+n));
n = 0; i = 0; while(i < 10000, n++; if(isA246281(n), i++; write("b246281.txt", i, " ", n)));

;; With Antti Karttunen's IntSeq-library.
(define A246281 (MATCHING-POS 1 1 (lambda (n) (<= (A048673 n) n))))

A new shorter version of name prepended by Antti Karttunen, Aug 27 2020

A037019 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 2^(p_1 - 1)3^(p_2 - 1)...A000040(k)^(p_k - 1).

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 64, 30, 36, 48, 1024, 60, 4096, 192, 144, 210, 65536, 180, 262144, 240, 576, 3072, 4194304, 420, 1296, 12288, 900, 960, 268435456, 720, 1073741824, 2310, 9216, 196608, 5184, 1260, 68719476736, 786432, 36864, 1680, 1099511627776

Offset: 1

Wouter Meeussen

This is an easy way to produce a number with exactly n divisors and it usually produces the smallest such number (A005179(n)). The references call n "ordinary" if A005179(n) = a(n) and "exceptional" or "extraordinary" otherwise. - David Wasserman, Jun 12 2002

			12 = 3*2*2, so a(12) = 2^2*3*5 = 60.

T. D. Noe, Table of n, a(n) for n = 1..1000
R. Brown, The minimal number with a given number of divisors, Journal of Number Theory 116 (2006) 150-158.
M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
Anna K. Savvopoulou and Christopher M. Wedrychowicz, On the smallest number with a given number of divisors, The Ramanujan Journal, 2015, Vol. 37, pp. 51-64.

Cf. A005179, A000040, A072066 (exceptional (or extraordinary) numbers).

Cf. A027746.

a037019 = product .
   zipWith (^) a000040_list . reverse . map (subtract 1) . a027746_row
-- Reinhard Zumkeller, Nov 25 2012

a:= n-> (l-> mul(ithprime(i)^(l[i]-1), i=1..nops(l)))(
        sort(map(i-> i[1]$i[2], ifactors(n)[2]), `>`)):
seq(a(n), n=1..60);  # Alois P. Heinz, Feb 28 2019

(Times@@(Prime[ Range[ Length[ # ] ] ]^Reverse[ #-1 ]))&@Flatten[ FactorInteger[ n ]/.{ a_Integer, b_}:>Table[ a, {b} ] ]

A037019(n,p=1)=prod(i=1,#f=Vecrev(factor(n)~),prod(j=1,f[i][2],(p=nextprime(p+1))^(f[i][1]-1))) \\ M. F. Hasler, Oct 14 2014

from math import prod
from sympy import factorint, prime
def a(n):
    pf = factorint(n, multiple=True)
    return prod(prime(i)**(pi-1) for i, pi in enumerate(pf[::-1], 1))
print([a(n) for n in range(1, 42)]) # Michael S. Branicky, Jul 24 2022

More terms from David Wasserman, Jun 12 2002

A128677 Least k>p such that (kp)^3 divides (p-1)^(kp)^2+1 for prime p = A000040(n).

Original entry on oeis.org

19, 41, 29, 23, 79, 41617, 20939, 47, 40427, 4093, 4441, 2543, 1033, 659, 2612032921, 394502321, 14958421, 17957, 569, 14747, 12641, 167, 174263, 100493, 285629

Offset: 2

Alexander Adamchuk, Mar 30 2007, Mar 31 2007, Apr 09 2007

For every prime p>2, p^3 divides (p-1)^(p^2)+1 and furthermore p divides all numbers n>1 such that n^3 divides (p-1)^(n^2)+1.
a(27)>10^15. - Max Alekseyev, Nov 30 2017
Some further terms: a(28)-a(36) = {857, 3271, 7243979, 509, 263, 43019, 38921, 2683, 312055091}. a(38)-a(43) = {7499, 88588425539, 9689, 359, 1087, 383}. a(45)-a(61) = {931417, 40597, 2111, 2677, 14983, 261061, 1302937, 479, 17935703, 503, 4227137, 39398453, 2153, 1627, 1109, 28663, 1699}. a(63)-a(69) = {1229, 1867, 78877, 500861, 1987, 62683, 2777}. a(71)-a(75) = {275884327, 719, 44041, 3122698559, 15161}. a(77)-a(80) = {907927, 202471, 5788837, 16361}.
a(n) <= A177996(n).
A000040(n) divides (a(n) - 1)/2. The quotients (a(n)-1)/2/A000040(n) are listed in A136374.

			a(2) = A127263(3)/3 = 57/3 = 19.

Cf. A127263, A128678, A128679, A128680, A128681, A128682, A128683, A128684, A128685, A136374.

a[n_] := Module[{p, k}, p = Prime[n]; k = p + 1;
   While[! Divisible[(p - 1)^(k p)^2 + 1, (k p)^3], k++]; k];
Table[a[n], {n, 2, 15}] (* Robert Price, Mar 23 2020 *)

a(n) = smallest prime divisor of (p-1)^(p^2)+1 other than p, where p=A000040(n).

a(16)-a(26), a(39), a(74) from Max Alekseyev, May 16 2010

A246377 Permutation of natural numbers: a(1) = 1, a(p_n) = 2a(n)+1, a(c_n) = 2a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n).

Original entry on oeis.org

1, 3, 7, 2, 15, 6, 5, 14, 4, 30, 31, 12, 13, 10, 28, 8, 11, 60, 29, 62, 24, 26, 9, 20, 56, 16, 22, 120, 61, 58, 63, 124, 48, 52, 18, 40, 25, 112, 32, 44, 27, 240, 21, 122, 116, 126, 57, 248, 96, 104, 36, 80, 17, 50, 224, 64, 88, 54, 23, 480, 121, 42, 244, 232, 252, 114, 59, 496, 192, 208, 125, 72, 49, 160, 34, 100

Offset: 1

Antti Karttunen, Aug 27 2014

nonn

This permutation is otherwise like Katarzyna Matylla's A135141, except that the role of even and odd numbers (or alternatively: primes and composites) has been swapped.
Because 2 is the only even prime, it implies that, apart from a(2)=3, odd numbers occur in odd positions only (along with many even numbers that also occur in odd positions).
This also implies that for each odd composite (A071904) there exists a separate infinite cycle in this permutation, apart from 9 and 15 which are in the same infinite cycle: (..., 23, 9, 4, 2, 3, 7, 5, 15, 28, 120, 82, 46, ...).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Inverse: A246378.
Cf. A000720, A010051, A065855, A071904, A246348, A246369, A246370.
Other related or similar permutations: A135141, A054429, A246201, A245703, A246376, A246379, A243071, A246681, A236854.
Differs from A237427 for the first time at n=19, where a(19) = 29, while A237427(19) = 62.

a(1) = 1, and for n > 1, if A010051(n) = 1 [i.e. when n is a prime], a(n)  = 1+(2*a(A000720(n))), otherwise a(n) = 2*a(A065855(n)).
As a composition of related permutations:
a(n) = A054429(A135141(n)).
a(n) = A135141(A236854(n)).
a(n) = A246376(A246379(n)).
a(n) = A246201(A245703(n)).
a(n) = A243071(A246681(n)). [For n >= 1].
Other identities.
For all n > 1 the following holds:
A000035(a(n)) = A010051(n). [Maps primes to odd numbers > 1, and composites to even numbers, in some order. Permutations A246379 & A246681 have the same property].

A024697 a(n) = p(1)p(n) + p(2)p(n-1) + ... + p(k)p(n+1-k), where k = [ (n+1)/2 ], p = A000040 = the primes.

Original entry on oeis.org

4, 6, 19, 29, 68, 94, 177, 231, 400, 484, 753, 903, 1340, 1552, 2157, 2489, 3352, 3784, 5013, 5515, 7052, 7758, 9773, 10575, 13076, 14076, 17023, 18339, 21876, 23414, 27715, 29437, 34570, 36500, 42335, 44731, 51560, 54198, 61955, 65051, 73700, 77402, 87293

Offset: 1

Clark Kimberling

nonn

a(n) = A025129(n) for even n. - M. F. Hasler, Apr 06 2014

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

Cf. A014342, A000040.

a024697 n = a024697_list !! (n-1)
a024697_list = f (tail a000040_list) [head a000040_list] 2 where
   f (p:ps) qs k = sum (take (div k 2) $ zipWith (*) qs $ reverse qs) :
                   f ps (p : qs) (k + 1)
-- Reinhard Zumkeller, Apr 07 2014

A024697:=n->sum( ithprime(k)*ithprime(n-k+1), k=1..(n+1)/2 ); seq(A024697(n), n=1..50); # Wesley Ivan Hurt, Apr 06 2014

Table[Sum[Prime[k] Prime[n - k + 1], {k, (n + 1)/2}], {n, 50}] (* Wesley Ivan Hurt, Apr 06 2014 *)

A024697(n)=sum(k=1, (n+1)\2, prime(k)*prime(n-k+1)) \\ M. F. Hasler, Apr 06 2014

Name edited and values double-checked by M. F. Hasler, Apr 06 2014

A256236 Smallest b > 1 such that the first n primes p (i.e., A000040(1)-A000040(n)) all satisfy b^(p-1) == 1 (mod p^2), i.e., smallest base b larger than 1 such that any member of the set of first n primes is a base-b Wieferich prime.

Original entry on oeis.org

5, 17, 449, 557, 19601, 132857, 4486949, 126664001, 2363321449, 5229752849, 2486195039249, 16250570614349, 83322586961893, 39699586259362801, 8042447016668335049, 449320365877347849601, 4376479338174582826793

Offset: 1

Felix Fröhlich, Mar 25 2015

There might be bases b where prime(n+1) is also a base-b Wieferich prime. This does not affect the membership of b in the sequence.
Are there any terms such that a(n) = a(n+1)?
Does b exist for all n?
All currently known terms satisfy a(n) >= A255901(n). Are there any terms such that a(n) < A255901(n)?
If it exists, a(12) > 6*10^12. - Robert Price, Oct 10 2019
a(n) <= prime(n)#^2+1 = A189409(n), since any prime p is a Wieferich prime in base k*p^2+1 for all k. - Jens Kruse Andersen, Dec 20 2020

			Values of bases b and the values of first Wieferich primes p to base b:
b             | p
-------------------------------------------------------------------------
5             | 2, 20771, 40487 ...
17            | 2, 3, 46021, 48947 ...
449           | 2, 3, 5, 1789 ...
557           | 2, 3, 5, 7, 23, 39829 ...
19601         | 2, 3, 5, 7, 11, 23, 47 ...
132857        | 2, 3, 5, 7, 11, 13, 73, 257 ...
4486949       | 2, 3, 5, 7, 11, 13, 17, 89, 197 ...
126664001     | 2, 3, 5, 7, 11, 13, 17, 19, 101, 2789 ...
2363321449    | 2, 3, 5, 7, 11, 13, 17, 19, 23 ...
5229752849    | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 881, 2246969 ...
2486195039249 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 ...

Cf. A255901.

b = 2; Table[While[fnd = True;
  For[i = 1, i <= n, i++,
   p = Prime[i];
   If[PowerMod[b, (p - 1), p^2] != 1 , fnd = False;  Break[]]];
b++; ! fnd]; b - 1, {n, 5}] (* Robert Price, Oct 10 2019 *)

a(n) = my(v=primes(n)); for(b=2, oo, for(k=1, #v, if(Mod(b, v[k]^2)^(v[k]-1)!=1, break, if(k==#v, return(b)))))

a(9)-a(11) from Robert Price, Oct 10 2019

a(12)-a(17) from Jens Kruse Andersen, Dec 28 2020

A080148 Positions of primes of the form 4*k+3 (A002145) among all primes (A000040).

Original entry on oeis.org

2, 4, 5, 8, 9, 11, 14, 15, 17, 19, 20, 22, 23, 27, 28, 31, 32, 34, 36, 38, 39, 41, 43, 46, 47, 48, 49, 52, 54, 56, 58, 61, 63, 64, 67, 69, 72, 73, 75, 76, 81, 83, 85, 86, 90, 91, 92, 93, 94, 95, 96, 99, 101, 103, 105, 107, 109, 111, 114, 115, 117, 118, 120, 124, 125, 128

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

It appears that a(n) = k such that binomial(prime(k),3) mod 2 = 1. See Maple code. - Gary Detlefs, Dec 06 2011
The above is correct (work mod 4). - Charles R Greathouse IV, Dec 06 2011
The asymptotic density of this sequence is 1/2 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

Zak Seidov, Table of n, a(n) for n = 1..10000

Almost complement of A080147 (1 is excluded from both).

Cf. A000040, A002145, A049084.

pos_of_primes_k_mod_n(300,3,4); # Given in A080147.
A080148 := proc(n)
        numtheory[pi](A002145(n)) ;
end proc:
seq(A080148(n),n=1..40) ; # R. J. Mathar, Dec 08 2011

Flatten[Position[Prime[Range[200]],?(IntegerQ[(#-3)/4]&)]] (* _Harvey P. Dale, Jun 06 2011 *)
Select[Range[135], Mod[Prime[#], 4] == 3 &] (* Amiram Eldar, Mar 01 2021 *)

i=0;forprime(p=2,1e3,i++;if(p%4==3,print1(i", "))) \\ Charles R Greathouse IV, Dec 06 2011

a(n) = A049084(A002145(n)). - R. J. Mathar, Oct 06 2008

A112049 a(n) = position of A112046(n) in A000040.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 5, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 6, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 4, 3, 1, 1, 2, 2, 1, 1

Offset: 1

Antti Karttunen, Aug 27 2005

nonn

A112051 gives the first positions of distinct new values in this sequence, that seem also to be the positions of the first occurrence of each n, and thus the positions of the records. Compare also to A084921. - Antti Karttunen, May 26 2017

Indranil Ghosh (terms 1..1000) & Antti Karttunen, Table of n, a(n) for n = 1..32768

Cf. A049084, A112046, A112051, A112060, A165471, A268829, A286465, A286466.

Cf. A286579 (ordinal transform).

a112046[n_]:=Block[{i=1},While[JacobiSymbol[i, 2n + 1]==1, i++]; i];a049084[n_]:=If[PrimeQ[n], PrimePi[n], 0]; Table[a049084[a112046[n]], {n, 102}] (* Indranil Ghosh, May 11 2017 *)

A112049(n) = for(i=1, (2*n), if((kronecker(i, (n+n+1)) < 1), return(primepi(i)))); \\ Antti Karttunen, May 26 2017

from sympy import jacobi_symbol as J, isprime, primepi
def a049084(n):
    return primepi(n) if isprime(n) else 0
def a112046(n):
    i=1
    while True:
        if J(i, 2*n + 1)!=1: return i
        else: i+=1
def a(n): return a049084(a112046(n))
print([a(n) for n in range(1, 103)]) # Indranil Ghosh, May 11 2017

a(n) = A049084(A112046(n)).

Unnecessary fallback-clause removed from the name by Antti Karttunen, May 26 2017

A242424 Bulgarian solitaire operation on partition list A112798: a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n).

Original entry on oeis.org

1, 2, 4, 3, 6, 6, 10, 5, 12, 9, 14, 10, 22, 15, 18, 7, 26, 20, 34, 15, 30, 21, 38, 14, 27, 33, 40, 25, 46, 30, 58, 11, 42, 39, 45, 28, 62, 51, 66, 21, 74, 50, 82, 35, 60, 57, 86, 22, 75, 45, 78, 55, 94, 56, 63, 35, 102, 69, 106, 42, 118, 87, 100, 13, 99, 70, 122, 65

Offset: 1

Antti Karttunen, May 13 2014

nonn

In "Bulgarian solitaire" a deck of cards or another finite set of objects is divided into one or more piles, and the "Bulgarian operation" is performed by taking one card from each pile, and making a new pile of them, which is added to the remaining set of piles. Essentially, this operation is a function whose domain and range are unordered integer partitions (cf. A000041) and which preserves the total size of a partition (the sum of its parts). This sequence is induced when the operation is implemented on the partitions as ordered by the list A112798.
Please compare to the definition of A122111, which conjugates the partitions encoded with the same system.
a(n) is even if and only if n is either a prime or a multiple of three.
Conversely, a(n) is odd if and only if n is a nonprime not divisible by three.

Martin Gardner, Colossal Book of Mathematics, Chapter 34, Bulgarian Solitaire and Other Seemingly Endless Tasks, pp. 455-467, W. W. Norton & Company, 2001.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Ethan Akin and Morton Davis, "Bulgarian solitaire", American Mathematical Monthly 92 (4): 237-250. (1985).
Wikipedia, Bulgarian solitaire
Index entries for sequences computed from indices in prime factorization

Row 1 of A243070 (table which gives successive "recursive iterates" of this sequence and converges towards A122111).
Fixed points: A002110 (primorial numbers).
Cf. A000041, A243051, A243072, A243073, A226062, A242422, A242423, A122111, A105560, A000040, A001222, A064989, A056239.

(define (A242424 n) (if (<= n 1) n (* (A000040 (A001222 n)) (A064989 n))))

a(1) = 1, a(n) = A000040(A001222(n)) * A064989(n) = A105560(n) * A064989(n).
a(n) = A241909(A243051(A241909(n))).
a(n) = A243353(A226062(A243354(n))).
a(A000079(n)) = A000040(n) for all n.
A056239(a(n)) = A056239(n) for all n.

cp's OEIS Frontend

A058698 a(n) = p(P(n)), P = primes (A000040), p = partition numbers (A000041).

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A246281 Numbers k for which A003961(k) < 2*k; Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then product_{k >= 1} (p_{k+1})^(c_k) < 2*n, where p_k indicates the k-th prime, A000040(k).

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A037019 Let n = p_1*p_2*...*p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 2^(p_1 - 1)*3^(p_2 - 1)*...*A000040(k)^(p_k - 1).

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A128677 Least k>p such that (kp)^3 divides (p-1)^(kp)^2+1 for prime p = A000040(n).

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A246377 Permutation of natural numbers: a(1) = 1, a(p_n) = 2*a(n)+1, a(c_n) = 2*a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n).

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A024697 a(n) = p(1)p(n) + p(2)p(n-1) + ... + p(k)p(n+1-k), where k = [ (n+1)/2 ], p = A000040 = the primes.

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A256236 Smallest b > 1 such that the first n primes p (i.e., A000040(1)-A000040(n)) all satisfy b^(p-1) == 1 (mod p^2), i.e., smallest base b larger than 1 such that any member of the set of first n primes is a base-b Wieferich prime.

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A246281 Numbers k for which A003961(k) < 2k; Numbers n such that if n = product_{k >= 1} (p_k)^(c_k), then product_{k >= 1} (p_{k+1})^(c_k) < 2n, where p_k indicates the k-th prime, A000040(k).

A037019 Let n = p_1p_2...p_k be the prime factorization of n, with the primes sorted in descending order. Then a(n) = 2^(p_1 - 1)3^(p_2 - 1)...A000040(k)^(p_k - 1).

A246377 Permutation of natural numbers: a(1) = 1, a(p_n) = 2a(n)+1, a(c_n) = 2a(n), where p_n = n-th prime = A000040(n), c_n = n-th composite number = A002808(n).