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 1-4 of 4 results.

A348507 a(n) = A003959(n) - n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

0, 1, 1, 5, 1, 6, 1, 19, 7, 8, 1, 24, 1, 10, 9, 65, 1, 30, 1, 34, 11, 14, 1, 84, 11, 16, 37, 44, 1, 42, 1, 211, 15, 20, 13, 108, 1, 22, 17, 122, 1, 54, 1, 64, 51, 26, 1, 276, 15, 58, 21, 74, 1, 138, 17, 160, 23, 32, 1, 156, 1, 34, 65, 665, 19, 78, 1, 94, 27, 74, 1, 360, 1, 40, 69, 104, 19, 90, 1, 406, 175, 44, 1, 204
Offset: 1

Views

Author

Antti Karttunen, Oct 30 2021

Keywords

Comments

a(p*(n/p)) - (n/p) = (p+1)*a(n/p) holds for all prime divisors p of n, which can be seen by expanding the left hand side as (A003959(p*(n/p)) - (p*(n/p))) - (n/p) = (p+1)*A003959(n/p)-((p+1)*(n/p)) = (p+1)*(A003959(n/p)-(n/p)) = (p+1)*a(n/p). This implies that a(n) >= A003415(n) for all n. (See also comments in A348970). - Antti Karttunen, Nov 06 2021

Links

Crossrefs

Cf. A001065, A003415, A003959, A020639, A032742, A065488, A348029, A348508, A348929 [= gcd(n,a(n))], A348950, A348970.
Cf. A348971 (Möbius transform) and A349139, A349140, A349141, A349142, A349143 (other Dirichlet convolutions).
Cf. also A168065 (the arithmetic mean of this and A322582), A168066.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n; Array[a, 100] (* Amiram Eldar, Oct 30 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
  • PARI
    A020639(n) = if(1==n,n,(factor(n)[1, 1]));
A348507(n) = { my(s=0, m=1, spf); while(n>1, spf = A020639(n); n /= spf; s += m*n; m *= (1+spf)); (s); }; \\ (Compare this with similar programs given in A003415 and in A322582) - Antti Karttunen, Nov 06 2021

Formula

a(n) = A003959(n) - n.
a(n) = A348508(n) + n.
a(n) = A001065(n) + A348029(n).
From Antti Karttunen, Nov 06 2021: (Start)
a(n) = Sum_{d|n} A348971(d).
a(n) = A003415(n) + A348970(n).
For all n >= 1, A322582(n) <= A003415(n) <= a(n).
For n > 1, a(n) = a(A032742(n))*(1+A020639(n)) + A032742(n). [See the comments above, and compare this with Reinhard Zumkeller's May 09 2011 recursive formula for A003415] (End)
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A065488 - 1. - Amiram Eldar, Jun 01 2025

A348970 a(n) = A003959(n) - A129283(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A129283(n) is sum of n and its arithmetic derivative.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 7, 1, 1, 0, 8, 0, 1, 1, 33, 0, 9, 0, 10, 1, 1, 0, 40, 1, 1, 10, 12, 0, 11, 0, 131, 1, 1, 1, 48, 0, 1, 1, 54, 0, 13, 0, 16, 12, 1, 0, 164, 1, 13, 1, 18, 0, 57, 1, 68, 1, 1, 0, 64, 0, 1, 14, 473, 1, 17, 0, 22, 1, 15, 0, 204, 0, 1, 14, 24, 1, 19, 0, 230, 67, 1, 0, 80, 1, 1, 1, 96, 0, 75, 1, 28
Offset: 1

Views

Author

Antti Karttunen, Nov 05 2021

Keywords

Comments

There are no negative terms. We prove this by induction over the prime factorization of n, showing that A348507(n) >= A003415(n) for all values of n >= 1. At n=1, both sequences have value 0, and at the primes both sequences obtain the value 1, so the base cases hold. We know that A348507(n)-(n/p) = (p+1)*A348507(n/p) for all prime factors p of n (see comment in A348507). With the arithmetic derivative we obtain respectively that A003415(n) = A003415(p*(n/p)) = A003415(p)*(n/p) + p*A003415(n/p) = (n/p) + p*A003415(n/p), for any prime factor p of n. Now A348507(p*(n/p)) >= A003415(p*(n/p)) iff A348507(p*(n/p)) - (n/p) >= A003415(p*(n/p)) - (n/p), that is, iff (p+1)*A348507(n/p) >= p*A003415(n/p), which indeed follows by the induction hypothesis, which assumes that A348507(x) >= A003415(x) for all proper divisors x of n.

Links

Crossrefs

Cf. A003415, A003959, A129283, A211991, A348029, A348507, A348971, A348972, A348973, A348974.
Cf. A008578 (positions of zeros), A001358 (positions of ones).

Programs

  • Mathematica
    d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); f[p_, e_] := (p + 1)^e; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - n - d[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2021 *)
  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348970(n) = (A003959(n) - (n+A003415(n)));

Formula

a(n) = A003959(n) - A129283(n) = A003959(n) - (n+A003415(n)).
a(n) = A348029(n) - A211991(n).
a(n) = A348507(n) - A003415(n).
For all n >= 1, a(A001358(n)) = 1.

A349140 a(n) = Sum_{d|n} d * A348507(n/d), where A348507(n) = A003959(n) - n, where A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 7, 1, 11, 1, 33, 10, 15, 1, 61, 1, 19, 17, 131, 1, 77, 1, 89, 21, 27, 1, 263, 16, 31, 67, 117, 1, 145, 1, 473, 29, 39, 25, 379, 1, 43, 33, 395, 1, 189, 1, 173, 137, 51, 1, 997, 22, 155, 41, 201, 1, 443, 33, 527, 45, 63, 1, 743, 1, 67, 177, 1611, 37, 277, 1, 257, 53, 265, 1, 1541, 1, 79, 187, 285, 37, 321
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Comments

Dirichlet convolution of A348507 with the identity function, A000027.
Dirichlet convolution of sigma with A348971.

Links

Crossrefs

Cf. A000027, A000203, A003959, A038040, A348507, A348971, A349141 (Möbius transform), A349142, A349143, A349170.
Cf. also A347130, A348980.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e; s[1] = 0; s[n_] := Times @@ f @@@ FactorInteger[n] - n; a[n_] := DivisorSum[n, #*s[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349140(n) = sumdiv(n,d,d*A348507(n/d));

Formula

a(n) = Sum_{d|n} d * A348507(n/d).
a(n) = Sum_{d|n} A000203(d) * A348971(n/d).
a(n) = Sum_{d|n} A349141(d).
For all n >= 1, a(n) >= A347130(n) >= A348980(n).
a(n) = A349170(n) - A038040(n). - Antti Karttunen, Nov 15 2021

A349143 a(n) = Sum_{d|n} A038040(d) * A348507(n/d), where A038040(n) = n*tau(n), A348507(n) = A003959(n) - n, and A003959 is fully multiplicative with a(p) = (p+1).

Original entry on oeis.org

0, 1, 1, 9, 1, 16, 1, 51, 13, 22, 1, 114, 1, 28, 25, 233, 1, 145, 1, 168, 31, 40, 1, 590, 21, 46, 106, 222, 1, 310, 1, 939, 43, 58, 37, 915, 1, 64, 49, 896, 1, 406, 1, 330, 262, 76, 1, 2570, 29, 297, 61, 384, 1, 1012, 49, 1202, 67, 94, 1, 2040, 1, 100, 340, 3489, 55, 598, 1, 492, 79, 574, 1, 4457, 1, 118, 360, 546, 55
Offset: 1

Views

Author

Antti Karttunen, Nov 08 2021

Keywords

Comments

Dirichlet convolution of A348507 with A038040, which is the Dirichlet convolution of the identity function (A000027) with itself.
Dirichlet convolution of the identity function (A000027) with A349140.
Dirichlet convolution of sigma (A000203) with A349141.
Dirichlet convolution of A060640 with A348971.

Links

Crossrefs

Cf. A000005, A000027, A000203, A003959, A038040, A060640, A348507, A348971, A349140, A349141, A349142.
Cf. also A349123, A348983.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; a[n_] := DivisorSum[n, #*DivisorSigma[0, #]*(s[n/#] - n/#) &]; Array[a, 100] (* Amiram Eldar, Nov 08 2021 *)
  • PARI
    A038040(n) = (n*numdiv(n));
A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348507(n) = (A003959(n) - n);
A349143(n) = sumdiv(n,d,A038040(d)*A348507(n/d));

Formula

a(n) = Sum_{d|n} A038040(n/d) * A348507(d).
a(n) = Sum_{d|n} d * A349140(n/d).
a(n) = Sum_{d|n} A000203(d) * A349141(n/d).
a(n) = Sum_{d|n} A060640(d) * A348971(n/d).
For all n >= 1, a(n) >= A349123(n) >= A348983(n).
Showing 1-4 of 4 results.

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