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 1-10 of 11 results. Next

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

A342671 a(n) = gcd(sigma(n), A003961(n)), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 21, 1, 3, 1, 15, 1, 3, 5, 1, 1, 3, 1, 9, 1, 3, 1, 1, 1, 3, 1, 9, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 15, 1, 3, 5, 3, 1, 21, 1, 3, 1, 1, 7, 3, 1, 9, 1, 3, 1, 15, 1, 3, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 5, 9, 1, 3, 1, 3, 1, 3, 1, 9, 1, 3, 13, 7, 1, 3, 1, 3, 1
Offset: 1

Views

Author

Antti Karttunen, Mar 20 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A161942, A286385, A341529, A342672, A342673, A348992, A349161, A349162, A349163, A349164, A349165 (positions of 1's), A349166 (of terms > 1), A349167, A349756, A350071 [= a(n^2)], A355828 (Dirichlet inverse).
Cf. A349169, A349745, A355833, A355924 (applied onto prime shift array A246278).
Cf. also A336850, A354827/A354828, A355932.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342671(n) = gcd(sigma(n), A003961(n));

Formula

a(n) = gcd(A000203(n), A003961(n)).
a(n) = gcd(A000203(n), A286385(n)) = gcd(A003961(n), A286385(n)).
a(n) = A341529(n) / A342672(n).
From Antti Karttunen, Jul 21 2022: (Start)
a(n) = A003961(n) / A349161(n).
a(n) = A000203(n) / A349162(n).
a(n) = A161942(n) / A348992(n).
a(n) = A003961(A349163(n)) = A003961(n/A349164(n)).
(End)

A336849 a(n) = A003961(n) / gcd(A003961(n), sigma(A003961(n))), where A003961 is the prime shift towards larger primes.

Original entry on oeis.org

1, 3, 5, 9, 7, 5, 11, 27, 25, 21, 13, 15, 17, 11, 35, 81, 19, 75, 23, 63, 55, 39, 29, 9, 49, 17, 125, 33, 31, 35, 37, 243, 65, 57, 77, 225, 41, 23, 85, 189, 43, 55, 47, 9, 175, 29, 53, 135, 121, 49, 19, 17, 59, 125, 13, 99, 115, 93, 61, 105, 67, 111, 275, 729, 119, 65, 71, 171, 29, 77, 73, 135, 79, 41, 245, 69, 143
Offset: 1

Views

Author

Antti Karttunen, Aug 06 2020

Keywords

Comments

If there are no more 1's in this sequence after the initial one, then there are no odd terms of A007691 (multiply perfect numbers) larger than one.
Denominator of the ratio A003973(n) / A003961(n), also denominator of the ratio (A341528(n)/A341529(n)) / (n / sigma(n)). - Antti Karttunen, Feb 16 2021

Links

Crossrefs

Cf. A003961, A003973, A007691, A017666, A336848, A336850, A341528, A341529, A341530.
Cf. A341525 (numerators).

Programs

  • Mathematica
    f[p_, e_] := NextPrime[p]^e; g[1] = 1; g[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := (gn = g[n])/GCD[gn, DivisorSigma[1, gn]]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336849(n) = { my(u=A003961(n)); (u/gcd(u, sigma(u))); };
\\ Or alternatively as:
A336849(n) = { my(u=A003961(n)); denominator(sigma(u)/u); };

Formula

a(n) = A003961(n) / A336850(n) = A003961(n) / gcd(A003961(n), A003973(n)).
a(n) = A017666(A003961(n)).

A341525 Numerator of A003973(n) / A003961(n).

Original entry on oeis.org

1, 4, 6, 13, 8, 8, 12, 40, 31, 32, 14, 26, 18, 16, 48, 121, 20, 124, 24, 104, 72, 56, 30, 16, 57, 24, 156, 52, 32, 64, 38, 364, 84, 80, 96, 403, 42, 32, 108, 320, 44, 96, 48, 14, 248, 40, 54, 242, 133, 76, 24, 26, 60, 208, 16, 160, 144, 128, 62, 208, 68, 152, 372, 1093, 144, 112, 72, 260, 36, 128, 74, 248, 80, 56
Offset: 1

Views

Author

Antti Karttunen, Feb 16 2021

Keywords

Comments

Also numerator of the ratio (A341528(n)/A341529(n)) / (n/sigma(n)).

Links

Crossrefs

Cf. A000203, A000290 (positions of odd terms), A003961, A003973, A017665, A151800, A336850, A341526, A341527, A341528, A341529, A341530.
Cf. A336849 (denominators).

Programs

  • Mathematica
    f[p_, e_] := (p^(e + 1) - 1)/((p - 1)*p^e); g[p_, e_] := f[NextPrime[p], e]; a[1] = 1; a[n_] := Numerator[Times @@ g @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Feb 17 2021 *)
  • PARI
    A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A341525(n) = { my(u=A003961(n), s=sigma(u)); (s/gcd(u, s)); };

Formula

a(n) = A017665(A003961(n)).
a(n) = A003973(n) / A336850(n) = A003973(n) / gcd(A003961(n), A003973(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} A341525(k)/A336849(k) = 1 / Product_{p prime} (1 - 1/(p*nextprime(p))) = 1.3766054560..., where nextprime(p) = A151800(p). - Amiram Eldar, Dec 28 2024

A336851 a(n) = sigma(A003961(n)) - A003961(n), where A003961 is a prime shift towards larger primes, sigma is the sum of divisors.

Original entry on oeis.org

0, 1, 1, 4, 1, 9, 1, 13, 6, 11, 1, 33, 1, 15, 13, 40, 1, 49, 1, 41, 17, 17, 1, 105, 8, 21, 31, 57, 1, 87, 1, 121, 19, 23, 19, 178, 1, 27, 23, 131, 1, 123, 1, 65, 73, 33, 1, 321, 12, 81, 25, 81, 1, 249, 21, 183, 29, 35, 1, 309, 1, 41, 97, 364, 25, 141, 1, 89, 35, 153, 1, 565, 1, 45, 97, 105, 25, 177, 1, 401, 156, 47
Offset: 1

Views

Author

Antti Karttunen, Aug 05 2020

Keywords

Comments

Even terms occur on square n, odd terms on nonsquare n.
Numbers k such that a(k) = 2^e for e >= 1, are: 4, 25, 841, 12769, 66896041, etc., i.e., terms of A073715 squared.

Links

Crossrefs

Cf. A000203, A001065, A003961, A003973, A073715, A252748, A286385, A326042, A336852.
Cf. A336850 [= gcd(a(n),A003961(n))].

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336851(n) = (sigma(A003961(n)) - A003961(n));

Formula

a(n) = A003973(n) - A003961(n) = A000203(A003961(n)) - A003961(n).
a(n) = A001065(A003961(n)).
a(n) = A336852(n) - A286385(n).

Extensions

Comments edited by Antti Karttunen, Jul 03 2023

A355932 a(n) = gcd(sigma(n), sigma(A003961(n))), where A003961 is fully multiplicative with a(p) = nextprime(p).

Original entry on oeis.org

1, 1, 2, 1, 2, 12, 4, 5, 1, 2, 2, 2, 2, 24, 24, 1, 2, 1, 4, 2, 8, 4, 6, 60, 1, 6, 4, 4, 2, 24, 2, 7, 12, 2, 48, 13, 2, 12, 4, 10, 2, 96, 4, 14, 2, 24, 6, 2, 19, 3, 24, 2, 6, 24, 8, 120, 16, 2, 2, 24, 2, 8, 4, 1, 12, 48, 4, 2, 12, 48, 2, 5, 2, 6, 2, 4, 24, 24, 4, 2, 11, 2, 6, 8, 4, 12, 24, 20, 2, 2, 8, 6, 4, 72, 24
Offset: 1

Views

Author

Antti Karttunen, Jul 22 2022

Keywords

Links

Crossrefs

Cf. A000203, A003961, A003973, A355933, A355934.
Cf. also A336850, A342671.

Programs

  • Mathematica
    f[p_, e_] := ((q = NextPrime[p])^(e + 1) - 1)/(q - 1); a[1] = 1; a[n_] := GCD[DivisorSigma[1, n], Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jul 22 2022 *)
  • PARI
    A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A355932(n) = gcd(sigma(n), A003973(n));

Formula

a(n) = gcd(A000203(n), A003973(n)).
a(n) = A003973(n) / A355933(n).
a(n) = A000203(n) / A355934(n).

A361468 a(n) = A249670(A003961(n)).

Original entry on oeis.org

1, 12, 30, 117, 56, 40, 132, 1080, 775, 672, 182, 390, 306, 176, 1680, 9801, 380, 9300, 552, 6552, 3960, 2184, 870, 144, 2793, 408, 19500, 1716, 992, 2240, 1406, 88452, 5460, 4560, 7392, 90675, 1722, 736, 9180, 60480, 1892, 5280, 2256, 126, 43400, 1160, 2862, 32670, 16093, 3724, 456, 442, 3540, 26000
Offset: 1

Views

Author

Antti Karttunen, Mar 20 2023

Keywords

Links

Crossrefs

Cf. A000290, A003961, A249670, A336850, A361467.

Programs

  • PARI
    A003961(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); }; \\ From A003961
A249670(n) = { my(ab = sigma(n)/n); numerator(ab)*denominator(ab); };
A361468(n) = A249670(A003961(n));

Formula

a(n) = A249670(A003961(n)) = A336849(n) * A341525(n).
a(n) = A361467(n) / A000290(A336850(n)).

A337335 a(n) = gcd(A048673(n), 1+A003973(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 29, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 19, 1, 13, 1, 1, 1, 13, 1, 1, 1
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2020

Keywords

Links

Crossrefs

Cf. A003961, A003973, A048673, A337337, A337342 (positions where equal to A048673).
Cf. also A336850.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337335(n) = { my(s=A003961(n)); gcd((1+s)/2, 1+sigma(s)); };

Formula

a(n) = gcd(A048673(n), 1+A003973(n)) = (n) = gcd((1+A003961(n))/2, 1+sigma(A003961(n))).
For all n>= 1, a(A000290(n)) = A337337(n).

A342673 a(n) = gcd(n, sigma(A003961(n))), where A003961 is fully multiplicative with a(prime(k)) = prime(k+1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 2, 3, 1, 1, 6, 1, 8, 1, 2, 1, 6, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 24, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 2, 3, 40, 1, 6, 1, 2, 1, 2, 1, 6, 7, 2, 3, 26, 1, 6, 1, 8, 3, 2, 1, 12, 1, 2, 3, 1, 1, 6, 1, 4, 3, 2, 1, 8, 1, 2, 3, 4, 7, 6, 1, 8, 1, 2, 1, 12, 5, 2, 3, 8, 1, 2, 1, 2, 3, 2, 1, 24, 1, 14, 1, 1, 1, 6, 1, 8, 3
Offset: 1

Views

Author

Antti Karttunen, Mar 20 2021

Keywords

Links

Crossrefs

Cf. A000203, A003961, A003973, A342671.
Cf. also A336850, A341528.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A342673(n) = gcd(n,sigma(A003961(n)));

Formula

a(n) = gcd(n, A003973(n)) = gcd(n, A000203(A003961(n))).

A378994 a(n) = gcd(2*n, sigma(2*n)).

Original entry on oeis.org

1, 1, 6, 1, 2, 4, 2, 1, 3, 2, 2, 12, 2, 28, 6, 1, 2, 1, 2, 10, 6, 4, 2, 4, 1, 2, 6, 8, 2, 12, 2, 1, 6, 2, 2, 3, 2, 4, 6, 2, 2, 28, 2, 4, 18, 4, 2, 12, 1, 1, 6, 2, 2, 4, 2, 8, 6, 2, 2, 120, 2, 4, 6, 1, 2, 12, 2, 2, 6, 28, 2, 1, 2, 2, 6, 4, 2, 4, 2, 2, 3, 2, 2, 24, 2, 4, 6, 4, 2, 6, 14, 8, 6, 4, 10, 4, 2, 7, 18, 5
Offset: 1

Views

Author

Antti Karttunen, Dec 13 2024

Keywords

Links

Crossrefs

Even bisection of A009194.
Topmost row of array A355925.
Cf. A000203, A062731, A378995, A378996.
Cf. also A336850.

Programs

  • Mathematica
    Table[GCD[2n,DivisorSigma[1,2n]],{n,100}] (* James C. McMahon, Dec 14 2024 *)
  • PARI
    A378994(n) = gcd(2*n, sigma(2*n));

Formula

a(n) = A009194(2*n) = gcd(2*n, A062731(n)).
a(n) = 2*n / A378996(n).
Showing 1-10 of 11 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.