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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A141467 a(1)=1; for n > 1, a(n) is the product of prime factors of the n-th composite, but with the largest prime incremented by 3 and the smallest decremented by 1.

Original entry on oeis.org

1, 6, 10, 12, 8, 12, 10, 16, 20, 18, 16, 20, 14, 24, 32, 16, 36, 20, 24, 40, 28, 20, 40, 36, 22, 32, 32, 30, 28, 48, 26, 48, 60, 40, 40, 32, 54, 56, 40, 44, 32, 48, 34, 60, 80, 64, 42, 40, 52, 50, 72, 40, 80, 44, 84, 48, 64, 108, 44, 60, 80, 46, 64, 56, 72, 96, 52, 68, 50, 88, 96, 70, 84
Offset: 1

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Author

Juri-Stepan Gerasimov, Aug 08 2008

Keywords

Comments

In the prime number decomposition of k=A002808(n), one instance of the largest prime, pmax=A052369(n), is replaced by pmax+3 and one instance of the smallest prime, pmin=A056608(n), is replaced by pmin-1. a(n) is the product of this modified set of factors if nonprime. The case of n=1, k=4, is the only case where this modified product (2+3)*(2-1)=5 is prime and listed as a(1)=1.

Examples

			     n-th composite
  n  & factorization      transformed product
  -  ---------------  --------------------------
  1      4 = 2*2      (2-1)*(2+3)   = 1*5   =  5 (prime)
  2      6 = 2*3      (2-1)*(3+3)   = 1*6   =  6 = a(2)
  3      8 = 2*2*2    (2-1)*2*(2+3) = 1*2*5 = 10 = a(3)
  4      9 = 3*3      (3-1)*(3+3)   = 2*6   = 12 = a(4)
  5     10 = 2*5      (2-1)*(5+3)   = 1*8   =  8 = a(5)
  6     12 = 2*2*3    (2-1)*2*(3+3) = 1*2*6 = 12 = a(6)
  7     14 = 2*7      (2-1)*(7+3)   = 1*10  = 10 = a(7)

Formula

a(n) = k*(pmax+3)*(pmin-1)/(pmin*pmax), n > 1, where k=A002808(n), pmin=A056608(n), pmax=A052369(n).

Extensions

Edited by R. J. Mathar, Aug 14 2008
Further edits by Jon E. Schoenfield, Feb 20 2021

A141552 Transformed products of prime factors of the composites, the largest prime and smallest prime incremented by 1.

Original entry on oeis.org

9, 12, 18, 16, 18, 24, 24, 24, 36, 36, 36, 32, 36, 48, 36, 42, 48, 48, 54, 72, 48, 54, 48, 72, 60, 56, 72, 72, 72, 72, 72, 96, 64, 90, 72, 84, 108, 72, 96, 80, 90, 108, 96, 96, 144, 84, 108, 108, 96, 120, 144, 114, 120, 120, 96, 126, 144, 144, 126, 144, 108, 132, 120, 144
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Aug 14 2008

Keywords

Comments

In the prime number decomposition of k=A002808(i), i=1,2,3,.., one instance of the largest prime, pmax=A052369(i), is replaced by pmax+1 and one instance of the smallest prime, pmin=A056608(i), is replaced by pmin+1. The product of this modified list of factors, k*(pmax+1)*(pmin+1)/(pmin*pmax), is added to the sequence.

Examples

			k(1)=4=(p(max)=2)*(p(min)=2), transformed (2+1)*(2+1)=3*3=9=a(1).
k(2)=6=(p(max)=3)*(p(min)=2), transformed (3+1)*(2+1)=4*3=12=a(2).
k(6)=12=(p(max)=3)*(p=2)*(p(min)=2), transformed (3+1)*2*(2+1)=4*2*3=24=a(6), etc.

Crossrefs

Cf. A141218, A141219, A141220, A141284.

Extensions

Edited and corrected by R. J. Mathar, Aug 18 2008

A161643 Sum of all numbers from the smallest up to the largest prime factor of the n-th composite.

Original entry on oeis.org

2, 5, 2, 3, 14, 5, 27, 12, 2, 5, 14, 25, 65, 5, 5, 90, 3, 27, 14, 2, 63, 152, 18, 5, 189, 88, 14, 27, 65, 12, 275, 5, 7, 14, 150, 90, 5, 56, 27, 187, 434, 14, 495, 25, 2, 81, 65, 152, 273, 27, 5, 702, 12, 189, 45, 90, 14, 3, 860, 27, 143, 945, 432, 65, 14, 70, 275, 493, 1127, 180, 5, 27
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 15 2009

Keywords

Examples

			For n = 5, the 5th composite is 10 = 2*5. a(5) = 2+3+4+5 = 14.

Crossrefs

Cf. A002808.

Formula

a(n) = A000217(A052369(n))-A000217(A056608(n)-1). - R. J. Mathar, Jun 16 2009

Extensions

Replaced 80 by 90 in two places, R. J. Mathar, Jun 16 2009

A161965 Composite numbers in A161849 in the order of appearance.

Original entry on oeis.org

4, 6, 4, 4, 6, 8, 4, 6, 6, 4, 10, 9, 4, 6, 4, 4, 4, 4, 8, 4, 10, 4, 12, 14, 4, 4, 10, 4, 6, 9, 12, 8, 6, 4, 4, 6, 6, 6, 4, 18, 8, 4, 9, 4, 4, 9, 4, 14, 9, 4, 6, 6, 18, 8, 4, 6, 20, 4, 6, 8, 15, 4, 10, 4, 4, 9, 8, 4, 16, 4, 6, 4, 6, 8, 6, 12, 4, 10, 4, 14, 4, 6, 6, 10, 10, 10, 12, 12, 4, 14, 4, 18, 16
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 23 2009

Keywords

Examples

			The sequence shows a(1) = A161849(55)=4, a(2)=A161849(66) = 6,
a(3) = A161849(70)=4 etc.

Crossrefs

Cf. A002808, A052369, A056608.

Extensions

Definition simplified, sequence extended by R. J. Mathar, Sep 11 2009

A161966 Primes in A161849 in order of appearance.

Original entry on oeis.org

2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 3, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 7, 2, 2, 3, 5, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 5, 2, 2, 3, 3, 2, 2, 5, 2, 2, 3, 2, 2, 11, 2, 2, 2, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 5, 3, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 3
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Jun 23 2009

Keywords

Comments

A052369(k) mod A056608(k) where prime.
The remainder of the division of the largest by the smallest prime divisor of the k-th composite, discarding all nonprime results, as k increases.

Examples

			The first seven entries of A161849 are not prime, and a(1)=A161849(8)=2 is the first one selected.
The entries A161849(9..20) are not prime, and a(2)=A161849(21) is the next one selected.

Crossrefs

Cf. A002808, A052369, A056608.

Extensions

Edited and extended by R. J. Mathar, Aug 02 2009
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