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.

A293514 a(n) = Product_{d|n, d>1} prime(A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 20, 6, 8, 2, 48, 2, 8, 8, 84, 2, 48, 2, 48, 8, 8, 2, 320, 6, 8, 20, 48, 2, 128, 2, 264, 8, 8, 8, 864, 2, 8, 8, 320, 2, 128, 2, 48, 48, 8, 2, 2688, 6, 48, 8, 48, 2, 320, 8, 320, 8, 8, 2, 3072, 2, 8, 48, 1560, 8, 128, 2, 48, 8, 128, 2, 11520, 2, 8, 48, 48, 8, 128, 2, 2688, 84, 8, 2, 3072, 8, 8, 8, 320
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2017

Keywords

Examples

			For n = 24, its divisors larger than one are: 2, 3, 4, 6, 8, 12, 24. Only 2 has valuation > 1, namely A286561(24,2) = 3 (as 2^3 divides 24), while the other six have valuation 1. Thus a(24) = prime(1)^6 * prime(3) = 64*5 = 320.
For n = 64, its divisors larger than one are: 2, 4, 8, 16, 32, 64. We see that 2^6 = 4^3 = 8^2 = 64, while valuation of the last three 16, 32 and 64 is 1. Thus a(64) = prime(1)^3 * prime(2) * prime(3) * prime(6) = 2^3 * 3 * 5 * 13 = 1560.

Links

Crossrefs

Cf. A000040, A286561.
Cf. also A293515, A293524, A294876, A293442.

Programs

  • PARI
    A293514(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(valuation(n,d)))); m; };

Formula

a(n) = Product_{d|n, d>1} A000040(A286561(n,d)).
Other identities. For all n >= 1:
A001222(a(n)) = A032741(n).
A007814(a(n)) = A056595(n) [See A046951.]
1+A056239(a(n)) = A169594(n).
A064989(a(n)) = A293515(n).

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).

A294876 a(n) = Product_{d|n, d>1} prime(gcd(d,n/d)).

Original entry on oeis.org

1, 2, 2, 6, 2, 8, 2, 18, 10, 8, 2, 72, 2, 8, 8, 126, 2, 200, 2, 72, 8, 8, 2, 648, 22, 8, 50, 72, 2, 128, 2, 882, 8, 8, 8, 23400, 2, 8, 8, 648, 2, 128, 2, 72, 200, 8, 2, 31752, 34, 968, 8, 72, 2, 5000, 8, 648, 8, 8, 2, 10368, 2, 8, 200, 16758, 8, 128, 2, 72, 8, 128, 2, 2737800, 2, 8, 968, 72, 8, 128, 2, 31752, 1150, 8, 2, 10368, 8, 8, 8, 648, 2, 80000, 8, 72
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2017

Keywords

Links

Crossrefs

Cf. A294877 (rgs-version of this filter).
Cf. A000040, A000188, A034444, A055155.
Cf. also A293442, A293514, A293524.

Programs

  • Mathematica
    A294876[n_] := Product[Prime[GCD[d, n/d]], {d, Rest[Divisors[n]]}];
Array[A294876, 100] (* Paolo Xausa, Feb 22 2024 *)
  • PARI
    A294876(n) = { my(m=1); fordiv(n,d,if(d>1, m *= prime(gcd(d,n/d)))); m; };

Formula

a(n) = Product_{d|n, d>1} A000040(gcd(d,n/d)).
Other identities. For all n >= 1:
1+A007814(a(n)) = A034444(n).
1+A056239(a(n)) = A055155(n).
For n > 1, A061395(a(n)) = A000188(n).

A294873 a(n) = Product_{d|n, d>1, d = x^(2k-1) for some maximal k >= 1} prime(k).

Original entry on oeis.org

1, 2, 2, 2, 2, 8, 2, 6, 2, 8, 2, 16, 2, 8, 8, 6, 2, 16, 2, 16, 8, 8, 2, 96, 2, 8, 6, 16, 2, 128, 2, 30, 8, 8, 8, 32, 2, 8, 8, 96, 2, 128, 2, 16, 16, 8, 2, 192, 2, 16, 8, 16, 2, 96, 8, 96, 8, 8, 2, 1024, 2, 8, 16, 30, 8, 128, 2, 16, 8, 128, 2, 384, 2, 8, 16, 16, 8, 128, 2, 192, 6, 8, 2, 1024, 8, 8, 8, 96, 2, 1024, 8, 16, 8, 8, 8, 1920, 2, 16, 16, 32, 2, 128, 2
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2017

Keywords

Links

Crossrefs

Cf. A000040, A056595, A052409, A183096.
Cf. A293524, A294874, A294875.

Programs

  • PARI
    A294873(n) = { my(m=1,e); fordiv(n,d, if(d>1, e = ispower(d); if(!e, m += m, if((e>1)&&(e%2), m *= prime((e+1)/2))))); m; };

Formula

a(n) = Product_{d|n, d>1, r = A052409(d) is odd} A000040((r+1)/2).
Other identities. For all n >= 1:
A001222(a(n)) = A056595(n).
A007814(a(n)) = A183096(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).
Showing 1-5 of 5 results.

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