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.

A075254 a(n) = n + (sum of prime factors of n taken with repetition).

Original entry on oeis.org

1, 4, 6, 8, 10, 11, 14, 14, 15, 17, 22, 19, 26, 23, 23, 24, 34, 26, 38, 29, 31, 35, 46, 33, 35, 41, 36, 39, 58, 40, 62, 42, 47, 53, 47, 46, 74, 59, 55, 51, 82, 54, 86, 59, 56, 71, 94, 59, 63, 62, 71, 69, 106, 65, 71, 69, 79, 89, 118, 72, 122, 95, 76, 76, 83, 82, 134, 89, 95, 84, 142
Offset: 1

Views

Author

Zak Seidov, Sep 10 2002

Keywords

Comments

a(n) = n + A001414(n).
Product of prime factors plus sum of prime factors of n. For minus instead of plus we have A075255, zeros A175787. - Gus Wiseman, Jan 26 2025

Examples

			a(6)=11 because 6=2*3, sopfr(6)=2+3=5 and 6+5=11.

Links

Crossrefs

A000027 gives product of prime factors, indices A003963.
A000040 lists the primes, differences A001223.
A001414 gives sum of prime factors, indices A056239.
A027746 lists prime factors, indices A112798, count A001222.
A075255 gives product of prime factors minus sum of prime factors.
Cf. A000720, A001055, A001222, A008472, A090340, A096461 (iteration), A175508, A319000, A325036, A325037, A325038, A379681, A379682.

Programs

  • Haskell
    a075254 n = n + a001414 n  -- Reinhard Zumkeller, Feb 27 2012
  • Magma
    [n eq 1 select 1 else (&+[p[1]*p[2]: p in Factorization(n)]) + n: n in [1..80]]; // G. C. Greubel, Jan 10 2019
  • Maple
    A075254 := proc(n)
    n+A001414(n) ;
end proc: # R. J. Mathar, Jul 27 2015
  • Mathematica
    Table[If[n==1,1, n +Plus@@Times@@@FactorInteger@n], {n, 80}] (* G. C. Greubel, Jan 10 2019 *)
  • PARI
    a(n) = my(f = factor(n)); n + sum(k=1, #f~, f[k,1]*f[k,2]); \\ Michel Marcus, Feb 22 2017
  • Sage
    [n + sum(factor(n)[j][0]*factor(n)[j][1] for j in range(0, len(factor(n)))) for n in range(1, 80)] # G. C. Greubel, Jan 10 2019

Formula

From Gus Wiseman, Jan 26 2025: (Start)
First differences are 1 - A090340(n).
a(n) = 2*n - A075255(n).
a(n) = 2*A001414(n) + A075255(n).
(End)

A090342 Difference between the sums of the prime factors, including multiplicity, of n and those of n + 3.

Original entry on oeis.org

-4, -3, -2, -3, -1, -1, 0, -5, -1, -6, 2, -1, 5, -8, 0, -11, 8, -2, 6, -14, 1, 3, 8, 0, -1, -14, -1, -20, 19, -4, 12, -2, 4, -18, -9, -6, 26, -20, 4, -32, 26, 1, 18, -32, 0, 11, 35, -9, -3, -41, 9, 1, 40, -11, -15, -46, 10, -30, 26, -1, 49, 15, -3, -55, -3, -10, 53, -50, 14, -59, 32, -1, 50, 21, -5, -56, 5, 6, 36, -70, -2, 21, 38
Offset: 1

Views

Author

Charles K. Layman (cklayman(AT)juno.com), Nov 25 2003

Keywords

Examples

			a(24)=0 because 24=2*2*2*3, 27=3*3*3 and (2+2+2+3)-(3+3+3)=0.

Links

Crossrefs

Cf. A001414, A090340, A090341, A090343.

Programs

  • Mathematica
    Join[{-4},#[[1]]-#[[4]]&/@(Partition[Table[Total[Flatten[Table[#[[1]], #[[2]]]&/@ FactorInteger[n]]],{n,2,90}],4,1])] (* Harvey P. Dale, Sep 16 2018 *)

Formula

a(n) = A001414(n) - A001414(n+3). - Wesley Ivan Hurt, Aug 29 2016

A090341 Difference between the sums of the prime factors, including multiplicity, of n and those of n + 2.

Original entry on oeis.org

-3, -2, -2, -1, -2, -1, 1, -1, -5, 0, -2, -2, 5, 1, -9, 0, -2, -1, 9, -4, -13, 4, 13, -6, 1, 4, -20, 1, -2, 0, 17, -9, 2, 9, -25, -11, 21, 10, -25, -1, -2, -3, 32, -10, -36, 14, 33, -1, -6, -5, -33, 6, 37, -2, -6, -18, -37, 19, -2, -21, 48, 21, -5, -4, -49, -5, 41, 7, -45, 2, -2, -27, 60, 16, -5, 5, -61, 5, 67, -30, -71, 29, 61, -31, -10
Offset: 1

Views

Author

Charles K. Layman (cklayman(AT)juno.com), Nov 25 2003

Keywords

Examples

			a(24)=-6 because 24=2*2*2*3, 26=2*13 and (2+2+2+3)-(2+13)=-6.

Crossrefs

Cf. A001414, A090340, A090342, A090343.

Programs

  • Mathematica
    Join[{-3},#[[1]]-#[[3]]&/@Partition[Table[Total[Flatten[Table[#[[1]], #[[2]]]& /@ FactorInteger[n]]],{n,2,90}],3,1]] (* Harvey P. Dale, Apr 06 2018 *)

Formula

a(n) = A001414(n) - A001414(n+2). - Wesley Ivan Hurt, Aug 29 2016

A090343 Difference between the sums of the prime factors, including multiplicity, of n and those of n + 4.

Original entry on oeis.org

-5, -3, -4, -2, -1, -2, -4, -1, -7, -2, 3, -1, -4, 1, -11, -1, 7, -5, -4, 0, 0, -2, 14, -2, -19, 5, -22, 1, 15, -9, 19, 0, -23, -2, -4, -1, -4, 9, -27, -4, 30, -13, -4, 4, -3, 13, 27, -6, -39, 1, 4, 4, 31, -20, -43, 1, -39, -2, 46, 0, 43, 17, -54, -9, -8, 2, -4, 9, -47, -25, 58, -11, 55, 21, -66, 10, 6, -25, -4, -1, -10, -2, 51, -3, -67
Offset: 1

Views

Author

Charles K. Layman (cklayman(AT)juno.com), Nov 25 2003

Keywords

Examples

			a(24)=-2 because 24=2*2*2*3, 28=2*2*7 and (2+2+2+3)-(2+2+7)=-2.

Crossrefs

Cf. A001414, A090340, A090341, A090342.

Programs

  • Mathematica
    Join[{-5},#[[1]]-#[[5]]&/@Partition[Table[Total[Times@@@FactorInteger[n]],{n,2,90}],5,1]] (* Harvey P. Dale, Jul 02 2022 *)

Formula

a(n) = A001414(n) - A001414(n+4). - Wesley Ivan Hurt, Aug 29 2016
Showing 1-4 of 4 results.

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