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.

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A018676 Divisors of 840.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

840 is a highly composite number: A002182(15)=840. - Reinhard Zumkeller, Jun 21 2010

Links

Crossrefs

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Programs

A178858 Divisors of 5040.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 21 2010

Keywords

Comments

5040 is a highly composite number: A002182(19)=5040;
the sequence is finite with A002183(19)=60 terms: a(60)=5040.

Links

Crossrefs

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178859, A178860, A178861, A178862, A178863, A178864.

Programs

A178859 Divisors of 7560.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42, 45, 54, 56, 60, 63, 70, 72, 84, 90, 105, 108, 120, 126, 135, 140, 168, 180, 189, 210, 216, 252, 270, 280, 315, 360, 378, 420, 504, 540, 630, 756, 840, 945, 1080, 1260, 1512, 1890
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 21 2010

Keywords

Comments

7560 is a highly composite number: A002182(20)=7560.
The sequence is finite with A002183(20)=64 terms: a(64)=7560.
Its primorial factorization is 6^2 * 210 and its representing polynomial p(x) of degree 6 with x=2 is x^6 + 18x^5 + 118x^4 + 348x^3 + 457x^2 + 210x. - Carlos Eduardo Olivieri, May 02 2015

Links

Crossrefs

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178860, A178861, A178862, A178863, A178864.

Programs

A178860 Divisors of 10080.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 96, 105, 112, 120, 126, 140, 144, 160, 168, 180, 210, 224, 240, 252, 280, 288, 315, 336, 360, 420, 480, 504, 560, 630, 672, 720, 840, 1008
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 21 2010

Keywords

Comments

10080 is a highly composite number: A002182(21)=10080.
The sequence is finite with A002183(21)=72 terms: a(72)=10080.

Links

Crossrefs

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178861, A178862, A178863, A178864.

Programs

A178861 Divisors of 15120.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 60, 63, 70, 72, 80, 84, 90, 105, 108, 112, 120, 126, 135, 140, 144, 168, 180, 189, 210, 216, 240, 252, 270, 280, 315, 336, 360, 378, 420, 432, 504, 540, 560, 630
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 21 2010

Keywords

Comments

15120 is a highly composite number: A002182(22)=15120;
the sequence is finite with A002183(22)=80 terms: a(80)=15120.
15120 is the smallest number with 80 divisors; 18480 is the next smallest; there are 84 such numbers less than 100,000. - Harvey P. Dale, Dec 17 2013

Links

Crossrefs

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178862, A178863, A178864.

Programs

A178862 Divisors of 20160.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 32, 35, 36, 40, 42, 45, 48, 56, 60, 63, 64, 70, 72, 80, 84, 90, 96, 105, 112, 120, 126, 140, 144, 160, 168, 180, 192, 210, 224, 240, 252, 280, 288, 315, 320, 336, 360, 420, 448, 480, 504, 560, 576
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 21 2010

Keywords

Comments

20160 is a highly composite number: A002182(23)=20160.
The sequence is finite with A002183(23)=84 terms: a(84)=20160.

Links

Crossrefs

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178863, A178864.

Programs

A178863 Divisors of 25200.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 28, 30, 35, 36, 40, 42, 45, 48, 50, 56, 60, 63, 70, 72, 75, 80, 84, 90, 100, 105, 112, 120, 126, 140, 144, 150, 168, 175, 180, 200, 210, 225, 240, 252, 280, 300, 315, 336, 350, 360, 400, 420, 450, 504
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 21 2010

Keywords

Comments

25200 is a highly composite number: A002182(24)=25200;
the sequence is finite with A002183(24)=90 terms: a(90)=25200.

Links

Crossrefs

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178864.

Programs

A178877 Divisors of 1260.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 21 2010

Keywords

Comments

1260 is a highly composite number: A002182(16)=1260;
the sequence is finite with A002183(16)=36 terms: a(36)=1260.

Links

Crossrefs

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Programs

A178878 Divisors of 1680.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60, 70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680
Offset: 1

Views

Author

Reinhard Zumkeller, Jun 21 2010

Keywords

Comments

1680 is a highly composite number: A002182(17)=1680;
the sequence is finite with A002183(17)=40 terms: a(40)=1680.

Links

Crossrefs

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Programs

A097621 In canonical prime factorization of n replace p^e with its index in A000961.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 7, 8, 10, 9, 12, 10, 12, 15, 11, 12, 16, 13, 20, 18, 18, 14, 21, 15, 20, 16, 24, 17, 30, 18, 19, 27, 24, 30, 32, 20, 26, 30, 35, 21, 36, 22, 36, 40, 28, 23, 33, 24, 30, 36, 40, 25, 32, 45, 42, 39, 34, 26, 60, 27, 36, 48, 28, 50, 54, 29, 48, 42, 60, 30, 56, 31
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 17 2004

Keywords

Comments

The definition of the sequence has been corrected, given that it uses A095874, the indices in the list A000961 of "powers of primes" starting with A000961(1) = 1, rather than A322981, index of p^e in the list of prime powers A246655, as written in the original definition. See A333235 for the variant of this sequence which uses A322981 and A246655 instead, maybe the more natural choice given that the factorization of integers consists of prime powers > 1. - M. F. Hasler, Jun 15 2021

Examples

			n=600 = 2^3 * 3 * 5^2 -> A095874(8)*A095874(3)*A095874(25) = 7 * 3 * 15 = 315.

Links

Crossrefs

Cf. A000961 (powers of primes), A246655 (prime powers), A003963, A018266, A095874 (index of n = p^e in A000961).
Cf. A097622, A097623, A097624.
Cf. A322981 (index of n = p^e in A246655), A333235 (variant of this sequence).

Programs

  • Maple
    N:= 1000: # to get a(1) to a(N)
Primes:= select(isprime,[2, seq(2*i+1,i=1..(N-1)/2)]):
PP:= sort([1,seq(seq(p^k, k=1..floor(log[p](N))),p=Primes)]):
for n from 1 to nops(PP) do B[PP[n]]:= n od:
seq(mul(B[f[1]^f[2]],f=ifactors(n)[2]),n=1..N); # Robert Israel, Sep 02 2015
  • Mathematica
    pp = Select[Range@100, Length[FactorInteger[#]] == 1 &]; a = Table[Times @@ (Position[pp, #][[1, 1]] & /@ (#[[1]]^#[[2]] & /@ FactorInteger[n])), {n, 73}] (* Ivan Neretin, Sep 02 2015 *)
  • PARI
    f(n) = if(isprimepower(n), sum(i=1, logint(n, 2), primepi(sqrtnint(n, i)))+1, n==1); \\ A095874
a(n) = my(fr=factor(n)); for (k=1, #fr~, fr[k,1] = f(fr[k,1]^fr[k,2]); fr[k,2] = 1); factorback(fr); \\ Michel Marcus, May 29 2021
A097621(n)=vecprod([A095874(f[1]^f[2])|f<-factor(n)~]) \\ M. F. Hasler, Jun 15 2021
  • Python
    from math import prod
from sympy import primepi, integer_nthroot, factorint
def A097621(n): return prod(1+int(primepi(m:=p**e)+sum(primepi(integer_nthroot(m,k)[0]) for k in range(2,m.bit_length()))) for p,e in factorint(n).items()) # Chai Wah Wu, Jan 19 2025

Formula

Multiplicative with: p^e -> A095874(p^e), p prime.
a(A000961(n)) = n; a(a(n)) = A097622(n); a(a(a(n))) = A097623(n);
a(n) <= n; a(n) = n iff 60 mod n = 0: a(A018266(n)) = A018266(n);
a(A097624(n)) = n and a(m) <> n for n < A097624(n).

Extensions

Definition corrected by M. F. Hasler, Jun 16 2021
Example corrected by Ray Chandler, Jun 30 2021
Previous Showing 11-20 of 22 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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