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 1-4 of 4 results.

A287056 a(n) is such that A100827(n) = A082917(n - a(n)) - 1, or -1 if there is no corresponding term.

Original entry on oeis.org

-1, -1, -1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, 0, -1, 1, 1, 1, 1, 1, 1, -1, 2, 2, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 2, 2, 2, -1, 3, 3, 3, 3, -1, 4, 4, -1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Jud McCranie, May 18 2017

Keywords

Comments

Most of the known terms of A100827 are 1 less than a term in A082917, and conversely. This sequences looks at the location in the sequence of the corresponding terms. Negative terms do not occur among the known terms of this sequence. When a(n+1) is different from a(n) (and both are nonnegative), there are |a(n+1)-a(n)| terms in one of the sequences that aren't in the other. With some irregularities, this sequence generally gradually increases at first, reaching a(49)=5. Then there are 9 a(n)=5, followed by 20 a(n)=4, followed by 30 a(n)=3, and then a(n)=2 for n=108 to 229. What is the behavior of the rest of the sequence? Does it stay at a(n)=2?

Examples

			Examples: A100827(6)=47, A082917(5)=47+1, so a(6) = 6-5 = 1. A100827(23)=779, A082917(21)=779+1, so a(23) = 23-21 = 2.

Links

Crossrefs

Cf. A082917, A100827.

A100827 Highly cototient numbers: records for a(n) in A063741.

Original entry on oeis.org

2, 4, 8, 23, 35, 47, 59, 63, 83, 89, 113, 119, 167, 209, 269, 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889, 2099, 2309, 2729, 3149, 3359, 3569, 3989, 4199, 4289, 4409, 4619, 5249, 5459, 5879, 6089, 6509, 6719, 6929
Offset: 1

Views

Author

Alonso del Arte, Jan 06 2005

Keywords

Comments

Each number k on this list has more solutions to the equation x - phi(x) = k (where phi is Euler's totient function, A000010) than any preceding k except 1.
This sequence is a subset of A063741. As noted in that sequence, there are infinitely many solutions to x - phi(x) = 1. Unlike A097942, the highly totient numbers, this sequence has many odd numbers besides 1.
With the expection of 2, 4, 8, all of the known terms are congruent to -1 mod a primorial (A002110). The specific primorial satisfying this congruence would result in a sequence similar to A080404 a(n)=A007947[A055932(n)]. - Wilfredo Lopez (chakotay147138274(AT)yahoo.com), Dec 28 2006
Because most of the solutions to x - phi(x) = k are semiprimes p*q with p+q=k+1, it appears that this sequence eventually has terms that are one less than the Goldbach-related sequence A082917. In fact, terms a(108) to a(176) are A082917(n)-1 for n=106..174. [T. D. Noe, Mar 16 2010] This holds through a(229). [Jud McCranie, May 18 2017]

Examples

			a(3) = 8 since x - phi(x) = 8 has three solutions, {12, 14, 16}, one more than a(2) = 4 which has two solutions, {6, 8}.

Links

Crossrefs

Cf. A063741, A097942, A007374, A101373.

Programs

  • Mathematica
    searchMax = 4000; coPhiAnsYldList = Table[0, {searchMax}]; Do[coPhiAns = m - EulerPhi[m]; If[coPhiAns <= searchMax, coPhiAnsYldList[[coPhiAns]]++ ], {m, 1, searchMax^2}]; highlyCototientList = {2}; currHigh = 2; Do[If[coPhiAnsYldList[[n]] > coPhiAnsYldList[[currHigh]], highlyCototientList = {highlyCototientList, n}; currHigh = n], {n, 2, searchMax}]; Flatten[highlyCototientList]

Extensions

More terms from Robert G. Wilson v, Jan 08 2005

A082918 Record-setting entries in A002375 (number of decompositions into two odd primes).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 19, 21, 24, 27, 30, 32, 41, 44, 51, 52, 57, 58, 68, 73, 76, 83, 91, 97, 114, 128, 138, 154, 163, 165, 171, 190, 198, 218, 222, 241, 268, 274, 292, 303, 329, 330, 340, 362, 393, 394, 433, 446, 447, 466, 477, 517, 530, 571, 615
Offset: 1

Views

Author

Hugo Pfoertner, Apr 15 2003

Keywords

Examples

			11 is not in the sequence because the first number that can be decomposed into a sum of two odd primes in 12 different ways is 120, whereas the first number with 11 decompositions is 144.

Links

Crossrefs

Cf. A002375, A082917 (numbers for which a new record is achieved).

Programs

  • Julia
    # See links.

A109679 Smallest even number which is the unordered sum of two primes in more ways than any previous even number.

Original entry on oeis.org

2, 4, 10, 22, 34, 48, 60, 78, 84, 90, 114, 120, 168, 180, 210, 300, 330, 390, 420, 510, 630, 780, 840, 990, 1050, 1140, 1260, 1470, 1650, 1680, 1890, 2100, 2310, 2730, 3150, 3570, 3990, 4200, 4410, 4620, 5250, 5460, 6090, 6510, 6930, 7980, 8190, 9030, 9240
Offset: 1

Views

Author

Gilmar Rodriguez Pierluissi (gilmarlily(AT)yahoo.com), Aug 30 2005

Keywords

Comments

Record value of A023036 or A045917.
a(n)== 0 (mod 30) for n > 13.

Links

Crossrefs

Essentially the same as A082917. Cf. A082918, A002375, A023036, A045917, A000954.

Programs

  • Mathematica
    f[n_] := Length[ Select[n - Prime@ Range@ PrimePi[n/2], PrimeQ]]; t = {}; mxm = -1; Do[ If[ f[n] > mxm, AppendTo[t, n]; mxm = f[n]], {n, 2, 9000, 2}]; t

Extensions

Edited and extended by Robert G. Wilson v, Sep 08 2005
Changed offset from 0 to 1 by Vincenzo Librandi, Apr 18 2013
Showing 1-4 of 4 results.

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