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.

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

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)).

A337339 Denominator of (1+sigma(s)) / ((s+1)/2), where s is the square of n prime-shifted once (s = A003961(n)^2 = A003961(n^2)).

Original entry on oeis.org

1, 5, 13, 41, 25, 113, 61, 365, 313, 221, 85, 1013, 145, 109, 613, 3281, 181, 2813, 265, 1985, 1513, 761, 421, 9113, 1201, 1301, 7813, 377, 481, 5513, 685, 29525, 2113, 1625, 2965, 25313, 841, 2381, 3613, 17861, 925, 13613, 1105, 6845, 15313, 3785, 1405, 82013, 7321, 10805, 4513, 11705, 1741, 70313, 4141, 8821, 6613, 865
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2020

Keywords

Comments

All terms are members of A007310, because all terms of A337336 and A337337 are.
No 1's after the initial one at a(1) => No quasiperfect numbers. See comments in A336700 & A337342.
If any quasiperfect numbers qp exist, they must occur also in A325311.
Question: Is there any reliable lower bound for this sequence? See A337340, A337341.
Duplicate values are rare, but at least two cases exist: a(21) = a(74) = 1513 and a(253) = a(554) = 71065. - Antti Karttunen, Jan 03 2024

Links

Crossrefs

Cf. A003961, A003973, A007310, A048673, A074627, A074630, A209922, A324899, A325311, A336697, A336700, A336844, A337194, A337336, A337337, A337340, A337341, A337342.
Cf. A337338 (numerators).
Cf. also A336848, A336849.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A337339(n) = { my(s=(A003961(n)^2),u=(s+1)/2); (u/gcd(1+sigma(s), u)); };
\\ Or alternatively as:
A337339(n) = { my(s=A003961(n^2)); denominator((1+sigma(s))/((s+1)/2)); };

Formula

a(n) = A337336(n) / A337337(n) = A048673(n^2) / gcd(A048673(n^2), A336844(n^2)).
a(n) = A337336(n) / gcd(A337336(n), 1+A003973(n^2)).

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)).

A336847 a(n) = A003973(n) - A336846(n).

Original entry on oeis.org

0, 2, 4, 12, 6, 12, 10, 36, 30, 28, 12, 72, 16, 36, 44, 120, 18, 122, 22, 102, 68, 52, 28, 120, 54, 60, 152, 150, 30, 168, 36, 362, 80, 76, 92, 402, 40, 84, 104, 312, 42, 264, 46, 156, 246, 108, 52, 720, 132, 222, 100, 216, 58, 600, 84, 456, 140, 124, 60, 612, 66, 148, 366, 1092, 140, 312, 70, 258, 160, 360, 72
Offset: 1

Views

Author

Antti Karttunen, Aug 06 2020

Keywords

Links

Crossrefs

Cf. A000005, A000203, A003961, A003973, A336845, A336846, A336848.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336847(n) = { my(u=A003961(n),s=sigma(u)); (s-gcd(s, numdiv(n)*u)); };

Formula

a(n) = A003973(n) - A336846(n).

A348941 a(n) = n / gcd(n, A326042(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 7, 15, 16, 17, 18, 19, 20, 21, 22, 23, 4, 25, 13, 27, 14, 29, 15, 31, 32, 33, 34, 35, 36, 37, 19, 39, 40, 41, 21, 43, 4, 45, 23, 47, 24, 49, 25, 17, 13, 53, 27, 11, 28, 57, 58, 59, 30, 61, 62, 63, 64, 65, 33, 67, 68, 23, 35, 71, 24, 73, 37, 75, 38, 77, 39, 79, 80, 81, 82
Offset: 1

Views

Author

Antti Karttunen, Nov 04 2021

Keywords

Comments

Denominator of ratio A326042(n) / n.
If there are no more 1's in this sequence after the initial one, then there are no odd terms of A336702 (numbers whose abundancy index is a power of 2) larger than one, and neither there are odd terms in A005820 or in A046060. Compare to similar conditions given in A336848, A336849 and A337339.

Links

Crossrefs

Cf. A005820, A046060, A326042, A336702, A336848, A336849, A337339, A348736, A348940, A348942 (numerators).

Programs

  • Mathematica
    f1[2, e_] := 1; f1[p_, e_] := NextPrime[p, -1]^e; s[n_] := Times @@ f1 @@@ FactorInteger[n]; f[p_, e_] := s[((q = NextPrime[p])^(e + 1) - 1)/(q - 1)]; s2[1] = 1; s2[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := n/GCD[n, s2[n]]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)
  • PARI
    A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A326042(n) = A064989(sigma(A003961(n)));
A348941(n) = (n / gcd(n, A326042(n)));

Formula

a(n) = n / A348940(n) = n / gcd(n, A326042(n)).

A348509 a(n) is the numerator of the harmonic mean of the divisors of A003961(n).

Original entry on oeis.org

1, 3, 5, 27, 7, 5, 11, 27, 75, 21, 13, 45, 17, 11, 35, 405, 19, 225, 23, 189, 55, 39, 29, 9, 49, 17, 125, 99, 31, 35, 37, 729, 65, 57, 77, 2025, 41, 23, 85, 189, 43, 55, 47, 27, 525, 29, 53, 675, 363, 147, 19, 51, 59, 125, 13, 99, 115, 93, 61, 315, 67, 111, 275, 5103, 119, 65, 71, 513, 29, 77, 73, 405, 79, 41, 245
Offset: 1

Views

Author

Antti Karttunen, Oct 31 2021

Keywords

Links

Crossrefs

Cf. A003961, A099377, A336845, A336846.
Cf. A336848 (denominators).

Programs

  • Mathematica
    f[p_, e_] := (e + 1)*(pn = NextPrime[p])^e*(pn - 1)/(pn^(e + 1) - 1); a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A099377(n) = { my(d=divisors(n)); numerator(#d/sum(k=1, #d, 1/d[k])); }; \\  From A099377
A348509(n) = A099377(A003961(n));

Formula

a(n) = A099377(A003961(n)).
a(n) = A336845(n) / A336846(n).
Showing 1-8 of 8 results.

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