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 1-9 of 9 results.

A322312 a(n) = Product_{d|n, d+1 is prime} prime(1+A286561(n,d+1)), where A286561(n,k) gives the k-valuation of n (for k > 1).

Original entry on oeis.org

2, 6, 2, 20, 2, 18, 2, 28, 2, 12, 2, 120, 2, 6, 2, 88, 2, 60, 2, 60, 2, 12, 2, 168, 2, 6, 2, 40, 2, 72, 2, 104, 2, 6, 2, 800, 2, 6, 2, 168, 2, 54, 2, 40, 2, 12, 2, 528, 2, 12, 2, 40, 2, 84, 2, 56, 2, 12, 2, 1440, 2, 6, 2, 136, 2, 72, 2, 20, 2, 24, 2, 2240, 2, 6, 2, 20, 2, 36, 2, 528, 2, 12, 2, 720, 2, 6, 2, 112, 2
Offset: 1

Views

Author

Antti Karttunen, Dec 03 2018

Keywords

Links

Crossrefs

Cf. A067513, A185633, A286561, A322313 (rgs-transform), A322314.
Cf. also A293514, A322310.

Programs

  • PARI
    A322312(n) = { my(m=1,p); fordiv(n,d,p=1+d; if(isprime(p), for(i=0,oo,if(n%(p^i),m *= prime(i);break)))); (m); };

Formula

a(n) = Product_{d|n} A000040(1+A286561(n,1+d))^A010051(1+d).
a(n) = A181819(A185633(n)).
For all n, A001222(a(n)) = A067513(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).

A309891 a(n) is the total number of trailing zeros in the representations of n over all bases b >= 2.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 6, 1, 3, 3, 8, 1, 6, 1, 6, 3, 3, 1, 9, 3, 3, 5, 6, 1, 7, 1, 10, 3, 3, 3, 11, 1, 3, 3, 9, 1, 7, 1, 6, 6, 3, 1, 13, 3, 6, 3, 6, 1, 9, 3, 9, 3, 3, 1, 12, 1, 3, 6, 14, 3, 7, 1, 6, 3, 7, 1, 15, 1, 3, 6, 6, 3, 7, 1, 13, 8, 3, 1, 12
Offset: 1

Views

Author

Rémy Sigrist, Aug 21 2019

Keywords

Comments

a(n) depends only on the prime signature of n.
a(n) is the sum of the k-adic valuations of n for k >= 2. - Friedjof Tellkamp, Jan 25 2025

Examples

			For n = 12: 12 has 2 trailing zeros in base 2 (1100), 1 trailing zero in bases 3, 4, 6 and 12 (110, 30, 20, 10) and no trailing zero in other bases, hence a(12) = 1*2 + 4*1 = 6.

Links

Crossrefs

Cf. A001221, A006218, A007814, A007949, A033093, A056239, A112765, A122841, A169594, A214411, A235127, A244413, A286561, A293514, A369180.
Cf. A111003.
First differences of A078632.

Programs

  • Mathematica
    Table[DivisorSum[n, IntegerExponent[n, #] &, # > 1 &], {n, 84}] (* Jon Maiga, Aug 25 2019 *)
  • PARI
    a(n) = sumdiv(n, d, if (d>1, valuation(n,d), 0))
  • PARI
    a(n) = {if(n == 1, return(0)); my(f = factor(n)[, 2], res = 0, t = 2, of = f, nf = f >> 1, nd(v) = prod(i = 1, #v, v[i] + 1)); while(Set(of) != [0], res += (nd(of) - nd(nf)) * (t-1); of = nf; t++; nf = f \ t); res} \\ David A. Corneth, Aug 22 2019

Formula

a(n) = Sum_{d|n, d>1} A286561(n,d), where A286561 gives the d-valuation of n.
a(p) = 1 for any prime number p.
a(p^k) = A006218(k) for any k >= 0 and any prime number p.
a(n) = 2^A001221(n) - 1 for any squarefree number n.
a(n) = 3 for any semiprime number n.
a(m*n) >= a(m) + a(n).
a(n) >= A007814(n) + A007949(n) + A235127(n) + A112765(n) + A122841(n) + A214411(n) + A244413(n).
a(n) = A056239(A293514(n)). - Antti Karttunen, Aug 22 2019
a(n) <= A033093(n). - Michel Marcus, Aug 22 2019
a(n) = A169594(n) - 1. - Jon Maiga, Aug 25 2019
From Friedjof Tellkamp, Feb 27 2024: (Start)
G.f.: Sum_{k>=2, j>=1} x^(k^j)/(1-x^(k^j)).
Dirichlet g.f.: zeta(s) * Sum_{k>=1} (zeta(k*s) - 1).
Sum_{n>=1} a(n)/n^2 = Pi^2/8 (A111003). (End)

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).

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).

A327167 a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 8, 1, 1, 1, 1, 2, 2, 1, 1, 1, 6, 1, 5, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 12, 1, 1, 1, 3, 6, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 8, 1, 1, 1, 1, 48, 1, 2, 1, 1, 2, 7, 1, 1, 2, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 6, 3, 3, 1, 1, 1, 1, 128
Offset: 1

Views

Author

Antti Karttunen, Sep 19 2019

Keywords

Links

Crossrefs

Cf. A008578, A276086, A286561, A327168.
Cf. also A293514, A322312, A323155, A327154, A327155, A327156.

Programs

  • PARI
    A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A327167(n) = { my(m=1,v); fordiv(A276086(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

Formula

a(n) = Product_{d|A276086(n), d>1} A008578(1+A286561(n,d)).
Other identities. For all n >= 1:
1+A001222(a(n)) = A327168(n).

A327154 a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 6, 1, 5, 2, 1, 1, 2, 1, 2, 1, 3, 1, 48, 1, 2, 1, 80, 1, 45, 1, 1, 2, 2, 1, 1, 1, 3, 1, 8, 1, 44, 1, 6, 2, 5, 1, 6, 1, 1, 3, 2, 1, 20, 1, 20, 1, 2, 1, 80, 1, 11, 1, 1, 1, 63, 1, 2, 2, 7, 1, 2, 1, 2, 1, 6, 1, 20, 1, 2, 1, 2, 1, 264, 1, 3, 2, 6, 1, 48, 2, 10, 1, 7, 2, 108, 1, 1, 2, 1, 1, 125, 1, 2, 2
Offset: 1

Views

Author

Antti Karttunen, Sep 18 2019

Keywords

Links

Crossrefs

Cf. A000040, A008578, A286561, A073802.
Cf. also A293514, A322312, A323155, A327155, A327156.

Programs

  • PARI
    A327154(n) = { my(m=1,s=sigma(n),v); fordiv(n,d,if((d>1) && ((v = valuation(s,d))>0), m *= prime(v))); (m); };

Formula

a(n) = Product_{d|n, d>1} A008578(1+A286561(sigma(n),d)), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A073802(n).

A327155 a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where A286561(n,d) gives the highest exponent of d dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 8, 1, 1, 1, 2, 1, 6, 1, 2, 2, 1, 1, 3, 1, 3, 1, 2, 1, 80, 1, 2, 1, 48, 1, 8, 1, 1, 2, 2, 1, 1, 1, 2, 1, 20, 1, 8, 1, 6, 3, 2, 1, 21, 1, 1, 2, 3, 1, 20, 1, 20, 1, 2, 1, 48, 1, 2, 1, 1, 1, 8, 1, 3, 2, 2, 1, 3, 1, 2, 1, 6, 1, 8, 1, 7, 1, 2, 1, 48, 1, 2, 2, 10, 1, 48, 2, 6, 1, 2, 2, 264, 1, 1, 3, 1, 1, 8, 1, 5, 2
Offset: 1

Views

Author

Antti Karttunen, Sep 18 2019

Keywords

Links

Crossrefs

Cf. A000040, A008578, A286561, A073802.
Cf. also A293514, A322312, A323155, A327154, A327156.

Programs

  • PARI
    A327155(n) = { my(m=1,v); fordiv(sigma(n),d,if((d>1) && ((v = valuation(n,d))>0), m *= prime(v))); (m); };

Formula

a(n) = Product_{d|sigma(n), d>1} A008578(1+A286561(n,d)), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A073802(n).

A327156 a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where A286561(n,x) gives the highest exponent of x dividing n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 8, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 5, 1, 8, 1, 1, 1, 16, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Sep 18 2019

Keywords

Links

Crossrefs

Cf. A000040, A008578, A286561, A173441.
Cf. also A293514, A322312, A323155, A327154, A327155.

Programs

  • PARI
    A327156(n) = { my(m=1,v); fordiv(n,d,if((d>1) && ((v = valuation(n,sigma(d)))>0), m *= prime(v))); (m); };

Formula

a(n) = Product_{d|n, d>1} A008578(1+A286561(n,sigma(d))), where sigma = A000203.
Other identities. For all n >= 1:
1+A001222(a(n)) = A173441(n).
Showing 1-9 of 9 results.

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