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 1-3 of 3 results.

A196934 a(n) is the first occurrence of n in sequence A078498.

Original entry on oeis.org

5, 8, 18, 14, 25, 38, 43, 50, 61, 48, 132, 167, 100, 88, 151, 217, 176, 216, 270, 214, 300, 785, 429, 687, 308, 1083, 374, 644, 713, 320, 840, 608, 654, 577, 1005, 1409, 1631, 1215, 928, 1386, 2304, 1984, 1203, 2336, 853, 1638, 1899, 1806, 1974, 1594, 1228
Offset: 1

Views

Author

Lei Zhou, Oct 07 2011

Keywords

Comments

Conjecture: Any prime number greater than 11 (p) can be the center term of arithmetic progressions prime chain p-6k, p, p+6k, while k>0.
a(n) is also the maximum number k that is needed to find a p(i)-6k, p(i), p(i)+6k kind of arithmetic progressions prime chain for all i <= n, while p(i) is the i-th prime number.
The Mathematica program gives the first 51 items.

Examples

			A078498(5)=1 (take the offset 5),  so a(1)=5;
2 first occurs as A078498(8), so a(2)=8;

Links

Crossrefs

Cf. A078498, A078497, A001748.

Programs

  • Mathematica
    max = 51; Array[fa, max]; Do[fa[i] = 0, {i, 1, max}]; ct = 0; i = 4; While[ct < max, i++; p = Prime[i]; j = 0; While[j++; df = 6*j; ! ((PrimeQ[p + df]) && (PrimeQ[p - df]))]; If[j <= max, If[fa[j] == 0, fa[j] = i; ct++]]]; Table[fa[i], {i, 1, max}]

A078497 The member r of a triple of primes (p,q,r) in arithmetic progression which sum to 3*prime(n) = A001748(n) = p + q + r.

Original entry on oeis.org

7, 11, 17, 19, 23, 31, 29, 41, 43, 43, 53, 67, 53, 59, 71, 79, 73, 83, 79, 97, 107, 107, 127, 113, 109, 113, 139, 137, 151, 149, 167, 151, 167, 163, 163, 199, 197, 179, 191, 199, 233, 223, 227, 241, 223, 283, 257, 277, 239, 251, 271, 263, 263, 269, 281, 313
Offset: 3

Views

Author

Serhat Sevki Dincer (sevki(AT)ug.bilkent.edu.tr), Nov 27 2002

Keywords

Comments

In case more than one triple of primes p, q=p+d and r=p+2*d exists, we take r=a(n) from the triple with the smallest d. This shows the difference from A092940, which would take the maximum r over all triples. - R. J. Mathar, May 19 2007

Examples

			a(1) = 7 because 3+5+7 = 15;
a(2) = 11 because 3+7+11 = 21;
a(3) = 17 because 5+11+17= 33.

Crossrefs

Cf. A078496, A078498, A001748, A092940, A071681, A078611.

Programs

  • Maple
    A078497 := proc(n) local p3, i,d,r,p; p3 := ithprime(n) ; i := n+1 ; while true do r := ithprime(i) ; d := r-p3 ; p := p3-d ; if isprime(p) then RETURN(r) ; fi ; i := i+1 ; od ; RETURN(-1) ; end: for n from 3 to 60 do printf("%d, ",A078497(n)) ; od ; # R. J. Mathar, May 19 2007
  • Mathematica
    f[n_] := Block[{p = Prime[n], k}, k = p + 1; While[ !PrimeQ[k] || !PrimeQ[2p - k], k++ ]; k]; Table[ f[n], {n, 3, 60}]

Extensions

Edited and extended by Robert G. Wilson v, Nov 29 2002
Further edited by N. J. A. Sloane, Jul 03 2008 at the suggestion of R. J. Mathar

A196935 a(n) is the number of arithmetic progressions prime chains in the form of p(n)-6k, p(n), p(n)+6k, while k > 0 and p(n) > 6k.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 3, 3, 4, 4, 5, 3, 4, 6, 5, 4, 4, 6, 5, 7, 6, 6, 6, 5, 7, 8, 9, 6, 10, 8, 7, 6, 9, 8, 9, 6, 8, 10, 10, 6, 9, 10, 11, 8, 11, 10, 9, 13, 13, 13, 13, 9, 10, 13, 11, 12, 14, 15, 11, 12, 12, 14, 17, 13, 18, 14, 14, 16, 14, 16, 14, 16, 15, 16
Offset: 5

Views

Author

Lei Zhou, Oct 07 2011

Keywords

Comments

Conjecture: a(n) > 0 for all n >= 5.
The Mathematica program gives term 5 through 80.

Examples

			n = 5, p(5) = 11; {5, 11, 17} forms a difference 6 Arithmetic Progressions Prime chain. And this is the only occurrence for 11.  So a(5) = 1;
n = 6, p(6) = 13; {7, 13, 19} forms a difference 6 Arithmetic Progressions Prime chain. And this is the only occurrence for 11.  So a(6) = 1;
...
n = 10, p(10) = 29; {17, 29, 41}, {11, 29, 47}, {5, 29, 53} form Arithmetic Progressions Prime chains with difference 12, 18, 24 respectively.  So a(10) = 3;

Links

Crossrefs

Cf. A196934, A078498, A078497, A001748.

Programs

  • Mathematica
    Table[ct = 0; p = Prime[i]; j = 0; While[j++; df = 6*j; df < p, If[(PrimeQ[p + df]) && (PrimeQ[p - df]), ct++]]; ct, {i, 5, 80}]
Showing 1-3 of 3 results.

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

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