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 1-5 of 5 results.

A304408 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*(k_j + 1)).

Original entry on oeis.org

1, 2, 4, 3, 8, 8, 12, 4, 6, 16, 20, 12, 24, 24, 32, 5, 32, 12, 36, 24, 48, 40, 44, 16, 12, 48, 8, 36, 56, 64, 60, 6, 80, 64, 96, 18, 72, 72, 96, 32, 80, 96, 84, 60, 48, 88, 92, 20, 18, 24, 128, 72, 104, 16, 160, 48, 144, 112, 116, 96, 120, 120, 72, 7, 192, 160, 132, 96, 176, 192
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(20) = a(2^2*5) = (2 - 1)*(2 + 1) * (5 - 1)*(1 + 1) = 24.

Links

Crossrefs

Cf. A000005, A000010, A000026, A000040, A000302 (numbers n such that a(n) is odd), A001221, A006093, A007947, A023900, A034444, A059975, A062355, A173557, A304407, A304409, A304411, A304412.

Programs

  • Maple
    a:= n-> mul((i[1]-1)*(i[2]+1), i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 05 2021
  • Mathematica
    a[n_] := Times @@ ((#[[1]] - 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 70}]
Table[DivisorSigma[0, n] EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]], {n, 70}]
  • PARI
    a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*(e+1))} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A000005(n)*abs(A023900(n)) = A000005(n)*A173557(n) = A000005(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*phi(n) if n is a squarefree (A005117), where omega() = A001221 and phi() = A000010.

A304409 If n = Product (p_j^k_j) then a(n) = Product (p_j*(k_j + 1)).

Original entry on oeis.org

1, 4, 6, 6, 10, 24, 14, 8, 9, 40, 22, 36, 26, 56, 60, 10, 34, 36, 38, 60, 84, 88, 46, 48, 15, 104, 12, 84, 58, 240, 62, 12, 132, 136, 140, 54, 74, 152, 156, 80, 82, 336, 86, 132, 90, 184, 94, 60, 21, 60, 204, 156, 106, 48, 220, 112, 228, 232, 118, 360, 122, 248, 126, 14, 260
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(12) = a(2^2*3) = 2*(2 + 1) * 3*(1 + 1) = 36.

Links

Crossrefs

Cf. A000005, A000026, A000040, A001221, A005117, A007947, A016754 (numbers n such that a(n) is odd), A034444, A038040, A064549, A299822, A304407, A304408, A304410 (fixed points), A304411, A304412.

Programs

  • Mathematica
    a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 65}]
Table[DivisorSigma[0, n] Last[Select[Divisors[n], SquareFreeQ]], {n, 65}]
  • PARI
    a(n)={numdiv(n)*factorback(factorint(n)[, 1])} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A000005(n)*A007947(n).
a(p^k) = p*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*n if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 2/p^s - 1/p^(2*s-1) + 1/p^(2*s)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, Jun 06 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(2*s-1) + 2/p^(s-1) + 1/p^(2*s) - 2/p^s) * ((p^s - p)/(p^s - 1))^2.
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = Product_{p prime} (1 - (3*p^2 + p - 1)/(p^2 * (p+1)^2)) = 0.40693068229776748114138817391056656864938379...,
f'(2) = f(2) * Sum_{p prime} 2*(3*p^4-3*p^2+1) * log(p) / ((p-1)*(p+1)*(p^4+2*p^3-2*p^2-p+1)) = f(2) * 2.2612432627709318567813765271568350301741329636853...
and gamma is the Euler-Mascheroni constant A001620. (End)

A304411 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*k_j).

Original entry on oeis.org

1, 3, 4, 6, 6, 12, 8, 9, 8, 18, 12, 24, 14, 24, 24, 12, 18, 24, 20, 36, 32, 36, 24, 36, 12, 42, 12, 48, 30, 72, 32, 15, 48, 54, 48, 48, 38, 60, 56, 54, 42, 96, 44, 72, 48, 72, 48, 48, 16, 36, 72, 84, 54, 36, 72, 72, 80, 90, 60, 144, 62, 96, 64, 18, 84, 144, 68, 108, 96, 144, 72, 72
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(24) = a(2^3*3) = (2 + 1)*3 * (3 + 1)*1 = 36.

Links

Crossrefs

Cf. A000005, A000026, A000040, A000203, A003959, A001615, A003961, A005117, A005361, A007947, A008864, A045965, A048250, A064478, A081294 (numbers n such that a(n) is odd), A181797, A304407, A304408, A304409, A304412.
Cf. A072691, A330596.

Programs

  • Mathematica
    a[n_] := Times @@ ((#[[1]] + 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 72}]
Table[Total[Select[Divisors[n], SquareFreeQ]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 72}]
  • PARI
    a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p+1)*e)} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A005361(n)*A048250(n) = A000005(n/A007947(n))*A000203(A007947(n)).
a(p^k) = (p + 1)*k where p is a prime and k > 0.
a(n) = Product_{p|n} (p + 1) if n is a squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A072691 * A330596 = 0.6156455744... . - Amiram Eldar, Nov 30 2022

A304407 If n = Product (p_j^k_j) then a(n) = Product ((p_j - 1)*k_j).

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 3, 4, 4, 10, 4, 12, 6, 8, 4, 16, 4, 18, 8, 12, 10, 22, 6, 8, 12, 6, 12, 28, 8, 30, 5, 20, 16, 24, 8, 36, 18, 24, 12, 40, 12, 42, 20, 16, 22, 46, 8, 12, 8, 32, 24, 52, 6, 40, 18, 36, 28, 58, 16, 60, 30, 24, 6, 48, 20, 66, 32, 44, 24, 70, 12, 72, 36, 16
Offset: 1

Views

Author

Ilya Gutkovskiy, May 12 2018

Keywords

Examples

			a(60) = a(2^2*3*5) = (2 - 1)*2 * (3 - 1)*1 * (5 - 1)*1 = 16.

Links

Crossrefs

Cf. A000010, A000026, A000040, A002110, A003958, A005117, A005361, A005867, A006093, A007947, A023900, A048250, A059975, A068997, A081294 (numbers n such that a(n) is odd), A087656, A090624, A152649, A173557, A304117, A304408, A304409, A304411, A304412.
Cf. A001221, A076479.

Programs

  • Maple
    seq(mul((p-1)*padic[ordp](n, p), p in numtheory[factorset](n)), n=1..100); # Ridouane Oudra, Jun 06 2025
  • Mathematica
    a[n_] := Times @@ ((#[[1]] - 1) #[[2]] & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 75}]
Table[EulerPhi[Last[Select[Divisors[n], SquareFreeQ]]] DivisorSigma[0, n/Last[Select[Divisors[n], SquareFreeQ]]], {n, 75}]
  • PARI
    a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); (p-1)*e)} \\ Andrew Howroyd, Jul 24 2018

Formula

a(n) = A005361(n)*abs(A023900(n)) = A005361(n)*A173557(n) = A005361(n)*A000010(A007947(n)).
a(p^k) = (p - 1)*k where p is a prime and k > 0.
a(n) = phi(n) if n is a squarefree (A005117), where phi() = A000010.
a(A002110(k)) = A005867(k).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^4/72) * Product_{p prime} (1 - 4/p^2 + 3/p^3 + 1/p^4 - 1/p^5) = 0.2644703894... . - Amiram Eldar, Nov 30 2022
a(n) = (-1)^A001221(n) * (Sum_{d1|n} Sum_{d2|n} mu(d1)*gcd(d1,d2)). - Ridouane Oudra, Jun 06 2025

A322021 Lexicographically earliest such sequence a that a(i) = a(j) => A046523(i) = A046523(j) and A048250(i) = A048250(j), for all i, j.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 14, 15, 16, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 26, 42, 43, 44, 45, 18, 42, 46, 47, 22, 42, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 54, 58, 61, 62, 63, 64, 26, 65, 54, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 52, 78, 79, 80, 81, 75, 82, 83, 26
Offset: 1

Views

Author

Antti Karttunen, Nov 29 2018

Keywords

Comments

Restricted growth sequence transform of A291758, which means that this is the lexicographically least sequence a, such that for all i, j: a(i) = a(j) <=> A291758(i) = A291758(j) <=> A046523(i) = A046523(j) and A048250(i) = A048250(j). That this is equal to the definition given in the title follows because any such lexicographically least sequence satisfying relation <=> is also the least sequence satisfying relation => with the same parameters.
For all i, j:
A295300(i) = A295300(j) => a(i) = a(j),
a(i) = a(j) => A304411(i) = A304411(j),
a(i) = a(j) => A304412(i) = A304412(j).

Links

Crossrefs

Cf. A046523, A048250, A291751, A291758, A294877, A295300, A322022.
Cf. also A304411, A304412.

Programs

  • PARI
    up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A048250(n) = if(n<1, 0, sumdiv(n, d, if(core(d)==d, d)));
v322021 = rgs_transform(vector(up_to, n, [A046523(n), A048250(n)]));
A322021(n) = v322021[n];
Showing 1-5 of 5 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.