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.

A294875 a(n) = Product_{d|n, d = x^k, with x,k > 1} prime(A052409(d)-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 6, 2, 1, 1, 2, 1, 1, 1, 30, 1, 2, 1, 2, 1, 1, 1, 6, 2, 1, 6, 2, 1, 1, 1, 210, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 30, 2, 2, 1, 2, 1, 6, 1, 6, 1, 1, 1, 2, 1, 1, 2, 2310, 1, 1, 1, 2, 1, 1, 1, 24, 1, 1, 2, 2, 1, 1, 1, 30, 30, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 210, 1, 2, 2, 8, 1, 1, 1, 6, 1, 1, 1, 24, 1, 1, 1, 30, 1, 1, 1, 2
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2017

Keywords

Comments

For all i, j:
a(i) = a(j) => A294874(i) = A294874(j) => A046951(i) = A046951(j).
a(i) = a(j) => A061704(i) = A061704(j).

Links

Crossrefs

Cf. A000040, A001597, A052409.
Cf. A293524, A294873, A294874.
Cf. A046951, A061704, A091050 (some of the matched sequences).

Programs

  • PARI
    A294875(n) = { my(m=1,e); fordiv(n,d, if(d>1, e = ispower(d); if(e>1, m *= prime(e-1)))); m; };

Formula

a(n) = Product_{d|n, d>1} A008578(A052409(d)).
a(n) = A064989(A293524(n)).
Other identities. For all n >= 1:
1 + A001222(a(n)) = A091050(n).

A293524 a(n) = Product_{d|n, d>1} prime(A052409(d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 30, 6, 8, 2, 48, 2, 8, 8, 210, 2, 48, 2, 48, 8, 8, 2, 480, 6, 8, 30, 48, 2, 128, 2, 2310, 8, 8, 8, 864, 2, 8, 8, 480, 2, 128, 2, 48, 48, 8, 2, 6720, 6, 48, 8, 48, 2, 480, 8, 480, 8, 8, 2, 3072, 2, 8, 48, 30030, 8, 128, 2, 48, 8, 128, 2, 17280, 2, 8, 48, 48, 8, 128, 2, 6720, 210, 8, 2, 3072, 8, 8, 8, 480
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2017

Keywords

Links

Crossrefs

Cf. A000040, A052409, A294873, A294874, A294875.
Cf. also A293514, A293442.

Programs

  • PARI
    A293524(n) = { my(m=1,e); fordiv(n,d, if(d>1, e = ispower(d); if(!e, m += m, m *= prime(e)))); m; };

Formula

a(n) = Product_{d|n, d>1} A000040(A052409(d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A183096(n).
A064989(a(n)) = A294875(n).

A294874 a(n) = Product_{d|n, d>1, d = x^(2k) for some maximal k} prime(k).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 1, 1, 1, 6, 1, 1, 1, 8, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 6, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 30, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 2, 2, 1, 1, 1, 6, 6, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 6, 1, 2, 2, 8, 1, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 6, 1, 1, 1, 2, 2, 1, 1, 2
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2017

Keywords

Examples

			For n = 36, it has three square-divisors: 4 = 2^(2*1), 9 = 3^(2*1) and 36 = 6^(2*1). Thus a(36) = prime(1) * prime(1) * prime(1) = 2*2*2 = 8.
For n = 64, it has three square-divisors: 4 = 2^(2*1), 16 = 2^(2*2) and 64 = 2^(2*3). Thus a(64) = prime(1) * prime(2) * prime(3) = 2*3*5 = 30.

Links

Crossrefs

Cf. A000040, A046951, A052409, A071325.
Cf. A293524, A294873, A294875.

Programs

  • PARI
    A294874(n) = { my(m=1,e); fordiv(n,d, if(d>1, e = ispower(d); if((e>1)&&!(e%2), m *= prime(e/2)))); m; };

Formula

a(n) = Product_{d|n, d>1, r = A052409(d) is even} A000040(r/2).
Other identities. For all n >= 1:
A001222(a(n)) = A071325(n).
1 + A001222(a(n)) = A046951(n).

A295878 Multiplicative with a(p^(2e)) = 1, a(p^(2e-1)) = prime(e).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 3, 1, 4, 2, 2, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 6, 1, 4, 3, 2, 2, 8, 2, 5, 4, 4, 4, 1, 2, 4, 4, 6, 2, 8, 2, 2, 2, 4, 2, 2, 1, 2, 4, 2, 2, 6, 4, 6, 4, 4, 2, 4, 2, 4, 2, 1, 4, 8, 2, 2, 4, 8, 2, 3, 2, 4, 2, 2, 4, 8, 2, 2, 1, 4, 2, 4, 4, 4, 4, 6, 2, 4, 4, 2, 4, 4, 4, 10, 2, 2, 2, 1, 2, 8, 2, 6, 8, 4, 2, 3, 2, 8, 4, 2, 2, 8, 4, 2, 2, 4, 4, 12
Offset: 1

Views

Author

Antti Karttunen, Nov 29 2017

Keywords

Comments

This sequence can be used as a filter. It matches at least to the following sequence, as for all i, j:
a(i) = a(j) => A162642(i) = A162642(j), as A162642(n) = A001222(a(n)).
a(i) = a(j) => A056169(i) = A056169(j), as A056169(n) = A007814(a(n)).
a(i) = a(j) => A295883(i) = A295883(j), as A295883(n) = A007949(a(n)).
a(i) = a(j) => A295662(i) = A295662(j).
a(i) = a(j) => A295664(i) = A295664(j).

Links

Crossrefs

Cf. A294873, A295879.

Programs

  • Mathematica
    Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, EvenQ@ e, 1, True, Prime[(e + 1)/2]]] &, 120] (* Michael De Vlieger, Nov 29 2017 *)

Formula

a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime((e(i)+1)/2)^A000035(e(i)).
Showing 1-4 of 4 results.

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