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.

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).

A294897 a(n) = Product_{d|n, gcd(d,n/d)>1} prime(A101296(gcd(d,n/d))-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 4, 1, 1, 1, 12, 1, 4, 1, 4, 1, 1, 1, 16, 2, 1, 4, 4, 1, 1, 1, 36, 1, 1, 1, 80, 1, 1, 1, 16, 1, 1, 1, 4, 4, 1, 1, 144, 2, 4, 1, 4, 1, 16, 1, 16, 1, 1, 1, 16, 1, 1, 4, 252, 1, 1, 1, 4, 1, 1, 1, 1600, 1, 1, 4, 4, 1, 1, 1, 144, 12, 1, 1, 16, 1, 1, 1, 16, 1, 16, 1, 4, 1, 1, 1, 1296, 1, 4, 4, 80, 1, 1, 1, 16, 1
Offset: 1

Views

Author

Antti Karttunen, Nov 21 2017

Keywords

Links

Crossrefs

Cf. A005117 (the positions of ones).
Cf. A101296, A294895.
Cf. also A292258 (A292259), A293515, A294875 for similar filter sequences.

Programs

  • PARI
    up_to = 16384
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]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
v101296 = rgs_transform(vector(up_to, n, A046523(n)));
A101296(n) = v101296[n];
A294897(n) = { my(m=1); fordiv(n,d,if(gcd(d,n/d)>1, m *= prime(A101296(gcd(d,n/d))-1))); m; };

Formula

a(n) = Product_{d|n} A008578(A101296(gcd(d,n/d))).
For n >= 1, A001222(a(n)) = A048105(n).

A293515 a(n) = Product_{d^k|n, d>1, k>1} prime(A286561(n,d)-1), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 10, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 14, 1, 1, 1, 8, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 10, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 66, 1, 1, 1, 2, 1, 1, 1, 12, 1, 1, 2, 2, 1, 1, 1, 10, 10, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 14, 1, 2, 2, 8, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 1, 10, 1, 1, 1, 2, 2
Offset: 1

Views

Author

Antti Karttunen, Nov 11 2017

Keywords

Links

Crossrefs

Cf. A000040, A008578, A286561.
Cf. A293514, A294875.
Cf. A046951.

Programs

  • PARI
    A293515(n) = { my(m=1,v); fordiv(n,d,if(d>1, v = valuation(n,d); if(v>1, m *= prime(v-1)))); m; };

Formula

a(n) = Product_{d|n, d>1} A008578(A286561(n,d)).
a(n) = A064989(A293514(n)).
Other identities. For all n >= 1:
1 + A001222(a(n)) = A046951(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).

A295879 Multiplicative with a(p) = 1, a(p^e) = prime(e-1) if e > 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 7, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 3, 2, 1, 13
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 sequences related to the counting of various non-unitary prime divisors:
For all i, j:
a(i) = a(j) => A056170(i) = A056170(j), as A056170(n) = A001222(a(n)).
a(i) = a(j) => A162641(i) = A162641(j).
a(i) = a(j) => A295659(i) = A295659(j).
a(i) = a(j) => A295662(i) = A295662(j).
a(i) = a(j) => A295883(i) = A295883(j), as A295883(n) = A007949(a(n)).
a(i) = a(j) => A295884(i) = A295884(j).
An encoding of the prime signature of A057521(n), the powerful part of n. - Peter Munn, Apr 06 2024

Links

Crossrefs

Cf. A003557, A008578, A057521, A064989, A181819.
Cf. also A293515, A294875, A294895, A294897, A295878, A351944.
Differs from A000688 for the first time at n=128, where a(128) = 13, while A000688(128) = 15.

Programs

  • Mathematica
    Array[Apply[Times, FactorInteger[#] /. {p_, e_} /; p > 0 :> Which[p == 1, 1, e == 1, 1, True, Prime[e - 1]]] &, 128] (* Michael De Vlieger, Nov 29 2017 *)
  • PARI
    a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, prime(f[i,2]-1)));} \\ Amiram Eldar, Nov 18 2022

Formula

a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product A008578(e(i)).
a(n) = A064989(A181819(n)).
a(n) = A181819(A003557(n)). - Antti Karttunen, Apr 03 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/p^2 + Sum_{k>=1} (prime(k+1)-prime(k))/p^(k+2)) = 2.208... . - Amiram Eldar, Nov 18 2022

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 1-6 of 6 results.

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