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.

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A348929 a(n) = gcd(n, A003959(n)), where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 12, 1, 2, 3, 1, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 1, 36, 1, 2, 1, 2, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 72, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 2, 3, 4, 1, 6, 1, 2, 3
Offset: 1

Views

Author

Antti Karttunen, Nov 07 2021

Keywords

Links

Crossrefs

Cf. A003959, A085731, A126865, A323166, A348507, A348928, A348999 [= a(A276086(n))].
Differs from similar A126795 for the first time at n=36, where a(36) = 36, while A126795(36) = 12.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e; a[n_] := GCD[n, Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Nov 07 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A348929(n) = gcd(n, A003959(n));

Formula

a(n) = gcd(n, A003959(n)) = gcd(n, A348507(n)) = gcd(A003959(n), A348507(n)).

A074107 a(n) = Product of (prime + 1) for first n primes - primorial (n); Sum of proper divisors of the n-th primorial.

Original entry on oeis.org

0, 1, 6, 42, 366, 4602, 66738, 1231314, 25136790, 612982650, 18612572370, 602072009070, 23079296834790, 976751205195990, 43281303292150770, 2090585319354906990, 113506497027753468870, 6842978980142398176930, 426187457118982899608730, 29098035465450244144376910, 2102916875063497845451016610, 156173789584825539524342644530
Offset: 0

Views

Author

Amarnath Murthy, Aug 22 2002

Keywords

Examples

			a(3) = (2+1)*(3+1)*(5+1) - 2*3*5 = 72 - 30 = 42.

Links

Crossrefs

Cf. A001065, A002110, A003415, A024451, A051674, A054640, A070826, A348507.

Programs

  • Maple
    for n from 1 to 25 do a[n] := product(ithprime(i)+1,i=1..n)-product(ithprime(i),i=1..n): od:seq(a[j],j=1..25);
  • Mathematica
    Module[{nn=20,p,pr,pr1},p=Prime[Range[nn]];pr=FoldList[Times,1,p];pr1= FoldList[Times,1,p+1];#[[2]]-#[[1]]&/@Rest[Thread[{pr,pr1}]]](* Harvey P. Dale, Feb 07 2015 *)
  • PARI
    A074107(n) = (prod(i=1,n,1+prime(i))-prod(i=1,n,prime(i))); \\ Antti Karttunen, Nov 19 2024

Formula

From Antti Karttunen, Nov 19 2024: (Start)
a(n) = A348507(A002110(n)) = A054640(n) - A002110(n) = A001065(A002110(n)).
a(n) >= A024451(n), because A348507(n) >= A003415(n).
For n >= 1, a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n).
(End)

Extensions

More terms from Sascha Kurz, Feb 01 2003
Term a(0)=0 prepended, data section further extended, and secondary definition added by Antti Karttunen, Nov 19 2024

A348732 a(n) = A003959(n) - A034448(n), where A003959 is multiplicative with a(p^e) = (p+1)^e and A034448 (usigma) is multiplicative with a(p^e) = (p^e)+1.

Original entry on oeis.org

0, 0, 0, 4, 0, 0, 0, 18, 6, 0, 0, 16, 0, 0, 0, 64, 0, 18, 0, 24, 0, 0, 0, 72, 10, 0, 36, 32, 0, 0, 0, 210, 0, 0, 0, 94, 0, 0, 0, 108, 0, 0, 0, 48, 36, 0, 0, 256, 14, 30, 0, 56, 0, 108, 0, 144, 0, 0, 0, 96, 0, 0, 48, 664, 0, 0, 0, 72, 0, 0, 0, 342, 0, 0, 40, 80, 0, 0, 0, 384, 174, 0, 0, 128, 0, 0, 0, 216, 0, 108
Offset: 1

Views

Author

Antti Karttunen, Oct 31 2021

Keywords

Links

Crossrefs

Cf. A003959, A005117 (positions of zeros), A034448, A034460, A048146, A348029, A348507.

Programs

  • Mathematica
    f1[p_, e_] := (p + 1)^e; f2[p_, e_] := p^e + 1; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)
  • PARI
    A003959(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]++); factorback(f); };
A034448(n) = { my(f = factor(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A348732(n) = (A003959(n)-A034448(n));

Formula

a(n) = A003959(n) - A034448(n).
a(n) = A348507(n) - A034460(n).
a(n) = A048146(n) + A348029(n).

A348508 a(n) = A003959(n) - 2*n, where A003959 is multiplicative with a(p^e) = (p+1)^e.

Original entry on oeis.org

-1, -1, -2, 1, -4, 0, -6, 11, -2, -2, -10, 12, -12, -4, -6, 49, -16, 12, -18, 14, -10, -8, -22, 60, -14, -10, 10, 16, -28, 12, -30, 179, -18, -14, -22, 72, -36, -16, -22, 82, -40, 12, -42, 20, 6, -20, -46, 228, -34, 8, -30, 22, -52, 84, -38, 104, -34, -26, -58, 96, -60, -28, 2, 601, -46, 12, -66, 26, -42, 4, -70, 288
Offset: 1

Views

Author

Antti Karttunen, Oct 30 2021

Keywords

Links

Crossrefs

Cf. A003959, A033879, A138636, A168036, A252748, A348029, A348970.

Programs

  • Mathematica
    f[p_, e_] := (p + 1)^e; a[1] = -1; a[n_] := Times @@ f @@@ FactorInteger[n] - 2*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); };
A348508(n) = (A003959(n) - 2*n);

Formula

a(n) = A003959(n) - 2*n.
a(n) = A348507(n) - n.
a(n) = A348029(n) - A033879(n).
From Antti Karttunen, Dec 05 2021: (Start)
a(n) = A168036(n) + A348970(n).
For all n >= 1, a(A138636(n)) = 12.
(End)
a(p) = 1 - p if p prime. - Bernard Schott, Feb 17 2022
Previous Showing 11-14 of 14 results.

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