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 1-6 of 6 results.

A063956 Sum of unitary prime divisors of n.

Original entry on oeis.org

0, 2, 3, 0, 5, 5, 7, 0, 0, 7, 11, 3, 13, 9, 8, 0, 17, 2, 19, 5, 10, 13, 23, 3, 0, 15, 0, 7, 29, 10, 31, 0, 14, 19, 12, 0, 37, 21, 16, 5, 41, 12, 43, 11, 5, 25, 47, 3, 0, 2, 20, 13, 53, 2, 16, 7, 22, 31, 59, 8, 61, 33, 7, 0, 18, 16, 67, 17, 26, 14, 71, 0, 73, 39, 3, 19, 18, 18, 79, 5, 0
Offset: 1

Views

Author

Labos Elemer, Sep 04 2001

Keywords

Examples

			The prime divisors of 420 = 2^2 * 3 * 5 * 7. Among them, those that have exponent 1 (i.e., unitary prime divisors) are {3, 5, 7}, so a(420) = 3 + 5 + 7 = 15.

Links

Crossrefs

Cf. A008472, A034387, A034444, A056169, A056170, A056171, A056172, A063958.

Programs

  • Mathematica
    Table[DivisorSum[n, # &, And[PrimeQ@ #, GCD[#, n/#] == 1] &], {n, 81}] (* Michael De Vlieger, Feb 17 2019 *)
f[p_, e_] := If[e == 1, p, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)
Join[{0},Table[Total[Select[FactorInteger[n],#[[2]]==1&][[;;,1]]],{n,2,100}]] (* Harvey P. Dale, Jan 26 2025 *)
  • PARI
    { for (n=1, 1000, f=factor(n)~; a=0; for (i=1, length(f), if (f[2, i]==1, a+=f[1, i])); write("b063956.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009

Formula

a(n*m) = a(n) + a(m) - a(gcd(n^2, m)) - a(gcd(n, m^2)) for all n and m > 0 (conjecture). - Velin Yanev, Feb 17 2019
From Amiram Eldar, Jul 24 2024: (Start)
a(n) = A008472(n) - A063958(n).
Additive with a(p^e) = p is e = 1, and 0 otherwise. (End)

Extensions

Example clarified by Harvey P. Dale, Jan 26 2025

A345266 a(n) = Sum_{p|n, p prime} gcd(p,n/p).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 3, 2, 2, 1, 3, 5, 2, 3, 3, 1, 3, 1, 2, 2, 2, 2, 5, 1, 2, 2, 3, 1, 3, 1, 3, 4, 2, 1, 3, 7, 6, 2, 3, 1, 4, 2, 3, 2, 2, 1, 4, 1, 2, 4, 2, 2, 3, 1, 3, 2, 3, 1, 5, 1, 2, 6, 3, 2, 3, 1, 3, 3, 2, 1, 4, 2, 2, 2, 3, 1, 5, 2, 3, 2, 2, 2, 3, 1, 8, 4, 7, 1, 3, 1, 3, 3
Offset: 1

Views

Author

Wesley Ivan Hurt, Jun 13 2021

Keywords

Examples

			a(18) = Sum_{p|18} gcd(p,18/p) = gcd(2,9) + gcd(3,6) = 1 + 3 = 4.

Links

Crossrefs

Cf. A001221 (omega), A007947 (rad), A008472 (sopf), A345302.
Cf. A055155, A056169, A063958.

Programs

  • Mathematica
    Table[Sum[GCD[k, n/k] (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[n/k] + Floor[n/k]), {k, n}], {n, 100}]
  • PARI
    a(n) = my(f=factor(n), p); sum(k=1, #f~, p=f[k, 1]; gcd(p,n/p)); \\ Michel Marcus, Jun 16 2021
  • PARI
    A345266(n) = vecsum(apply(p->gcd(p,n/p), factor(n)[,1])); \\ Antti Karttunen, Nov 13 2021

Formula

a(p) = 1 for p prime.
From Wesley Ivan Hurt, Nov 21 2021: (Start)
a(n) = A056169(n) + A063958(n).
If n is squarefree, then a(n) = omega(n).
a(p^k) = p for primes p and k >= 2. (End)

Extensions

Data section extended up to 105 terms by Antti Karttunen, Nov 13 2021

A071327 Sum of the squared primes dividing n.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 4, 9, 0, 0, 4, 0, 0, 0, 4, 0, 9, 0, 4, 0, 0, 0, 4, 25, 0, 9, 4, 0, 0, 0, 4, 0, 0, 0, 13, 0, 0, 0, 4, 0, 0, 0, 4, 9, 0, 0, 4, 49, 25, 0, 4, 0, 9, 0, 4, 0, 0, 0, 4, 0, 0, 9, 4, 0, 0, 0, 4, 0, 0, 0, 13, 0, 0, 25, 4, 0, 0, 0, 4, 9, 0, 0, 4, 0, 0, 0, 4, 0, 9, 0
Offset: 1

Views

Author

Reinhard Zumkeller, May 18 2002

Keywords

Links

Crossrefs

Cf. A000196, A010051, A010052, A056170, A063958, A071326.

Programs

  • Mathematica
    Array[DivisorSum[#, # &, PrimeQ@ Sqrt@ # &] &, 91] (* Michael De Vlieger, Nov 18 2017 *)
  • PARI
    A071327(n) = { my(r); sumdiv(n,d,(issquare(d,&r)&&isprime(r)) * d); } \\ Antti Karttunen, Nov 19 2017

Formula

a(n) = Sum_{d|n} A010052(d)*A010051(A000196(d))*d. - Antti Karttunen, Nov 18 2017
G.f.: Sum_{k>=1} prime(k)^2 * x^(prime(k)^2) / (1 - x^(prime(k)^2)). - Ilya Gutkovskiy, Apr 06 2020
a(n) = Sum_{p^2|n} p^2. - Wesley Ivan Hurt, Feb 21 2022
Additive with a(p^e) = p^2 if e >= 2, and 0 otherwise. - Amiram Eldar, May 15 2025

A063960 Sum of non-unitary prime divisors of n!: sum of those prime divisors for which the exponent in the prime factorization exceeds 1.

Original entry on oeis.org

0, 0, 0, 2, 2, 5, 5, 5, 5, 10, 10, 10, 10, 17, 17, 17, 17, 17, 17, 17, 17, 28, 28, 28, 28, 41, 41, 41, 41, 41, 41, 41, 41, 58, 58, 58, 58, 77, 77, 77, 77, 77, 77, 77, 77, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 129, 129, 129, 129, 160, 160, 160, 160
Offset: 1

Views

Author

Labos Elemer, Sep 04 2001

Keywords

Comments

Sum of the prime numbers among the smallest parts of the partitions of n into two parts. For example, a(8)=5; the partitions of 8 into two parts are (7,1), (6,2), (5,3) and (4,4). The prime numbers among the smallest parts are 2 and 3, so 2 + 3 = 5. - Wesley Ivan Hurt, Nov 01 2017
Number of distinct rectangles with integer length and prime width such that L + W = n, W <= L. For a(14)=17; the rectangles are 2 X 12, 3 X 11, 5 X 9, and 7 X 7. The sum of the lengths are then 2+3+5+7 = 17. - Wesley Ivan Hurt, Nov 08 2017

Examples

			20! = (2^18)*(3^8)*(5^4)*(7^2)*11*13*17*19, the non-unitary prime divisors are {2, 3, 5, 7}, so a(20) = 2 + 3 + 5 + 7 = 17.

Links

Crossrefs

Cf. A010051, A034387, A034444, A056169, A056170, A056171, A056172, A063958.

Programs

  • Maple
    seq(add(j, j=select(isprime, [$1..iquo(n,2)])), n=1..65); # Peter Luschny, Nov 28 2022
  • Mathematica
    Join[{0,0,0},Table[Total[Transpose[Select[FactorInteger[n!], Last[#]>1&]][[1]]],{n,4,70}]] (* Harvey P. Dale, Jun 19 2013 *)
  • PARI
    { for (n=1, 1000, f=factor(n!)~; a=0; for (i=1, length(f), if (f[2, i]>1, a+=f[1, i])); write("b063960.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009

Formula

a(n) = Sum_{i=1..floor(n/2)} i * A010051(i). - Wesley Ivan Hurt, Oct 31 2017
a(n) = A034387(floor(n/2)) for n >= 2. - Georg Fischer, Nov 28 2022
a(n) = A063958(n!). - Amiram Eldar, Jul 24 2024

A064146 Sum of non-unitary prime divisors of binomial(n,floor(n/2)).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 0, 3, 5, 0, 2, 2, 2, 3, 3, 0, 2, 0, 2, 2, 2, 0, 2, 7, 7, 10, 10, 5, 5, 3, 3, 3, 5, 5, 7, 7, 7, 3, 5, 2, 2, 2, 2, 10, 10, 8, 10, 12, 12, 12, 12, 9, 9, 2, 2, 2, 2, 2, 2, 2, 2, 10, 10, 7, 9, 7, 9, 5, 5, 0, 2, 2, 2, 7, 7, 14, 14, 7, 9, 12, 12, 5, 5, 10, 10, 10, 10, 5, 5, 12, 12, 12
Offset: 1

Views

Author

Labos Elemer, Sep 11 2001

Keywords

Links

Crossrefs

Cf. A008472, A001405, A063958, A034444, A056169, A046098, A048243.

Programs

  • Maple
    a:= n-> add(`if`(i[2]>1, i[1], 0), i=ifactors(binomial(n, iquo(n,2)))[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 24 2018
  • Mathematica
    a[n_] := Sum[If[i[[2]] > 1, i[[1]], 0], {i, FactorInteger[ Binomial[n, Quotient[n, 2]]]}];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)
  • PARI
    a(n) = { my(f=factor(binomial(n, n\2))); sum(i=1, #f~, if (f[i, 2]>1, f[i,1])) } \\ Harry J. Smith, Sep 09 2009

Formula

a(n) = A063958(A001405(n)).

A333842 G.f.: Sum_{k>=1} k * x^(prime(k)^2) / (1 - x^(prime(k)^2)).

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 3, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 4, 3, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 3, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 1, 0, 4, 2, 4, 0, 0, 0, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Apr 07 2020

Keywords

Comments

Sum of indices of non-unitary prime factors of n (prime factors for which the exponent exceeds 1).

Examples

			a(450) = a(2 * 3^2 * 5^2) = a(prime(1) * prime(2)^2 * prime(3)^2) = 2 + 3 = 5.

Links

Crossrefs

Cf. A000720, A003557, A005117 (positions of 0's), A056170, A056239, A063958, A066328, A071773.

Programs

  • Mathematica
    nmax = 104; CoefficientList[Series[Sum[k x^(Prime[k]^2)/(1 - x^(Prime[k]^2)), {k, 1, nmax}], {x,0, nmax}], x] // Rest
f[p_, e_] := If[e == 1, 0, PrimePi[p]]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jul 24 2024 *)
  • PARI
    A333842(n) = { my(f=factor(n)); sum(k=1, #f~, if(1==f[k, 2],0,1)*primepi(f[k, 1])); }; \\ Antti Karttunen, Jun 12 2020

Formula

a(n) = A056239(A071773(n)) = A066328(A003557(n)). - Peter Munn and Antti Karttunen, Jun 13 2020
Additive with a(p^e) = primepi(p) = A000720(p) if e >= 2, and 0 otherwise. - Amiram Eldar, Jul 24 2024
Showing 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.