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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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

A118359 Primes for which the weight as defined in A117078 is 7 and the gap as defined in A001223 is 6.

Original entry on oeis.org

83, 167, 251, 433, 503, 587, 601, 727, 1063, 1217, 1231, 1553, 1777, 1861, 1973, 1987, 2281, 2351, 2393, 2897, 3541, 4073, 4283, 4451, 4507, 4591, 4871, 5081, 5431, 5557, 5641, 5683
Offset: 1

Views

Author

Rémi Eismann, May 24 2006, May 04 2007

Keywords

Comments

The prime numbers in this sequence are of the form (14i-1) with i=(level(n)+1)/2, level(n) defined in A117563. level(n) is not multiple of 3.

Examples

			prime(24) = prime (23) + prime(23)mod(7) = prime (23) + prime(23)mod(77)
89 = 83 + 83mod(7) = 83 + 83mod(77)
k=7, level = 77/7 = 11

Links

Crossrefs

Cf. A117078, A117563, A001359, A074822, A118922, A118924, A119504, A119597, A119596, A119595.

A119504 Primes for which the weight as defined in A117078 is 23.

Original entry on oeis.org

631, 773, 2467, 2833, 3121, 3203, 3347, 3617, 4219, 4733, 4909, 4951, 5273, 6619, 7027, 7129, 7529, 8263, 8783, 9049, 9413, 9643, 9649, 10891, 11483, 11719, 12541, 13093, 13183, 13841, 14243, 14293, 14851, 15121, 15629, 15667, 15671, 15761
Offset: 1

Views

Author

Rémi Eismann, May 27 2006, May 04 2007

Keywords

Comments

The prime numbers in this sequence are of the form (56i-23+gap) with i=(level(n)+1)/2, level(n) defined in A117563.

Examples

			a(1) = prime(115) = 631 because prime(116) = prime(115) + (prime(115) mod 53) = 641
g(n) = 641 - 631 = 10
Prime(115) + 23 - 10 = 644, 644/46 = 14

Links

Crossrefs

Cf. A117078, A118359, A001359, A118380, A118924, A118922.

Formula

A117078 : a(n) = smallest k such that prime(n+1) = prime(n) + (prime(n) mod k), or 0 if no such k exists. prime(n) for which k=23.

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.

A130703 a(n) = smallest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 9, 14, 10, 27, 35, 22, 18, 65, 77, 18, 26, 119, 27, 38, 34, 27, 209, 46, 28, 55, 299, 36, 35, 377, 45, 62, 58, 45, 527, 40, 54, 629, 95, 54, 74, 779, 63, 86, 82, 63, 989, 94, 54, 161, 235, 68, 91, 265, 81, 65, 106, 81, 145, 118, 90, 1769, 1829
Offset: 1

Views

Author

Rémi Eismann, Aug 16 2007 - Jan 10 2011

Keywords

Comments

a(n) is the weight of triangular numbers.
The decomposition of triangular numbers into weight * level + gap is A000217(n) = a(n) * A184219(n) + (n + 1) if a(n) > 0.

Examples

			For n = 1 we have A000217(n) = 1, A000217(n+1) = 3; there is no k such that 3 - 1 = 2 = (1 mod k), hence a(1) = 0.
For n = 5 we have A000217(n) = 15, A000217(n+1) = 21; 9 is the smallest k such that 21 - 15 = 6 = (15 mod k), hence a(5) = 9.
For n = 22 we have A000217(n) = 253, A000217(n+1) = 276; 46 is the smallest k such that 276 - 253 = 23 = (253 mod k), hence a(22) = 46.

Links

Crossrefs

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368.

A125623 Primes p=prime(i) of level (1,16), i.e., such that A118534(i)=prime(i-16).

Original entry on oeis.org

356604959, 613768081, 709208323, 950803363, 979872743, 1174872271, 1186433617, 1625945609, 1796767963, 1840621901, 2348698453, 2547482281, 3385901059, 3446679371, 3512406283, 3735873397, 4080198391, 4106437259, 4319987921, 4695419887, 5285414713, 5288810297
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 (1,16): 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(48470200) - prime(48470199) = 950803519 - 950803363 = 950803363 - 950803207 = prime(48470199) - prime(48470199-16) and prime(48470199) has level 1 in A117563, so prime(48470199) = 950803363 has level (1,16).

Links

Crossrefs

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

Programs

  • PARI
    lista(nn) = my(v=primes(17)); forprime(p=61, nn, if(2*v[17]-p==v[1], print1(v[17], ", ")); v=concat(v[2..17], p)); \\ Jinyuan Wang, Jun 18 2021

Extensions

Definition and comment reworded following suggestions from the authors. - M. F. Hasler, Nov 30 2009
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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