cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A073255 Sum of divisors of n-th composite number.

Original entry on oeis.org

7, 12, 15, 13, 18, 28, 24, 24, 31, 39, 42, 32, 36, 60, 31, 42, 40, 56, 72, 63, 48, 54, 48, 91, 60, 56, 90, 96, 84, 78, 72, 124, 57, 93, 72, 98, 120, 72, 120, 80, 90, 168, 96, 104, 127, 84, 144, 126, 96, 144, 195, 114, 124, 140, 96, 168, 186, 121, 126, 224, 108, 132
Offset: 1

Views

Author

Labos Elemer, Jul 22 2002

Keywords

Examples

			First composite is 4, sigma[4]=1+2+4=7=a(1).

Links

Crossrefs

Cf. A000203, A002808, A035004.

Programs

  • Maple
    map(numtheory:-sigma, remove(isprime,[$4..100])); # Robert Israel, Aug 27 2018
  • Mathematica
    c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] Table[DivisorSigma[1, c[w]], {w, 1, 128}]
DivisorSigma[1,#]&/@(Select[Range[100],CompositeQ]) (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 10 2019 *)
  • PARI
    lista(nn) = forcomposite(c=1, nn, print1(sigma(c), ", ")); \\ Michel Marcus, Feb 21 2016

Formula

a(n) = A000203(A002808(n)).

A073256 a(n) = phi(n-th composite number).

Original entry on oeis.org

2, 2, 4, 6, 4, 4, 6, 8, 8, 6, 8, 12, 10, 8, 20, 12, 18, 12, 8, 16, 20, 16, 24, 12, 18, 24, 16, 12, 20, 24, 22, 16, 42, 20, 32, 24, 18, 40, 24, 36, 28, 16, 30, 36, 32, 48, 20, 32, 44, 24, 24, 36, 40, 36, 60, 24, 32, 54, 40, 24, 64, 42, 56, 40, 24, 72, 44, 60, 46, 72, 32, 42, 60
Offset: 1

Views

Author

Labos Elemer, Jul 22 2002

Keywords

Examples

			100th composite is 133; phi(133) = 108 = a(100).

Links

Crossrefs

Cf. A000010, A002808, A035004, A073255.

Programs

  • Mathematica
    c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x]; Table[EulerPhi[c[w]], {w, 1, 128}]
With[{nn=100},EulerPhi[#]&/@Complement[Range[2,nn], Prime[Range[ PrimePi[ nn]]]]] (* Harvey P. Dale, Apr 28 2014 *)
EulerPhi[Select[Range[100],CompositeQ]] (* Harvey P. Dale, Jul 05 2024 *)

Formula

a(n) = A000010(A002808(n)).

A158338 Composite numbers k such that k - number of divisors of k = prime.

Original entry on oeis.org

6, 15, 16, 21, 27, 33, 35, 51, 57, 65, 77, 87, 93, 105, 111, 135, 141, 143, 155, 161, 165, 177, 183, 185, 189, 201, 203, 215, 231, 237, 245, 267, 275, 285, 287, 321, 335, 341, 345, 357, 371, 375, 377, 393, 413, 425, 429, 437, 447, 453, 465, 471
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 16 2009, Nov 14 2009

Keywords

Comments

Subsequence of A067531. - Michel Marcus, Dec 22 2014

Examples

			6 is composite and has 4 divisors (1, 2, 3, 6); 6 - 4 = 2, which is prime, so 6 is in the sequence.
15 is composite and has 4 divisors (1, 3, 5, 15); 15 - 4 = 11, which is prime, so 15 is in the sequence.
16 is composite and has 5 divisors (1, 2, 4, 8, 16); 16 - 5 = 11, which is prime, so 16 is in the sequence.

Links

Crossrefs

Cf. A000005, A000040, A002808, A035004, A067531.

Programs

  • Magma
    [k:k in [1..500]|not IsPrime(k) and IsPrime(k-#Divisors(k))]; // Marius A. Burtea, Jul 16 2019
  • Mathematica
    Select[Range[500], CompositeQ[#] && PrimeQ[# - DivisorSigma[0, #]] &] (* Amiram Eldar, Jul 16 2019 *)

Extensions

Extended by R. J. Mathar, May 19 2010

A166281 Number of ordered factorizations of the nonprimes (A018252).

Original entry on oeis.org

1, 2, 3, 4, 2, 3, 8, 3, 3, 8, 8, 8, 3, 3, 20, 2, 3, 4, 8, 13, 16, 3, 3, 3, 26, 3, 3, 20, 13, 8, 8, 3, 48, 2, 8, 3, 8, 20, 3, 20, 3, 3, 44, 3, 8, 32, 3, 13, 8, 3, 13, 76, 3, 8, 8, 3, 13, 48, 8, 3, 44, 3, 3, 3, 20, 44, 3, 8, 3, 3, 3, 112, 8, 8, 26, 13, 20, 13, 3
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 10 2009

Keywords

Links

Crossrefs

Cf. A002033, A018252, A035004, A074206.

Programs

  • Mathematica
    f[1] = 1; f[n_] := f[n] = DivisorSum[n, f[#] &, # < n &]; f /@ Select[Range[100], ! PrimeQ[#] &] (* Amiram Eldar, May 01 2025 *)

Formula

a(n) = A002033(A018252(n)-1). - R. J. Mathar, Oct 14 2009
a(n) = A074206(A018252(n)). - Amiram Eldar, May 01 2025

Extensions

Entries checked by R. J. Mathar, Oct 14 2009
Name corrected by Amiram Eldar, May 01 2025

A166983 The n-th composite minus the number of its divisors.

Original entry on oeis.org

1, 2, 4, 6, 6, 6, 10, 11, 11, 12, 14, 17, 18, 16, 22, 22, 23, 22, 22, 26, 29, 30, 31, 27, 34, 35, 32, 34, 38, 39, 42, 38, 46, 44, 47, 46, 46, 51, 48, 53, 54, 48, 58, 57, 57, 61, 58, 62, 65, 62, 60, 70, 69, 70, 73, 70, 70, 76, 78, 72, 81, 82, 83, 80, 78, 87, 86, 89, 90, 91, 84
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 26 2009

Keywords

Examples

			a(1)=4-3=1, a(2)=6-4=2, a(3)=8-4=4.

Links

Crossrefs

Cf. A000005, A002808.

Programs

  • Mathematica
    With[{comps=Rest[Complement[Range[100],Prime[Range[PrimePi[100]]]]]}, #-DivisorSigma[0,#]&/@comps] (* Harvey P. Dale, Dec 16 2011 *)
#-DivisorSigma[0,#]&/@Select[Range[100],CompositeQ] (* Harvey P. Dale, Aug 20 2025 *)

Formula

a(n)=A002808(n)-A000005(A002808(n)) = A002808(n)-A035004(n+1).

Extensions

Formula and two entries corrected by R. J. Mathar, May 21 2010.

A167133 Primes of the form (number of prime factors of k-th composite) plus (number of divisors of k-th composite).

Original entry on oeis.org

5, 7, 5, 5, 7, 11, 11, 13, 11, 5, 13, 11, 11, 17, 11, 13, 11, 11, 17, 11, 11, 5, 7, 11, 11, 11, 11, 5, 11, 11, 23, 11, 11, 11, 11, 13, 17, 11, 13, 11, 11, 11, 11, 11, 23, 11, 17, 11, 11, 11, 11, 11, 11, 5, 11, 23, 11, 11, 11, 7, 11, 11, 11, 5, 11, 11, 11, 11, 17, 23, 11, 11, 11, 11
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 28 2009

Keywords

Comments

Contains every prime > 3 infinitely many times, as A000005(p^k)+A001222(p^k)=2*k+1 for prime p. - Robert Israel, Sep 30 2020

Examples

			a(1) = 2+3 = 5 (for 1st composite=4); a(2) = 3+4 = 7 (for 3rd composite=8).

Links

Crossrefs

Cf. A000005, A000040, A002808, A001222, A069346.
Cf. A035004, A062502.

Programs

  • Maple
    f:= proc(n) local F,x;
  if isprime(n) then return NULL fi;
  F:= ifactors(n)[2];
  x:= add(t[2],t=F) + mul(1+t[2],t=F);
  if isprime(x) then x fi
end proc:
map(f, [$4..1000]); # Robert Israel, Sep 30 2020

Extensions

Corrected and extended by R. J. Mathar, Oct 29 2009

A269065 Irregular triangle read by rows: row n lists divisors of n-th composite number.

Original entry on oeis.org

1, 2, 4, 1, 2, 3, 6, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 2, 3, 4, 6, 12, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 2, 3, 6, 9, 18, 1, 2, 4, 5, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 2, 3, 4, 6, 8, 12, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27, 1, 2, 4, 7, 14, 28, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 4, 8, 16, 32, 1, 3, 11, 33, 1, 2, 17, 34
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 21 2016

Keywords

Comments

Subsequence of A027750.
Row sums give A073255.
Right border gives A002808.

Examples

			Triangle begins:
1,  2,  4;
1,  2,  3,  6;
1,  2,  4,  8;
1,  3,  9;
1,  2,  5,  10;
1,  2,  3,  4,  6,  12;
1,  2,  7,  14;
1,  3,  5,  15
1,  2,  4,  8,  16;
1,  2,  3,  6,  9,  18;
1,  2,  4,  5,  10, 20;
1,  3,  7,  21;
1,  2,  11, 22;
1,  2,  3,  4,  6,  8,  12, 24;
1,  5,  25;
1,  2,  13, 26;
1,  3,  9,  27;
1,  2,  4,  7,  14, 28;
1,  2,  3,  5,  6,  10, 15, 30;
1,  2,  4,  8,  16, 32;
1,  3,  11, 33;
1,  2,  17, 34;
...

Links

Crossrefs

Cf. A002808, A027750, A035004 (row length), A133021, A133031, A138881.

Programs

  • Mathematica
    Flatten[Table[Divisors[Composite[n]], {n, 22}]]
  • PARI
    tabf(nn) =  forcomposite(c=1, nn, print(divisors(c), ", ")); \\ Michel Marcus, Feb 21 2016

A297217 Most common value of the number of divisors function among all composites up to composite(n) inclusive, or 0 if there is a tie.

Original entry on oeis.org

3, 0, 4, 0, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4
Offset: 1

Views

Author

Felix Fröhlich, Mar 02 2018

Keywords

Comments

Is a(n) = 4 for all n > 4?

Examples

			          n  |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 | 10 | 11 | 12
-------------------------------------------------------------------------
  A002808(n) |  4 |  6 |  8 |  9 | 10 | 12 | 14 | 15 | 16 | 18 | 20 | 21
-------------------------------------------------------------------------
A035004(n+1) |  3 |  4 |  4 |  3 |  4 |  6 |  4 |  4 |  5 |  6 |  6 |  4
-------------------------------------------------------------------------
        a(n) |  3 |  0 |  4 |  0 |  4 |  4 |  4 |  4 |  4 |  4 |  4 |  4

Crossrefs

Cf. A002808, A035004.

Programs

  • PARI
    composite(n) = my(i=0); forcomposite(c=1, , i++; if(i==n, return(c)))
mcv(v) = my(w=vecsort(v, , 8), count=vector(#w), ind=0, i=0); for(x=1, #w, for(y=1, #v, if(w[x]==v[y], count[x]++))); for(k=1, #count, if(count[k]==vecmax(count), ind=k; i++)); if(i > 1, return(0), return(w[ind]))
a(n) = my(v=[]); for(k=1, n, v=concat(v, numdiv(composite(k)))); mcv(v)

A158340 Composite numbers k such that (number of prime factors of k, counted with multiplicity) + (number of divisors of k) is a prime.

Original entry on oeis.org

4, 8, 9, 25, 27, 30, 32, 36, 42, 49, 64, 66, 70, 72, 78, 100, 102, 105, 108, 110, 114, 121, 125, 130, 138, 154, 165, 169, 170, 174, 180, 182, 186, 190, 195, 196, 200, 222, 225, 230, 231, 238, 243, 246, 252, 255, 256, 258, 266, 273, 282, 285, 286, 289, 290, 300
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Mar 16 2009, Nov 14 2009

Keywords

Examples

			4 is a term: 4 = 2*2 has 2 prime factors (counted with multiplicity) and 3 divisors (1, 2, and 4), and 2 + 3 = 5 (a prime).
8 is a term: 8 = 2*2*2 has 3 prime factors and 4 divisors (1, 2, 4, and 8), and 3 + 4 = 7 (a prime).
9 is a term: 9 = 3*3 has 2 prime factors and 3 divisors (1, 3, and 9), and 2 + 3 = 5 (a prime).

Crossrefs

Cf. A000040, A002808, A035004.

Extensions

Corrected (148 replaced with 138) by R. J. Mathar, May 19 2010
Showing 1-9 of 9 results.

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