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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A146071 Consider A145834 as the first step of the sieving (subtracting the sum of its prime factors with repetition from the composite numbers). This sequence is the result of the subsequent application of above described sieving - thus all terms of this sequence arise as prime numbers.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 5, 7, 2, 3, 11, 11, 3, 7, 7, 11, 3, 17, 11, 3, 19, 7, 23, 11, 17, 23, 29, 11, 29, 7, 11, 37, 23, 17, 31, 23, 43, 23, 43, 23, 3, 37, 29, 17, 23, 47, 17, 47, 43, 43, 37, 23, 29, 53, 59, 37, 67, 43, 23, 43, 17, 41, 23, 71, 59, 71, 47, 59, 7, 71, 83, 23, 23, 41, 67, 17, 59
Offset: 1

Views

Author

Alexander R. Povolotsky, Oct 26 2008

Keywords

Comments

Florentin Smarandache (in a Sunday, Nov 02 2008 email exchange) asks: how many times does each prime (> 3) appear in this sequence? This question can also be asked about A029909. - Alexander R. Povolotsky, Nov 07 2008

Crossrefs

Cf. A145834, A001414, A002808, A075255.

Programs

Extensions

More terms from M. F. Hasler, Nov 02 2008

A075255 a(n) = n - (sum of prime factors of n (with repetition)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 3, 3, 0, 5, 0, 5, 7, 8, 0, 10, 0, 11, 11, 9, 0, 15, 15, 11, 18, 17, 0, 20, 0, 22, 19, 15, 23, 26, 0, 17, 23, 29, 0, 30, 0, 29, 34, 21, 0, 37, 35, 38, 31, 35, 0, 43, 39, 43, 35, 27, 0, 48, 0, 29, 50, 52, 47, 50, 0, 47, 43, 56, 0, 60, 0, 35, 62, 53, 59, 60
Offset: 1

Views

Author

Zak Seidov, Sep 10 2002

Keywords

Examples

			a(6) = 1 because 6 = 2 * 3, sopfr(6) = 2 + 3 = 5 and 6 - 5 = 1.

Links

Crossrefs

Cf. A001414, A008472, A075254, A075653.
Cf. A145834 (= 0 followed by the nonzero terms of this sequence). - M. F. Hasler, Oct 31 2008

Programs

  • Magma
    [n eq 1 select 1 else n-(&+[p[1]*p[2]: p in Factorization(n)]): n in [1..80]]; // G. C. Greubel, Jan 11 2019
  • Maple
    a:= n-> n-add(i[1]*i[2], i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 07 2015
  • Mathematica
    Join[{1}, Table[n - Total[Times@@@FactorInteger[n]], {n, 2, 80}]] (* Harvey P. Dale, Sep 20 2011 *)
  • PARI
    A075255(n)=n-sum(i=1,#n=factor(n)~,n[1,i]*n[2,i]) \\ M. F. Hasler, Oct 31 2008
  • Python
    from sympy import factorint
def A075255(n): return n - sum(factorint(n,multiple=True)) # Chai Wah Wu, May 19 2022
  • 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 11 2019

Formula

a(n) = n - A001414(n).
a(n) = 0 if n is prime or if n = 4. - Alonso del Arte, Jul 31 2018

A146764 Primes not in A075255.

Original entry on oeis.org

13, 61, 73, 109, 151, 181, 229, 241, 257, 293, 307, 313, 349, 353, 373, 397, 409, 487, 509, 557, 571, 577, 601, 613, 643, 653, 661, 709, 727, 733, 739, 751, 761, 773, 811, 823, 937, 941, 977
Offset: 1

Views

Author

M. F. Hasler, Nov 04 2008

Keywords

Comments

It has been asked whether A146071 contains all primes. The answer is no: since A075255(n) > n/2-2 for nonprime n, any prime p that did not appear until the rank 2(p+2) is not in A075255. This is a sufficient condition for not being in A146071, but unless proved otherwise, there may be primes in A075255, i.e., not listed here, which nevertheless do not appear in A146071.

Crossrefs

Cf. A075255, A146071, A145834.

Programs

  • PARI
    A146764( END=999 )=local( n=1, t=0, k); forprime( p=1,END, while( n<2*(p+2), isprime( k=A075255(n++)) || next; t=bitor(1<
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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