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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A162174 Primes classified by level.

Original entry on oeis.org

5, 13, 19, 23, 31, 37, 43, 47, 53, 61, 73, 97, 113, 127, 131, 139, 151, 157, 163, 173, 181, 199, 211, 223, 233, 257, 263, 271, 293, 307, 313, 317, 337, 353, 373, 389, 397, 401, 421, 457, 479, 509, 523, 541, 547, 563, 571, 593, 607, 619, 647, 653, 661, 673, 691
Offset: 1

Views

Author

Rémi Eismann, Jun 27 2009

Keywords

Comments

Conjecture : primes classified by level are rarefying among prime numbers.
A000040(n) = 2, 3, 7, A162175(n), a(n) [From Rémi Eismann, Jun 27 2009]

Examples

			For prime(3)=5, A117078(3)=3 > A117563(3)=1 ; prime(3)=5 is classified by level. For prime(172)=1021, A117078(172)=337 > A117563(172)=3 ; prime(172)=1021 is classified by level.

Links

Crossrefs

Cf. A117078, A117563, A000040.
Cf. A162175. [From Rémi Eismann, Jun 27 2009]

Formula

If for prime(n), A117078(n) (the weight) > A117563(n) (the level) then prime(n) is classified by level.
If for prime(n), A117078(n) (the weight) <= A117563(n) (the level) and A117078(n) <> 0 then prime(n) is classified by weight. [From Rémi Eismann, Jun 27 2009]

A118464 Primes p=prime(i) of level (1,5), i.e., such that A118534(i) = prime(i-5).

Original entry on oeis.org

13933, 23633, 28229, 49223, 71363, 79633, 81239, 90547, 96857, 97613, 108827, 115363, 117443, 126781, 130657, 133733, 153533, 157679, 176819, 186799, 197389, 206651, 221327, 222199, 228139, 246947, 266297, 272203, 276049, 279221, 282493, 290627, 292493, 296299
Offset: 1

Views

Author

Rémi Eismann, May 04 2006

Keywords

Comments

This subsequence of A125830 and of A162174 gives primes of level (1,5): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

Examples

			prime(5061) = 49223 has level (1,5): prime(5062) = 49253 = 2*prime(5061) - prime(5061-5) = 2*prime(5061) - prime(5056).

Links

Crossrefs

Cf. A117078, A117563, A118534, A125830, A162174.

Programs

  • Mathematica
    With[{m = 5}, Prime@ Select[Range[m + 1, 3*10^4], If[MemberQ[{1, 2, 4}, #], 0, 2 Prime[#] - Prime[# + 1]] == Prime[# - m] &]] (* Michael De Vlieger, Jul 16 2017 *)
  • PARI
    lista(nn) = my(c=6, v=primes(6)); forprime(p=17, nn, if(2*v[c]-p==v[c=c%6+1], print1(precprime(p-1), ", ")); v[c]=p); \\ Jinyuan Wang, Jun 18 2021

Extensions

Edited by N. J. A. Sloane, May 14 2006
More terms from Rémi Eismann, May 21 2006
Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

A118467 Primes p = prime(i) of level (1,3), i.e., such that A118534(i) = prime(i-3).

Original entry on oeis.org

619, 1069, 1459, 1499, 1759, 1789, 2861, 3331, 3931, 4177, 4801, 4831, 5419, 6229, 6397, 8431, 8893, 9067, 9631, 11003, 11131, 11789, 12619, 14251, 15331, 15889, 16661, 17683, 17939, 18269, 18553, 19219, 19391, 19507, 20029, 20759, 22039, 22159, 22171, 22549
Offset: 1

Views

Author

Rémi Eismann, May 24 2006

Keywords

Comments

If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

Examples

			prime(115) - prime(114) = 631 - 619 = 619 - 607 = prime(114) - prime(114-3).

Links

Crossrefs

Subsequence of A125830 and A162174.
Cf. A006562 (primes of level (1,1)), A117078, A117563, A117876, A118464.

Programs

  • Mathematica
    Select[Partition[Prime[Range[2600]],5,1],#[[5]]-#[[4]]==#[[4]]-#[[1]]&][[All,4]] (* Harvey P. Dale, Aug 28 2021 *)

Extensions

Definition and comment reworded, following author's suggestions, by M. F. Hasler, Nov 30 2009

A119402 Primes p=prime(i) of level (1,11), i.e., such that A118534(i)=prime(i-11).

Original entry on oeis.org

576791, 3361517, 9433859, 10460719, 11630503, 11707537, 12080027, 19743677, 28716287, 33384517, 34961923, 36627659, 37776967, 38087983, 40794049, 45650359, 49152757, 52230229, 53152907, 53240927, 55036789, 56167103, 56177783, 57717749, 58804483, 71849423, 76119269
Offset: 1

Views

Author

Rémi Eismann and Fabien Sibenaler, Jul 25 2006

Keywords

Comments

This subsequence of A125830 and of A162174 gives primes of level (1,11): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

Examples

			prime(240963) - prime(240962) = 3361601 - 3361517 = 3361517 - 3361433 = prime(240962) - prime(240962-11) and prime(240962) has level 1 in A117563, so prime(240962)=3361517 has level (1,11).

Links

Crossrefs

Cf. A006562 (primes of level (1,1)), A117078, A117563, A006562, A117876, A118464, A118467.

Extensions

More terms from Fabien Sibenaler, Oct 20 2006
Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

A119403 Primes p=prime(i) of level (1,10), i.e., such that A118534(i)=prime(i-10).

Original entry on oeis.org

745757, 1103639, 1583369, 1895359, 2124049, 3327419, 4234537, 4437779, 5071973, 6287647, 7702573, 8470927, 8675923, 9493151, 9750079, 10868203, 11213843, 14244173, 14796253, 14978893, 15611909, 16489273, 17528681, 18280771, 19125163, 19403831, 19631411, 21975167
Offset: 1

Views

Author

Rémi Eismann and Fabien Sibenaler, Jul 25 2006

Keywords

Comments

This subsequence of A125830 and of A162174 gives primes of level (1,10): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

Examples

			prime(353166) - prime(353165) = 5072057 - 5071973 = 5071973 - 5071889 = prime(353165) - prime(353165-10) and prime(353165) has level 1 in A117563, so prime(353165)=5071973 has level (1,10).

Links

Crossrefs

Cf. A117078, A117563, A006562 (primes of level (1,1)), A117876, A118464, A118467.

Programs

  • PARI
    lista(nn) = my(c=11, v=primes(11)); forprime(p=37, nn, if(2*v[c]-p==v[c=c%11+1], print1(precprime(p-1), ", ")); v[c]=p); \\ Jinyuan Wang, Jun 18 2021

Extensions

More terms from Fabien Sibenaler, Oct 20 2006
Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

A119404 Primes p=prime(i) of level (1,9), i.e., such that A118534(i)=prime(i-9).

Original entry on oeis.org

678659, 855739, 1403981, 2366543, 2744783, 2830657, 3027539, 3317033, 4525909, 4676851, 5341463, 5819563, 7087123, 7181897, 8815663, 9324257, 9878929, 9976937, 10403251, 10440641, 10447457, 10766411, 10787377, 11829151, 11881957, 12539389, 14026433, 14087179
Offset: 1

Views

Author

Rémi Eismann and Fabien Sibenaler, Jul 25 2006

Keywords

Comments

This subsequence of A125830 and of A162174 gives primes of level (1,9): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

Examples

			prime(780815) - prime(780814) = 11882071 - 11881957 = 11881957 - 11881843 = prime(780814) - prime(780814-9) and prime(780814) has level 1 in A117563, so prime(780814)=11881957 has level (1,9).

Links

Crossrefs

Cf. A117078, A117563, A006562 (primes of level (1,1)), A117876, A118464, A118467.

Extensions

Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009

A090369 Smallest divisor of 2n that is > 2, or 0 if no such divisor exists.

Original entry on oeis.org

0, 4, 3, 4, 5, 3, 7, 4, 3, 4, 11, 3, 13, 4, 3, 4, 17, 3, 19, 4, 3, 4, 23, 3, 5, 4, 3, 4, 29, 3, 31, 4, 3, 4, 5, 3, 37, 4, 3, 4, 41, 3, 43, 4, 3, 4, 47, 3, 7, 4, 3, 4, 53, 3, 5, 4, 3, 4, 59, 3, 61, 4, 3, 4, 5, 3, 67, 4, 3, 4, 71, 3, 73, 4, 3, 4, 7, 3, 79, 4, 3, 4, 83, 3, 5, 4, 3, 4, 89, 3, 7, 4, 3, 4, 5
Offset: 1

Views

Author

Lekraj Beedassy, Nov 27 2003

Keywords

Links

Crossrefs

Cf. A090368, A020639, A117078, A117563, A118534.

Programs

  • Maple
    A090369 := proc(n) local lf,i ; lf := numtheory[divisors](2*n) ; for i from 1 to nops(lf) do if op(i,lf) > 2 then RETURN( op(i,lf) ) ; fi ; od ; RETURN(0) ; end : for n from 0 to 100 do printf("%d,",A090369(n)) ; od ; # R. J. Mathar, Jun 02 2006
  • Mathematica
    Join[{0},Table[SelectFirst[Divisors[2n],#>2&],{n,2,120}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 24 2017 *)

Extensions

More terms from Ray Chandler, Dec 02 2003
Edited by N. J. A. Sloane at the suggestion of Rémi Eismann, Sep 15 2007

A125576 Primes p=prime(i) of level (1,15), i.e., such that A118534(i)=prime(i-15).

Original entry on oeis.org

264426203, 295902073, 361949821, 704544167, 1075639757, 1259347393, 1290546427, 1301756207, 1335396547, 1370742383, 1460811643, 1497078991, 1514647247, 1643839649, 1783137281, 2142070103, 2424093281, 2471124197, 2494743721, 2577014057, 2706824389, 2951139253
Offset: 1

Views

Author

Rémi Eismann and Fabien Sibenaler, Jan 27 2007

Keywords

Comments

This subsequence of A125830 and of A162174 gives primes of level (1,15): If prime(i) has level 1 in A117563 and 2*prime(i) - prime(i+1) = prime(i-k), then we say that prime(i) has level (1,k).

Examples

			prime(16042282) - prime(16042281) = 295902247 - 295902073 = 295902073 - 295901899 = prime(16042281) - prime(16042281-15) and prime(16042281) has level 1 in A117563, so prime(16042281)=295902073 has level (1,15).

Links

Crossrefs

Cf. A117078, A117563, A006562 (primes of level (1,1)), A117876, A118464, A118467, A119402, A119403, A119404.

Programs

  • PARI
    lista(nn) = my(c=16, v=primes(16)); forprime(p=59, nn, if(2*v[c]-p==v[c=c%16+1], print1(precprime(p-1), ", ")); v[c]=p); \\ Jinyuan Wang, Jun 18 2021

Extensions

Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009
Terms a(5) and beyond from b-file by Andrew Howroyd, Feb 05 2018

A130533 a(n) = smallest k such that A001358(n+1) = A001358(n) + (A001358(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 2, 6, 13, 9, 2, 19, 2, 19, 2, 3, 4, 37, 8, 43, 47, 47, 53, 2, 6, 59, 61, 8, 71, 6, 79, 2, 5, 83, 89, 2, 3, 12, 101, 107, 4, 3, 3, 2, 11
Offset: 1

Views

Author

Rémi Eismann, Aug 16 2007 - Jan 20 2011

Keywords

Comments

a(n) is the "weight" of semiprimes.
The decomposition of semiprimes into weight * level + gap is A001358(n) = a(n) * A184729(n) + A065516(n) if a(n) > 0.

Examples

			For n = 1 we have A001358(n) = 4, A001358(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A001358(n) = 9, A001358(n+1) = 10; 2 is the smallest k such that 10 - 9 = 1 = (9 mod k), hence a(3) = 2.
For n = 19 we have A001358(n) = 55, A001358(n+1) = 57; 53 is the smallest k such that 57 - 55 = 2 = (55 mod k), hence a(19) = 53.

Links

Crossrefs

Cf. A001358, A065516, A184729, A184728, A117078, A117563, A001223, A118534.

A130650 a(n) = smallest k such that A014612(n+1) = A014612(n) + (A014612(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 4, 13, 2, 13, 18, 4, 43, 8, 3, 41, 4, 4, 3, 13, 2, 37, 16, 43, 97, 4, 9, 10, 53, 4, 5, 10, 3, 6, 61, 43, 2, 11, 2, 12, 163, 8, 13, 2, 5, 173, 8, 89, 4, 3, 37, 61, 101, 101, 107, 229, 113
Offset: 1

Views

Author

Rémi Eismann, Aug 16 2007 - Jan 21 2011

Keywords

Comments

a(n) is the "weight" of 3-almost primes.
The decomposition of 3-almost primes into weight * level + gap is A014612(n) = a(n) * A184753(n) + A114403(n) if a(n) > 0.

Examples

			For n = 1 we have A014612(1) = 8, A014612(2) = 12; there is no k such that 12 - 8 = 4 = (8 mod k), hence a(1) = 0.
For n = 3 we have A014612(3) = 18, A014612(4) = 20; 4 is the smallest k such that 20 - 18 = 2 = (18 mod k), hence a(3) = 4.
For n = 21 we have A014612(21) = 98, A014612(22) = 99; 97 is the smallest k such that 99 - 98 = 1 = (97 mod k), hence a(21) = 97.

Links

Crossrefs

Cf. A014612, A114403, A184753, A184752, A117078, A117563, A001223, A118534.
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