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 1-4 of 4 results.

A276075 a(1) = 0, a(n) = (e1*i1! + e2*i2! + ... + ez*iz!) for n = prime(i1)^e1 * prime(i2)^e2 * ... * prime(iz)^ez, where prime(k) is the k-th prime, A000040(k).

Original entry on oeis.org

0, 1, 2, 2, 6, 3, 24, 3, 4, 7, 120, 4, 720, 25, 8, 4, 5040, 5, 40320, 8, 26, 121, 362880, 5, 12, 721, 6, 26, 3628800, 9, 39916800, 5, 122, 5041, 30, 6, 479001600, 40321, 722, 9, 6227020800, 27, 87178291200, 122, 10, 362881, 1307674368000, 6, 48, 13, 5042, 722, 20922789888000, 7, 126, 27, 40322, 3628801, 355687428096000, 10, 6402373705728000, 39916801, 28, 6, 726, 123
Offset: 1

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Author

Antti Karttunen, Aug 18 2016

Keywords

Comments

Additive with a(p^e) = e * (PrimePi(p)!), where PrimePi(n) = A000720(n).
a(3181) has 1001 decimal digits. - Michael De Vlieger, Dec 24 2017

Links

Crossrefs

Cf. A000040, A000142, A000720, A002110, A007489, A019565, A028234, A033312, A055396, A059590, A067029, A076954, A206296, A260443, A276080, A276081.
Left inverse of A276076.
Cf. also A048675.

Programs

  • Mathematica
    Array[If[# == 1, 0, Total[FactorInteger[#] /. {p_, e_} /; p > 1 :> e PrimePi[p]!]] &, 66] (* Michael De Vlieger, Dec 24 2017 *)
  • Python
    from sympy import factorint, factorial as f, primepi
def a(n):
    F=factorint(n)
    return 0 if n==1 else sum(F[i]*f(primepi(i)) for i in F)
print([a(n) for n in range(1, 121)]) # Indranil Ghosh, Jun 21 2017

Formula

a(1) = 0; for n > 1, a(n) = a(A028234(n)) + (A067029(n) * A000142(A055396(n))).
Other identities.
For all n >= 0:
a(A276076(n)) = n.
a(A002110(n)) = A007489(n).
a(A019565(n)) = A059590(n).
a(A206296(n)) = A276080(n).
a(A260443(n)) = A276081(n).
For all n >= 1:
a(A000040(n)) = n! = A000142(n).
a(A076954(n-1)) = A033312(n).

A276081 a(n) = A276075(A260443(n)).

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 8, 9, 24, 11, 14, 13, 30, 17, 32, 33, 120, 35, 38, 25, 54, 27, 44, 43, 144, 47, 62, 49, 150, 65, 152, 153, 720, 155, 158, 73, 174, 63, 92, 79, 264, 81, 98, 71, 198, 87, 188, 187, 840, 191, 206, 109, 294, 111, 212, 199, 864, 215, 302, 217, 870, 305, 872, 873, 5040, 875, 878, 313, 894, 231, 332, 247, 984, 237, 266, 155, 438, 171
Offset: 0

Views

Author

Antti Karttunen, Aug 18 2016

Keywords

Links

Crossrefs

Cf. A000142, A002487, A260443, A276075.
Cf. also A276080.

Programs

  • Python
    from sympy import factorint, factorial as f, prime, primepi
from operator import mul
from functools import reduce
def a003961(n):
    F=factorint(n)
    return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
def a260443(n): return n + 1 if n<2 else a003961(a260443(n//2)) if n%2==0 else a260443((n - 1)//2)*a260443((n + 1)//2)
def a276075(n):
    F=factorint(n)
    return 0 if n==1 else sum([F[i]*f(primepi(i)) for i in F])
def a(n): return a276075(a260443(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 21 2017
  • Scheme
    (define (A276081 n) (A276075 (A260443 n)))
;; A more practical standalone program, that uses memoization-macro definec:
(define (A276081 n) (sum_factorials_times_elements_in (A260443as_index_lists n)))
(definec (A260443as_index_lists n) (cond ((zero? n) (list)) ((= 1 n) (list 1)) ((even? n) (cons 0 (A260443as_index_lists (/ n 2)))) (else (add_two_lists (A260443as_index_lists (/ (- n 1) 2)) (A260443as_index_lists (/ (+ n 1) 2))))))
(define (add_two_lists nums1 nums2) (let ((len1 (length nums1)) (len2 (length nums2))) (cond ((< len1 len2) (add_two_lists nums2 nums1)) (else (map + nums1 (append nums2 (make-list (- len1 len2) 0)))))))
(define (sum_factorials_times_elements_in nums) (let loop ((s 0) (nums nums) (i 2) (f 1)) (cond ((null? nums) s) (else (loop (+ s (* (car nums) f)) (cdr nums) (+ 1 i) (* i f))))))

Formula

a(n) = A276075(A260443(n)).

A370511 Expansion of Sum_{k>=0} k! * ( x/(1-x^3) )^k.

Original entry on oeis.org

1, 1, 2, 6, 25, 124, 738, 5137, 40926, 367236, 3664321, 40241168, 482282700, 6263450401, 87618831730, 1313438757210, 21003941166601, 356910563528412, 6422026243338846, 121980432505328545, 2438957859604373534, 51206341993873093608, 1126314833020691642497
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2024

Keywords

Crossrefs

Cf. A001339, A276080.
Cf. A358493.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x/(1-x^3))^k))
  • PARI
    a(n) = sum(k=0, n\3, (n-3*k)!*binomial(n-2*k-1, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (n-3*k)! * binomial(n-2*k-1,k).

A370669 Expansion of Sum_{k>=0} k! * ( x/(1+x^2) )^k.

Original entry on oeis.org

1, 1, 2, 5, 20, 103, 630, 4475, 36232, 329341, 3320890, 36787889, 444125628, 5803850515, 81625106990, 1229298774647, 19738870726160, 336627732586105, 6076590994501938, 115752541255203869, 2320456607696181220, 48833227436258924671, 1076420625931284514342
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2024

Keywords

Crossrefs

Cf. A000255, A370670.
Cf. A177249, A212580, A276080.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*(x/(1+x^2))^k))
  • PARI
    a(n) = sum(k=0, n\2, (-1)^k*(n-2*k)!*binomial(n-k-1, k));

Formula

a(n) = Sum_{k=0..floor(n/2)} (-1)^k * (n-2*k)! * binomial(n-k-1,k).
a(n) = n*a(n-1) + (n-4)*a(n-3) + a(n-4) for n > 4.
Showing 1-4 of 4 results.

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