A251239 -id:A251239 - cp's OEIS frontend

A253297 Primes p for which some multiple kp > 2p appears before p does in A098550.

5, 7, 13, 17, 23, 29, 41, 67, 83, 97

Offset: 1

L. Edson Jeffery, Dec 29 2014

nonn

It is conjectured (see A098550) that there are no more terms of the sequence.
Corresponding indices in A098550 are {9, 15, 23, 30, 51, 61, 87, 142, 175, 204}; indices in A000040 are {3, 4, 6, 7, 9, 10, 13, 19, 23, 25}.
A098550(A098551(a(n)) - 2) / a(n) > 2. - Reinhard Zumkeller, Dec 30 2014

Cf. A098550.

Cf. A010051, A098551, A251239, A251541.

a253297 n = a253297_list !! (n-1)
a253297_list = f a098550_list where
   f (u:vs@(:v:)) = if a010051' v == 1 && div u v > 2
                         then v : f vs else f vs
-- Reinhard Zumkeller, Dec 30 2014

A253573 Rectangular array a(n,k) read by upwards antidiagonals: row A(n) is the result of applying the function defined in A098550 to the set comprising row n of A253572, for n >= 2.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 9, 1, 2, 3, 4, 9, 8, 1, 2, 3, 4, 9, 8, 27, 1, 2, 3, 4, 9, 8, 15, 16, 1, 2, 3, 4, 9, 8, 15, 16, 81, 1, 2, 3, 4, 9, 8, 15, 14, 5, 32, 1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 243, 1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 64

Offset: 2

L. Edson Jeffery, Jan 03 2015

Let A(n) be the n-th row of this table and B(n) the n-th row of A253572. For n >= 2, A(n) is initialized with the first three entries of B(n). Let a(n,k) be the next entry of A(n) to be found. Then a(n,k) = m = the least number in B(n) such that m is not already in A(n), gcd(a(n,k-1),m) = 1 and gcd(a(n-2),m) > 1.
A(n) is not a permutation of B(n), for any n. Proof: Let a(n,k) be the k-th entry in A(n). By the definition of B(n) (see A253572), for k>1, there are distinct primes p,q in {prime(1),...,prime(n)} such that p divides a(n,k-1) and q divides a(n,k-2). It follows that we can never have a(n,k) = m*primorial(n), for any k>3 and any m>0. QED
Conjecture 1: Successive rows tend to A098550.
Conjecture 2: The main diagonal is A098550.
Conjecture 3: For each n, the primes prime(1), ..., prime(n) in row n appear in their natural order.
Conjecture 4: a(n,A251239(j)) = A000040(j), for j = 1,...,n inclusive.

			Array A starts:
{1, 2, 3, 4, 9, 8, 27, 16, 81, 32, 243, 64, 729, 128, 2187}
{1, 2, 3, 4, 9, 8, 15, 16,  5,  6,  25, 12, 125,  18,  625}
{1, 2, 3, 4, 9, 8, 15, 14,  5,  6,  25, 12,  35,  16,    7}
{1, 2, 3, 4, 9, 8, 15, 14,  5,  6,  25, 12,  35,  16,    7}
{1, 2, 3, 4, 9, 8, 15, 14,  5,  6,  25, 12,  35,  16,    7}
{1, 2, 3, 4, 9, 8, 15, 14,  5,  6,  25, 12,  35,  16,    7}

Cf. A253572, A098550 (and many cross-references therein).

r = 13; max = 300; prev = Table[2^j, {j, 0, max}]; Do[y[n] = {}; g = {-1}; next = Take[Union[Flatten[Table[Prime[n]^j*prev, {j, 0, max}]]], max]; prev = next; Do[AppendTo[y[n], next[[1]]]; next = Delete[next, 1], {3}]; While[g != {0}, a = y[n][[-1]]; b = y[n][[-2]]; g = FirstPosition[next, v_ /; GCD[a, v] == 1 && GCD[b, v] > 1, 0]; If[g != {0}, y[n] = Flatten[Append[y[n], next[[g]]]]; next = Delete[next, g]]], {n, 2, r}]; Flatten[Table[y[n - k + 1][[k]], {n, 2, r}, {k, n - 1}]] (* Array antidiagonals flattened *)

A256213 Indices of prime terms in A254077.

Original entry on oeis.org

2, 3, 10, 13, 21, 24, 33, 43, 46, 58, 61, 70, 75, 90, 97, 102, 111, 120, 133, 138, 141, 155, 162, 178, 187, 192, 200, 209, 214, 219, 247, 255, 262, 265, 286, 289, 302, 312, 319, 339, 346, 349, 366, 376, 392, 395, 413, 428, 435, 444, 449, 468, 471, 483, 496

Offset: 1

N. J. A. Sloane, Mar 26 2015

nonn

It would be nice to have a definition for this sequence which was independent of A254077.
From John Mason, Apr 15 2015: (Start)
Apparently, taking into account the first 675025 terms, corresponding to the first 20 million terms of A254077, a(n) divided by n-th prime A000040(n) is converging to 2. Here is the tail of this calculation:
n        a(n)   prime    ratio
675016 19999695 10167763 1.966971004
675017 19999723 10167779 1.966970663
675018 19999766 10167799 1.966971023
675019 19999771 10167803 1.966970741
675020 19999787 10167809 1.966971154
675021 19999790 10167811 1.966971062
675022 19999903 10167881 1.966968634
675023 19999974 10167917 1.966968652
675024 19999985 10167919 1.966969347
675025 19999988 10167923 1.966968869
(End)
From John Mason, May 26 2016: (Start)
With respect to the previous observation, apparently, taking into account the first 26694011 terms, corresponding to the first 10^9 terms of A254077, a(n) divided by n-th prime A000040(n) is converging to just under 2. Here is the tail of this calculation:
n          a(n)    prime     ratio
26694004 999999729 506784809 1.9732235679542636
26694005 999999770 506784833 1.9732235554097570
26694006 999999827 506784857 1.9732235744368345
26694007 999999857 506784881 1.9732235401868667
26694008 999999915 506784917 1.9732235144638293
26694009 999999941 506784919 1.9732235579804220
26694010 999999946 506784923 1.9732235522720947
26694011 999999967 506784937 1.9732235391992323
(End)

Ray Chandler and John P. Linderman, Table of n, a(n) for n = 1..42315 [First 10000 terms from Ray Chandler]
John Mason, The first 675025 terms (zipped file)

Cf. A254077, A251239, A010051.

a256213 n = a256213_list !! (n-1)
a256213_list = filter ((== 1) . a010051' . a254077) [1..]
-- Reinhard Zumkeller, May 05 2015

f[n_] := Block[{s = Range@ n, j, k}, For[k = 4, k <= n, k++, j = 4; While[Nand[GCD[j, s[[k - 2]]] > GCD[j, s[[k - 1]]], !MemberQ[Take[s, k - 1], j]], j++]; s[[k]] = j]; s]; Position[f@ 500, ?PrimeQ] // Flatten (* _Michael De Vlieger, Apr 15 2015 *)

A253609 Array read by upwards antidiagonals: A(n, k) = index of prime(k)^n in A098550.

Original entry on oeis.org

2, 4, 3, 6, 5, 9, 14, 19, 11, 15, 29, 57, 94, 40, 22, 65, 171, 483, 269, 124, 23, 137, 549, 2549, 1996, 1071, 187, 30, 277, 1786, 13468, 14547, 12661, 1810, 273, 43, 546, 5563, 69298, 105091, 144229, 24916, 4142, 313, 51, 1109, 17088, 353423, 750571, 1624729, 335764, 74341, 5856, 505, 61

Offset: 1

L. Edson Jeffery, Jan 05 2015

Entry A(n, k), in row n and column k, is the index of the entry in A098550 such that A098550(A(n, k)) = prime(k)^n.

Conjecture: For all natural numbers i, j, k, prime(k)^i precedes prime(k)^(i+1) and prime(k)^j precedes prime(k+1)^j in A098550.

			Array begins:
   2,    3,     9,     15,       22,       23,        30,        43, ...
   4,    5,    11,     40,      124,      187,       273,       313, ...
   6,   19,    94,    269,     1071,     1810,      4142,      5856, ...
  14,   57,   483,   1996,    12661,    24916,     74341,    116524, ...
  29,  171,  2549,  14547,   144229,   335764,   1300310,   2276597, ...
  65,  549, 13468, 105091,  1624729,  4458533,  22501985,  43999361, ...
 137, 1786, 69298, 750571, 18146462, 58762243, 387122632, 845496081, ...

Cf. A098550, A251239 (row 1), A251240 (row 2), A251393 (column 1).

Cf. A251241 = {1} union {this array}.

More terms from Jinyuan Wang, Jan 26 2025

cp's OEIS Frontend

A253297 Primes p for which some multiple kp > 2p appears before p does in A098550.

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A253573 Rectangular array a(n,k) read by upwards antidiagonals: row A(n) is the result of applying the function defined in A098550 to the set comprising row n of A253572, for n >= 2.

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A256213 Indices of prime terms in A254077.

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A253609 Array read by upwards antidiagonals: A(n, k) = index of prime(k)^n in A098550.

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cp's OEIS Frontend

A253297 Primes p for which some multiple k*p > 2*p appears before p does in A098550.

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Haskell

A253573 Rectangular array a(n,k) read by upwards antidiagonals: row A(n) is the result of applying the function defined in A098550 to the set comprising row n of A253572, for n >= 2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A256213 Indices of prime terms in A254077.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

A253609 Array read by upwards antidiagonals: A(n, k) = index of prime(k)^n in A098550.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A253297 Primes p for which some multiple kp > 2p appears before p does in A098550.