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 1-8 of 8 results.

A336837 Numerator of ratio A336841(n)/A000005(n).

Original entry on oeis.org

0, 1, 2, 14, 3, 9, 5, 17, 44, 13, 6, 32, 8, 21, 23, 284, 9, 163, 11, 137, 37, 25, 14, 105, 30, 33, 86, 73, 15, 81, 18, 547, 44, 37, 53, 1622, 20, 45, 58, 149, 21, 129, 23, 260, 401, 57, 26, 1662, 230, 109, 65, 114, 29, 297, 63, 237, 79, 61, 30, 263, 33, 73, 213, 4010, 83, 153, 35, 383, 100, 183, 36, 1715, 39, 81
Offset: 1

Views

Author

Antti Karttunen, Aug 07 2020

Keywords

Links

Crossrefs

Cf. A000005, A003961, A003973, A336838, A336841, A336856.
Cf. A336839 (denominators).

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336841(n) = ((numdiv(n)*A003961(n)) - sigma(A003961(n)));
A336837(n) = numerator(A336841(n)/numdiv(n));

Formula

a(n) = A336841(n) / A336856(n) = A336841(n) / gcd(A000005(n), A336841(n)).

A336854 a(n) = A336841(n)/2.

Original entry on oeis.org

0, 1, 2, 7, 3, 18, 5, 34, 22, 26, 6, 96, 8, 42, 46, 142, 9, 163, 11, 137, 74, 50, 14, 420, 45, 66, 172, 219, 15, 324, 18, 547, 88, 74, 106, 811, 20, 90, 116, 596, 21, 516, 23, 260, 401, 114, 26, 1662, 115, 327, 130, 342, 29, 1188, 126, 948, 158, 122, 30, 1578, 33, 146, 639, 2005, 166, 612, 35, 383, 200, 732, 36, 3430
Offset: 1

Views

Author

Antti Karttunen, Aug 08 2020

Keywords

Links

Crossrefs

Cf. A003961, A003973, A048673, A336840, A336841.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336854(n) = { my(s=A003961(n)); sumdiv(s,d,(s-d))/2; };

Formula

a(n) = Sum_{d|n} (A048673(n)-A048673(d)).

A003973 Inverse Möbius transform of A003961; a(n) = sigma(A003961(n)), where A003961 shifts the prime factorization of n one step towards the larger primes.

Original entry on oeis.org

1, 4, 6, 13, 8, 24, 12, 40, 31, 32, 14, 78, 18, 48, 48, 121, 20, 124, 24, 104, 72, 56, 30, 240, 57, 72, 156, 156, 32, 192, 38, 364, 84, 80, 96, 403, 42, 96, 108, 320, 44, 288, 48, 182, 248, 120, 54, 726, 133, 228, 120, 234, 60, 624, 112, 480, 144, 128, 62, 624, 68
Offset: 1

Views

Author

Marc LeBrun

Keywords

Comments

Sum of the divisors of the prime shifted n, or equally, sum of the prime shifted divisors of n. - Antti Karttunen, Aug 17 2020

Links

Crossrefs

Cf. A000203, A000290 (positions of odd terms), A003961, A007814, A048673, A108228, A151800, A295664, A336840.
Permutation of A008438.
Used in the definitions of the following sequences: A326042, A336838, A336841, A336844, A336846, A336847, A336848, A336849, A336850, A336851, A336852, A336856, A336931, A336932.
Cf. also A003972.

Programs

  • Mathematica
    b[1] = 1; b[p_?PrimeQ] := b[p] = Prime[ PrimePi[p] + 1]; b[n_] := b[n] = Times @@ (b[First[#]]^Last[#] &) /@ FactorInteger[n]; a[n_] := Sum[ b[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]  (* Jean-François Alcover, Jul 18 2013 *)
  • PARI
    aPrime(p,e)=my(q=nextprime(p+1));(q^(e+1)-1)/(q-1)
a(n)=my(f=factor(n));prod(i=1,#f~,aPrime(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Jul 18 2013
  • PARI
    A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); }; \\ Antti Karttunen, Aug 06 2020
  • Python
    from math import prod
from sympy import factorint, nextprime
def A003973(n): return prod(((q:=nextprime(p))**(e+1)-1)//(q-1) for p,e in factorint(n).items()) # Chai Wah Wu, Jul 05 2022

Formula

Multiplicative with a(p^e) = (q^(e+1)-1)/(q-1) where q = nextPrime(p). - David W. Wilson, Sep 01 2001
From Antti Karttunen, Aug 06-12 2020: (Start)
a(n) = Sum_{d|n} A003961(d) = Sum_{d|A003961(n)} d.
a(n) = A000203(A003961(n)) = A000593(A003961(n)).
a(n) = 2*A336840(n) - A000005(n) = 2*Sum_{d|n} (A048673(d) - (1/2)).
a(n) = A008438(A108228(n)) = A008438(A048673(n)-1).
a(n) = A336838(n) * A336856(n).
a(n) is odd if and only if n is a square.
(End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} p^3/((p+1)*(p^2-nextprime(p))) = 3.39513795..., where nextprime is A151800. - Amiram Eldar, Dec 08 2022, May 30 2025

Extensions

More terms from David W. Wilson, Aug 29 2001
Secondary name added by Antti Karttunen, Aug 06 2020

A336840 Inverse Möbius transform of A048673.

Original entry on oeis.org

1, 3, 4, 8, 5, 14, 7, 22, 17, 18, 8, 42, 10, 26, 26, 63, 11, 65, 13, 55, 38, 30, 16, 124, 30, 38, 80, 81, 17, 100, 20, 185, 44, 42, 50, 206, 22, 50, 56, 164, 23, 148, 25, 94, 127, 62, 28, 368, 68, 117, 62, 120, 31, 316, 58, 244, 74, 66, 32, 318, 35, 78, 189, 550, 74, 172, 37, 133, 92, 196, 38, 626, 41, 86, 174, 159
Offset: 1

Views

Author

Antti Karttunen, Aug 07 2020

Keywords

Comments

Arithmetic mean of the number of divisors (A000005) and prime-shifted sigma (A003973), thus a(n) is the average between the number of and the sum of divisors of A003961(n).
The local minima occur on primes p, where p/2 < a(p) <= (p+1).

Links

Crossrefs

Cf. A000005, A003961, A003973, A007503, A048673, A113415, A336838, A336839, A336841, A349371, A349384, A349385, A349910 [= a(n)-A048673(n)], A364063, A378520 (Dirichlet inverse).

Programs

  • PARI
    A048673(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (factorback(f)+1)/2; };
A336840(n) = sumdiv(n,d,A048673(d));
  • PARI
    A336840(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); (1/2)*(numdiv(n)+sigma(factorback(f))); };

Formula

a(n) = Sum_{d|n} A048673(d).
a(n) = (1/2) * (A000005(n) + A003973(n)).
a(n) = A113415(A003961(n)). - Antti Karttunen, Jun 01 2022
a(n) = A349371(A003961(n)) = A364063(A048673(n)). - Antti Karttunen, Nov 30 2024

A336839 Denominator of the arithmetic mean of the divisors of A003961(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 07 2020

Keywords

Comments

Also denominator of A336841(n) / A000005(n).
All terms are odd because A336932(n) = A007814(A003973(n)) >= A295664(n) for all n.

Links

Crossrefs

Cf. A000005, A000225, A003961, A003973, A007814, A052940, A057021, A295664, A336840, A336841, A336856, A336931, A336932.
Cf. A336918 (positions of 1's), A336919 (of terms > 1).
Cf. A336837 and A336838 (numerators).

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336839(n) = denominator(sigma(A003961(n))/numdiv(n));

Formula

a(n) = denominator(A003973(n)/A000005(n)).
a(n) = d(n)/A336856(n) = d(n)/gcd(d(n),A003973(n)) = d(n)/gcd(d(n),A336841(n)), where d(n) is the number of divisors of n, A000005(n).
a(n) = A057021(A003961(n)).
For all primes p, and e >= 0, a(A000225(e)) = a(p^((2^e) - 1)) = 1. [See A336856]
It seems that for all odd primes p, and with the exponents e=5, 11, 17 or 23 (at least these), a(p^e) = 1.
It seems that a(27^((2^n)-1)) = A052940(n-1) for all n >= 1.

A336845 a(n) = A000005(n) * A003961(n), where A003961 is the prime shift towards larger primes, and A000005 gives the number of divisors of n, and also of A003961(n).

Original entry on oeis.org

1, 6, 10, 27, 14, 60, 22, 108, 75, 84, 26, 270, 34, 132, 140, 405, 38, 450, 46, 378, 220, 156, 58, 1080, 147, 204, 500, 594, 62, 840, 74, 1458, 260, 228, 308, 2025, 82, 276, 340, 1512, 86, 1320, 94, 702, 1050, 348, 106, 4050, 363, 882, 380, 918, 118, 3000, 364, 2376, 460, 372, 122, 3780, 134, 444, 1650, 5103, 476, 1560
Offset: 1

Views

Author

Antti Karttunen, Aug 06 2020

Keywords

Comments

Dirichlet convolution of A003961 with itself.
Sequence is not injective, as it has duplicate values, for example: a(162) = a(243) = 18750. See also comments in A336475.

Links

Crossrefs

Cf. A000005, A000203, A000290, A003961, A038040, A336475.
Cf. also A336841, A336846 [= gcd(a(n),A003973(n))], A336847, A336848.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336845(n) = (numdiv(n)*A003961(n))
  • PARI
    A336845(n) = { my(f = factor(n)); prod(i=1, #f~, (1+f[i,2]) * (nextprime(1+f[i, 1])^f[i,2])); };
  • PARI
    A336845(n) = sumdiv(n,d,A003961(d)*A003961(n/d));

Formula

Multiplicative with a(prime(i)^e) = (e+1) * prime(1+i)^e.
a(n) = A000005(n) * A003961(n).
a(n) = A038040(A003961(n)).
a(n) = A336841(n) + A003973(n).
a(n) is odd if and only if n is a square.

A336846 a(n) = gcd(sigma(A003961(n)), A000005(n)*A003961(n)).

Original entry on oeis.org

1, 2, 2, 1, 2, 12, 2, 4, 1, 4, 2, 6, 2, 12, 4, 1, 2, 2, 2, 2, 4, 4, 2, 120, 3, 12, 4, 6, 2, 24, 2, 2, 4, 4, 4, 1, 2, 12, 4, 8, 2, 24, 2, 26, 2, 12, 2, 6, 1, 6, 20, 18, 2, 24, 28, 24, 4, 4, 2, 12, 2, 4, 6, 1, 4, 24, 2, 2, 20, 24, 2, 20, 2, 12, 6, 6, 4, 24, 2, 2, 1, 4, 2, 36, 4, 12, 4, 8, 2, 4, 4, 6, 4, 12, 4, 12, 2, 2, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Aug 06 2020

Keywords

Links

Crossrefs

Cf. A000005, A000203, A000290 (positions of odd terms), A003961, A003973, A324121, A336476, A336841, A336845, A336847, A336848.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336846(n) = { my(u=A003961(n),s=sigma(u)); gcd(s, numdiv(n)*u); };

Formula

a(n) = gcd(A003973(n), A336845(n)) = gcd(A003973(n), A336841(n)).
a(n) = gcd(A000203(A003961(n)), A000005(n)*A003961(n)).
a(n) = A324121(A003961(n)).

A336856 Prime-shifted analog of gcd(d(n), sigma(n)): a(n) = gcd(A000005(n), A003973(n)).

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Aug 12 2020

Keywords

Links

Crossrefs

Cf. A000005, A003961, A003973, A009205, A336837, A336838, A336839, A336840, A336841.

Programs

  • PARI
    A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A336856(n) = gcd(numdiv(n), A003973(n));

Formula

a(n) = A009205(A003961(n)).
a(n) = gcd(A000005(n), A003973(n)) = gcd(A000005(n), A336841(n)).
a(n) = gcd(A000005(n), 2*A336840(n)).
a(n) = A003973(n) / A336838(n) = A000005(n) / A336839(n).
For n > 1, a(n) = A336841(n) / A336837(n).
For all primes p, and n >= 0, a(p^((2^n)-1)) = 2^n.
Showing 1-8 of 8 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.