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 1-6 of 6 results.

A167130 Primes of the form A002808(k) + A062502(k+1).

Original entry on oeis.org

11, 11, 17, 23, 23, 31, 37, 37, 41, 47, 53, 53, 53, 59, 67, 71, 71, 73, 79, 79, 89, 97, 101, 113, 113, 113, 127, 131, 139, 151, 157, 157, 163, 167, 167, 173, 173, 181, 179, 191, 193, 193, 199, 211, 223, 229, 233, 239, 239, 241, 251, 257, 269, 269, 269, 271, 277
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 28 2009

Keywords

Examples

			a(1)=8+3(2*2*2)=11, a(2)=9+2(3*3)=11, a(3)=15+2(3*5)=17, a(4)=20+3(2*2*5)=23.

Crossrefs

Cf. A000040, A002808, A062502.

Extensions

Edited (but not checked) by N. J. A. Sloane, Nov 01 2009
Added another 113, replaced 197 by 199 and 211, inserted 229 etc. R. J. Mathar, Oct 29 2009

A359637 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have at least n prime factors, counted with multiplicity.

Original entry on oeis.org

7, 97, 349, 13309, 33613, 5594749, 84477247, 1524981247, 60924074749
Offset: 2

Views

Author

Hugo Pfoertner, Jan 16 2023

Keywords

Examples

			a(2) = 7: 8 = 2^3, 9 = 3^2, 10 = 2*5 all have at least the minimum number of 2 prime factors;
a(3) = 97: 98 = 2*7^2, 99 = 3^2*11, 100 = 2^2*5^2 have a minimum of 3 prime factors;
a(4) = 349: 350 = 2*5^2*7, 351 = 3^3*13, 352 = 2^5*11 have a minimum of 4 prime factors.

Crossrefs

Cf. A001222, A001359, A062502, A359636.

Programs

  • PARI
    a359637(maxp) = {my (k=2, pp=3); forprime (p=5, maxp, my(mi=oo); if (p-pp>2, for (j=pp+1, p-1, my(mo=bigomega(j)); if (mo=k, print1(pp,", "); k++)); pp=p)};
a359637(10^8)

A167133 Primes of the form (number of prime factors of k-th composite) plus (number of divisors of k-th composite).

Original entry on oeis.org

5, 7, 5, 5, 7, 11, 11, 13, 11, 5, 13, 11, 11, 17, 11, 13, 11, 11, 17, 11, 11, 5, 7, 11, 11, 11, 11, 5, 11, 11, 23, 11, 11, 11, 11, 13, 17, 11, 13, 11, 11, 11, 11, 11, 23, 11, 17, 11, 11, 11, 11, 11, 11, 5, 11, 23, 11, 11, 11, 7, 11, 11, 11, 5, 11, 11, 11, 11, 17, 23, 11, 11, 11, 11
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Oct 28 2009

Keywords

Comments

Contains every prime > 3 infinitely many times, as A000005(p^k)+A001222(p^k)=2*k+1 for prime p. - Robert Israel, Sep 30 2020

Examples

			a(1) = 2+3 = 5 (for 1st composite=4); a(2) = 3+4 = 7 (for 3rd composite=8).

Links

Crossrefs

Cf. A000005, A000040, A002808, A001222, A069346.
Cf. A035004, A062502.

Programs

  • Maple
    f:= proc(n) local F,x;
  if isprime(n) then return NULL fi;
  F:= ifactors(n)[2];
  x:= add(t[2],t=F) + mul(1+t[2],t=F);
  if isprime(x) then x fi
end proc:
map(f, [$4..1000]); # Robert Israel, Sep 30 2020

Extensions

Corrected and extended by R. J. Mathar, Oct 29 2009

A359638 a(n) is the least odd prime not in A001359 such that all subsequent composites in the gap up to the next prime have exactly n prime factors, counted with multiplicity.

Original entry on oeis.org

601, 1429, 81547, 248749, 27140749, 310314157, 3566181247
Offset: 3

Views

Author

Hugo Pfoertner, Jan 16 2023

Keywords

Crossrefs

Cf. A001222, A001359, A062502, A359636, A359637.

Programs

  • PARI
    a359638(maxp) = {my (k=3, pp=3); forprime (p=5, maxp, my (mi=oo, ma=0); if (p-pp>2, for (j=pp+1, p-1, my(mo=bigomega(j)); if(mo

A109709 Triangle in which n-th row gives the prime factors of the n-th composite number with repetition.

Original entry on oeis.org

2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 5, 2, 2, 3, 2, 7, 3, 5, 2, 2, 2, 2, 2, 3, 3, 2, 2, 5, 3, 7, 2, 11, 2, 2, 2, 3, 5, 5, 2, 13, 3, 3, 3, 2, 2, 7, 2, 3, 5, 2, 2, 2, 2, 2, 3, 11, 2, 17, 5, 7, 2, 2, 3, 3, 2, 19, 3, 13, 2, 2, 2, 5, 2, 3, 7, 2, 2, 11, 3, 3, 5, 2, 23, 2, 2, 2, 2, 3, 7, 7, 2, 5, 5, 3, 17, 2, 2, 13, 2, 3, 3, 3
Offset: 1

Views

Author

Lior Manor, Aug 08 2005

Keywords

Examples

			Irregular triangle begins:
  2, 2;
  2, 3;
  2, 2, 2;
  3, 3;
  2, 5;
  ...

Crossrefs

Column k=1 gives A056608.
Last terms in rows give A052369.
Row lengths give A062502(n+1).
Cf. A002808, A027746.

Programs

  • Mathematica
    fn[{a_,b_}]:=Table[a,b];Flatten[Table[fn/@FactorInteger[ ResourceFunction["Composite"][n]],{n,37}]] (* James C. McMahon, Mar 29 2024 *)

A375160 Square array T(n, k), n >= 2 and k >= 1, read by antidiagonals in ascending order, give the smallest number that starts a sequence of exactly k consecutive numbers each having exactly n prime factors (counted with multiplicity), or -1 if no such number exists.

Original entry on oeis.org

4, 8, 9, 16, 27, 33, 32, 135, 170, -1, 64, 944, 1274, 603, -1, 128, 5264, 15470, 4023, 602, -1, 256, 29888, 33614, 57967, 12122, 2522, -1, 512, 50624, 3145310, 8706123, 632148, 204323, 211673, -1
Offset: 2

Views

Author

Jean-Marc Rebert, Aug 09 2024

Keywords

Comments

All positive terms are composite.

Examples

			T(2,3) = 33 = 3*11, because both 34 and 35 have the same number of prime factors. Thus, 33 is the starting number of a run of 3 numbers that each have 2 prime factors (counted with multiplicity). No lesser number has this property, so T(2,3) = 33.
Table begins (upper left corner = T(2,1)):
   4        9       33      -1 ...
   8       27      170     603 ...
  16      135     1274    4023 ...
  32      944    15470   57967 ...
  ...     ...      ...     ... ...

Crossrefs

Cf. A002808, A062502.
Cf. Numbers m through m+k have the same number of prime divisors (with multiplicity): A045920 (k=1), A045939 (k=2), A045940 (k=3), A045941 (k=4), A045942 (k=5), A123103 (k=6), A123201 (k=7), A358017 (k=8), A358018 (k=9), A358019 (k=10).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.