A071904 -id:A071904 - cp's OEIS frontend

A162022 Smallest prime factor of n-th odd composite integers A071904.

3, 3, 3, 5, 3, 3, 5, 3, 3, 7, 3, 5, 3, 3, 5, 3, 3, 7, 3, 5, 3, 7, 3, 5, 3, 3, 3, 5, 3, 7, 11, 3, 5, 3, 7, 3, 3, 11, 5, 3, 3, 5, 3, 7, 3, 13, 3, 5, 3, 3, 5, 11, 3, 3, 3, 7, 5, 3, 11, 3, 5, 7, 3, 13, 3, 3, 5, 3, 3, 5, 13, 3, 11, 3, 7, 3, 5, 3, 3, 5, 3, 3, 7, 17, 3, 5, 3, 13, 7, 3, 5, 3, 3, 11, 3, 17, 5, 3, 7, 3

Offset: 1

Zak Seidov, Jun 25 2009

nonn

Records are for n's such that A071904(n) = squares of primes.

a(n) = A020639(A071904(n)). [Reinhard Zumkeller, Oct 10 2011]

			A071904(1)=9, hence a(1)=3, A071904(4)=25, hence a(4)=5.

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

Cf. A014076, A071904.

nn=501;With[{ci=Complement[Range[9,nn,2],Prime[Range[PrimePi[nn]]]]}, FactorInteger[ #][[1,1]]&/@ci] (* Harvey P. Dale, Nov 30 2012 *)

from sympy import primepi, primefactors
def A162022(n):
    if n == 1: return 3
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return min(primefactors(m)) # Chai Wah Wu, Jul 31 2024

Corrected example a(4)=5 Francesco Antoni (francesco_antoni(AT)yahoo.com), Aug 04 2010

A164510 First differences of A071904 (Odd composite numbers).

Original entry on oeis.org

6, 6, 4, 2, 6, 2, 4, 6, 4, 2, 4, 2, 6, 2, 4, 6, 2, 4, 4, 2, 4, 2, 2, 4, 6, 6, 4, 2, 2, 2, 2, 2, 4, 4, 2, 6, 2, 2, 2, 6, 2, 4, 2, 4, 4, 2, 4, 2, 6, 2, 2, 2, 6, 6, 2, 2, 2, 2, 4, 2, 2, 2, 2, 4, 6, 4, 2, 6, 2, 2, 2, 4, 2, 4, 2, 4, 2, 6, 2, 4, 6, 2, 2, 2, 4, 2, 2, 2, 2, 2, 4, 6, 4, 2, 2, 2, 2, 2, 4, 2, 4, 2, 2, 2, 6

Offset: 1

Zak Seidov, Aug 14 2009

Are all terms <=6?
This is A067970 without its first term. [R. J. Mathar, Aug 17 2009]
Yes, all terms are at most 6. For a value of 8, we have to have p, p+2, p+4 all prime, and this is possible only for p=3. As a result, 1 would have to be an odd composite number, which it is not. Therefore all terms are <=6. [J. Lowell, Aug 17 2009]

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Joel E. Cohen and Dexter Senft, Gaps of size 2, 4, and (conditionally) 6 between successive odd composite numbers occur infinitely often, Notes Num. Theor. Disc. Math. (2025) Vol. 31, No. 3, 494-503. See p. 495.

Cf. A071904.

Differences@ Select[Range[1, 360, 2], CompositeQ] (* Michael De Vlieger, Aug 29 2025 *)

from sympy import primepi, isprime
def A164510(n):
    m, k = n, primepi(n+1) + n + (n+1>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    for d in range(2, 7, 2):
        if not isprime(m+d):
            return d # Chai Wah Wu, Aug 02 2024

A134229 Sum of the odd composites A071904 less than or equal to 1+2*10^n.

Original entry on oeis.org

45, 5975, 724952, 78848811, 8290599189, 857088171080, 87727442181950, 8924793000002668, 904326399306717962, 91382247906379573443, 9216035852504141985766, 928095944723211397518108, 93359489289526851301833406, 9383123076016179512328557690

Offset: 1

Enoch Haga, Oct 14 2007

nonn

A134228(n) + a(n) = A134230(n).

			a(1)=45 because that is the sum of the odd composites (9+15+21=45) less than or equal to 21.

Cf. A134228, A134230.

a(n) = sum{A071904(i): A071904(i) <= 1+2*10^n}. - R. J. Mathar, Oct 28 2007

Better definition from R. J. Mathar, Oct 28 2007

a(9)-a(14) from Hiroaki Yamanouchi, Jul 06 2014

A256252 Number of successive odd noncomposite numbers A006005 and number of successive odd composite numbers A071904, interleaved.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Mar 30 2015

nonn

See also A256253 and A256262 which contain similar diagrams.

			Consider an irregular array in which the odd-indexed rows list successive odd noncomposite numbers (A006005) and the even-indexed rows list successive odd composite numbers (A071904), in the sequence of odd numbers (A005408), as shown below:
1, 3, 5, 7;
9;
11, 13;
15;
17; 19;
21,
23;
25, 27;
39, 31;
...
a(n) is the length of the n-th row.
.
Illustration of the first 16 regions of the diagram of the symmetric representation of odd noncomposite numbers A006005 and odd composite numbers A071904:
.            _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
.           |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _  |   31
.           |_ _ _ _ _ _ _ _ _ _ _ _ _ _  | |   29
.           | | |_ _ _ _ _ _ _ _ _ _ _  | | |   23
.           | | | |_ _ _ _ _ _ _ _ _  | | | |   19
.           | | | |_ _ _ _ _ _ _ _  | | | | |   17
.           | | | | |_ _ _ _ _ _  | | | | | |   13
.           | | | | |_ _ _ _ _  | | | | | | |   11
.           | | | | | |_ _ _  | | | | | | | |    7
.           | | | | | |_ _  | | | | | | | | |    5
.           | | | | | |_  | | | | | | | | | |    3
.   A071904 | | | | | |_|_|_|_| | | | | | | |    1
.      9    | | | | |_ _ _ _ _|_|_| | | | | | A006005
.     15    | | | |_ _ _ _ _ _ _ _|_|_| | | |
.     21    | | |_ _ _ _ _ _ _ _ _ _ _|_| | |
.     25    | |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
.     27    |_ _ _ _ _ _ _ _ _ _ _ _ _ _|_|_|
.
a(n) is also the length of the n-th boundary segment in the zig-zag path of the above diagram, between the two types of numbers, as shown below for n = 1..9:
.                      _ _ _ _
.                             |_ _
.                                 |_ _
.                                     |_
.                                       |
.                                       |_ _
.
The sequence begins:      4,1,2,1,2,1,1,2,2,...
.

Cf. A005408, A006005, A071904, A256253, A256262.

lista(nn) = {my(nb = 1, isc = 0); forstep (n=3, nn, 2, if (bitxor(isc, isprime(n)), nb++, print1(nb, ", "); nb = 1; isc = ! isc););} \\ Michel Marcus, May 25 2015

a(n) = A256253(n+1), n >= 2.

A280285 Number of partitions of n into odd composite numbers (A071904).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 2, 0, 0, 2, 0, 0, 2, 1, 1, 3, 0, 0, 3, 1, 0, 4, 1, 1, 5, 1, 0, 5, 2, 2, 6, 2, 1, 8, 3, 1, 8, 3, 2, 11, 3, 2, 12, 5, 4, 13, 5, 3, 16, 8, 4, 18, 7, 6, 22, 9, 7, 24, 12, 9, 28, 12, 9, 33, 18, 11, 36, 18, 14, 45, 22, 16, 48, 26, 22, 54, 29, 23, 66, 38

Offset: 0

Ilya Gutkovskiy, Dec 31 2016

nonn

			a(36) = 3 because we have [27, 9], [21, 15] and [9, 9, 9, 9].

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Composite Number
Index entries for related partition-counting sequences

Cf. A002095, A002808, A023895, A071904, A204389, A280287.

a:= proc(n) option remember; `if`(n=0, 1, add(add(
      `if`(d>1 and d::odd and not isprime(d), d, 0),
       d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 31 2016

nmax = 100; CoefficientList[Series[(1 - x)/(1 - x^2) Product[(1 - x^(2 k)) (1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: ((1 - x)/(1 - x^2))*Product_{k>=1} (1 - x^(2*k))*(1 - x^prime(k))/(1 - x^k).

A281681 a(n) = A055396(A071904(n)) - 1.

Original entry on oeis.org

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

Offset: 1

Enrique Navarrete, Jan 26 2017

nonn

The sequence measures, in a sense, inversions in remainders of odd numbers upon factoring out their largest divisors (see A281680).
In A281680, we have A281680(4) = A281680(7) = A281680(10) = 3 (and there will be infinitely many 1's to the right after each one of them), so there is why a(1)=a(2)=a(3)=1. Then we have A281680(12) = 5 (and there will be infinitely many 1's and 3's to the right), so that's why a(4) = 2, and so forth. I used 1,2,3,... here to represent these inversions, but any other symbols could have been used.
Entries correspond to the position of the lowest prime factor of the odd composites, with prime=3 being position 1. - Bill McEachen, Jan 28 2018

Bill McEachen, Table of n, a(n) for n = 1..10000

Cf. A055396, A071904, A281680, A162022.

genit(maxx)={forcomposite(i5=9,maxx,if(i5%2==0,next);ptr=0;forprime(x=3,maxx,ptr+=1;if(i5%x==0,print1(ptr,",");break)));} \\ Bill McEachen, Jan 28 2018

from sympy import primepi, primefactors
def A281681(n):
    if n == 1: return 1
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return primepi(min(primefactors(m)))-1 # Chai Wah Wu, Aug 02 2024

Name changed by Robert Israel, Aug 03 2020

A292597 a(1) = 1; for n > 1, a(n) = c(n) + 2*a(floor(n/2)), where c(n) is the characteristic function of odd composites, A071904.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 4, 8, 9, 8, 8, 8, 8, 8, 9, 16, 16, 18, 18, 16, 17, 16, 16, 16, 17, 16, 17, 16, 16, 18, 18, 32, 33, 32, 33, 36, 36, 36, 37, 32, 32, 34, 34, 32, 33, 32, 32, 32, 33, 34, 35, 32, 32, 34, 35, 32, 33, 32, 32, 36, 36, 36, 37, 64, 65, 66, 66, 64, 65, 66, 66, 72, 72, 72, 73, 72, 73, 74, 74, 64, 65, 64, 64, 68, 69, 68, 69, 64, 64, 66, 67

Offset: 1

Antti Karttunen, Sep 27 2017

nonn

1-bits in base-2 expansion of a(n) indicate the positions of odd nonprimes in the sequence [n, floor(n/2), floor(n/4), ..., 1].

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to binary expansion of n

Cf. A000035, A010051, A014076, A071904, A292596, A292598.

a(1) = 1; for n > 1, a(n) = (A000035(n)*(1-A010051(n))) + 2*a(floor(n/2)).

For all n >= 1, a(n) + A292596(n) = n.

A276662 Iterative procedure in A316941 applied to the odd composite numbers (A071904) (a(n) = -1 if no prime is ever reached).

Original entry on oeis.org

311, 1129, 37, 773, 313, 311, 1129, 313, 3119014487, 31079, 317, 773, 1129, 3110647, 3103819425079, 310397, 5113, 31079, 3109, 3137, 310361, 31259, 331, 36389, 191176757654383, 31063, 337, 523, 324941, 31393, 127139, 33769, 31034567124791, 32369, 719, 5623, 347, 3371, 131777, 349, 31039, 34412909

Offset: 1

Bill McEachen, Sep 11 2016

a(n) = A316941(A071904).

			The first entry is from 9 = 3*3. 33 = 3*11, and 311 is prime.
A longer 10 step progression is a(9) from 45. Specifically, 45=3*15 concatenating to 315=3*105 concatenating to 3105=3*1035 concatenating to 31035=3*10345 concatenating to 310345=5*62069 concatenating to 562069=41*13709 concatenating to 4113709=19*216511 concatenating to 19216511=17*1130383 concatenating to 171130383 = 3*57043461 concatenating to 357043461=3*119014487 concatenating to 3119014487 which is prime. a(9) then is 3119014487.

Cf. A037274, A071904, A316941.

Map[NestWhile[Function[n, FromDigits@ Flatten@ IntegerDigits@ {#, n/#} &[FactorInteger[n][[1, 1]]]], #, ! PrimeQ@ # &] &, Select[Range[9, 157, 2], CompositeQ]] (* Michael De Vlieger, Sep 13 2016 *)

genit(iend)={i5=9;while(i5<=iend,n=i5;while(isprime(n),n+=2);i5=n;endless=0;while(endless<99999,dun=0;z=divisors(n);
a=z[2];b=n/a;k=length(digits(b));q=a*10^k+b;if(isprime(q),dun=1;break);endless+=1;n=q);if(dun>0,print1(q,","));i5+=2);}

from sympy import primepi, primefactors, factorint
def A276662(n):
    if n == 1: return 311
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    while sum((f:=factorint(m)).values()) > 1:
        m = int(str(p:=min(f))+str(m//p))
    return m # Chai Wah Wu, Aug 02 2024

Edited by N. J. A. Sloane, Oct 02 2016

A280287 Number of partitions of n into distinct odd composite numbers (A071904).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 0, 1, 1, 0, 2, 0, 1, 2, 1, 0, 3, 2, 1, 2, 1, 0, 3, 2, 1, 3, 2, 1, 5, 2, 1, 4, 3, 2, 4, 2, 1, 6, 4, 2, 6, 4, 3, 7, 4, 3, 6, 5, 4, 9, 5, 4, 10, 8, 4, 10, 6, 6, 12, 9, 5, 13, 9, 8, 14, 11, 7, 17, 13, 9, 16, 12, 11, 21

Offset: 0

Ilya Gutkovskiy, Dec 31 2016

nonn

			a(48) = 3 because we have [39, 9], [33, 15] and [27, 21].

Eric Weisstein's World of Mathematics, Composite Number
Index entries for related partition-counting sequences

Cf. A002095, A002808, A096258, A023895, A071904, A204389, A280285.

nmax = 105; CoefficientList[Series[(1 + x^2)/(1 + x) Product[(1 + x^k)/((1 + x^(2 k)) (1 + x^Prime[k])), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: ((1 + x^2)/(1 + x))*Product_{k>=1} (1 + x^k)/((1 + x^(2*k))*(1 + x^prime(k))).

A283801 Concatenation of the first n odd composite numbers (A071904).

Original entry on oeis.org

9, 915, 91521, 9152125, 915212527, 91521252733, 9152125273335, 915212527333539, 91521252733353945, 9152125273335394549, 915212527333539454951, 91521252733353945495155, 9152125273335394549515557, 915212527333539454951555763, 91521252733353945495155576365

Offset: 1

XU Pingya, Mar 17 2017

There are 3 primes in the first 5028 terms of this sequence, see A283802.

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A071904, A089933, A132932.

bb[1]=9;bb[n_]:=bb[n]=Which[PrimeQ[bb[n-1]+2]==False,bb[n-1]+2,PrimeQ[bb[n-1]+4]==False,bb[n-1]+4,True,bb[n-1]+6];coc[n_]:=FromDigits[Flatten[IntegerDigits[Table[bb[k],{k,1,n}]]]];Table[coc[n],14]
f[n_] := Block[{oc = cc = 0, k = 2}, While[oc <= n, If[ OddQ@ k && !PrimeQ@ k, cc = cc*10^IntegerLength[k] +k; oc++]; k++]; cc]; Array[f, 14] (* Robert G. Wilson v, Mar 17 2017 *)

cp's OEIS Frontend

A162022 Smallest prime factor of n-th odd composite integers A071904.

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A164510 First differences of A071904 (Odd composite numbers).

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A134229 Sum of the odd composites A071904 less than or equal to 1+2*10^n.

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A256252 Number of successive odd noncomposite numbers A006005 and number of successive odd composite numbers A071904, interleaved.

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PARI

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A280285 Number of partitions of n into odd composite numbers (A071904).

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Maple

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A281681 a(n) = A055396(A071904(n)) - 1.

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PARI

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A292597 a(1) = 1; for n > 1, a(n) = c(n) + 2*a(floor(n/2)), where c(n) is the characteristic function of odd composites, A071904.

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A276662 Iterative procedure in A316941 applied to the odd composite numbers (A071904) (a(n) = -1 if no prime is ever reached).

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