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.

A336841 Prime-shifted analog of A094471: a(n) = A336845(n) - A003973(n).

Original entry on oeis.org

0, 2, 4, 14, 6, 36, 10, 68, 44, 52, 12, 192, 16, 84, 92, 284, 18, 326, 22, 274, 148, 100, 28, 840, 90, 132, 344, 438, 30, 648, 36, 1094, 176, 148, 212, 1622, 40, 180, 232, 1192, 42, 1032, 46, 520, 802, 228, 52, 3324, 230, 654, 260, 684, 58, 2376, 252, 1896, 316, 244, 60, 3156, 66, 292, 1278, 4010, 332, 1224, 70, 766
Offset: 1

Views

Author

Antti Karttunen, Aug 06 2020

Keywords

Comments

All terms are even because A003973 and A336845 match parity-wise. Also in the sum formulas, only even terms are summed (only one of which is zero).

Links

Crossrefs

Cf. A000005, A000203, A003961, A048673, A094471, A336837, A336839, A336840, A336845, A336856.
Cf. A336846 [= gcd(a(n), A003973(n))].
Twice the terms of A336854.

Programs

Formula

a(n) = A336845(n) - A003973(n) = (A000005(n)*A003961(n)) - A000203(A003961(n)).
a(n) = A094471(A003961(n)).
a(n) = Sum_{d|n} (A003961(n)-A003961(d)) = Sum_{d|A003961(n)} (A003961(n)-d).
a(n) = 2*A336854(n) = 2*Sum_{d|n} (A048673(n)-A048673(d)).
a(n) = ((A003961(n)+1)*A000005(n)) - 2*A336840(n).
a(n) = 2 * ((A000005(n)*A048673(n)) - A336840(n)).
a(n) = A000005(n) * (A336837(n)/A336839(n)) = A336837(n) * A336856(n).

A336848 a(n) = A003973(n) / A336846(n).

Original entry on oeis.org

1, 2, 3, 13, 4, 2, 6, 10, 31, 8, 7, 13, 9, 4, 12, 121, 10, 62, 12, 52, 18, 14, 15, 2, 19, 6, 39, 26, 16, 8, 19, 182, 21, 20, 24, 403, 21, 8, 27, 40, 22, 12, 24, 7, 124, 10, 27, 121, 133, 38, 6, 13, 30, 26, 4, 20, 36, 32, 31, 52, 34, 38, 62, 1093, 36, 14, 36, 130, 9, 16, 37, 62, 40, 14, 57, 52, 42, 18, 42, 484, 781
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 A001599 (Ore's Harmonic Numbers) larger than one.

Links

Crossrefs

Cf. A000005, A000203, A001599, A003961, A003973, A336845, A336846, A336847, A336849.

Programs

  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A336848(n) = { my(u=A003961(n),s=sigma(u)); (s/gcd(s, numdiv(n)*u)); };

Formula

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

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).

A347136 a(n) = Sum_{d|n} d * A003961(n/d), where A003961 shifts the prime factorization of its argument one step towards larger primes.

Original entry on oeis.org

1, 5, 8, 19, 12, 40, 18, 65, 49, 60, 24, 152, 30, 90, 96, 211, 36, 245, 42, 228, 144, 120, 52, 520, 109, 150, 272, 342, 60, 480, 68, 665, 192, 180, 216, 931, 78, 210, 240, 780, 84, 720, 90, 456, 588, 260, 100, 1688, 247, 545, 288, 570, 112, 1360, 288, 1170, 336, 300, 120, 1824, 128, 340, 882, 2059, 360, 960, 138
Offset: 1

Views

Author

Antti Karttunen, Aug 24 2021

Keywords

Comments

Dirichlet convolution of the identity function (A000027) with the prime shifted identity (A003961). Multiplicative because both A000027 and A003961 are.
Dirichlet convolution of Euler phi (A000010) with the prime shifted sigma (A003973).
Dirichlet convolution of sigma (A000203) with the prime shifted phi (A003972).
Inverse Möbius transform of A347137.

Links

Crossrefs

Cf. A003961, A003972, A003973, A151800, A347121, A347137 (Möbius transform).
Cf. also A038040, A336845, A336475, A347236.

Programs

  • Mathematica
    f[p_, e_] := ((np = NextPrime[p])^(e + 1) - p^(e + 1))/(np - p); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 24 2021 *)
  • PARI
    A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A347136(n) = sumdiv(n,d,d*A003961(n/d));

Formula

a(n) = Sum_{d|n} d * A003961(n/d).
a(n) = Sum_{d|n} A000010(n/d) * A003973(d).
a(n) = Sum_{d|n} A000203(n/d) * A003972(d).
a(n) = Sum_{d|n} A347137(d).
For all primes p, a(p) = p + A003961(p).
a(n) = A347121(n) + 2*n.
Multiplicative with a(p^e) = (A151800(p)^(e+1) - p^(e+1))/(A151800(p)-p). - Amiram Eldar, Aug 24 2021

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 1-6 of 6 results.

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