A098719 -id:A098719 - cp's OEIS frontend

A341351 a(n) = A048673(A181815(n)).

1, 2, 5, 3, 14, 8, 41, 23, 4, 122, 13, 68, 11, 365, 38, 203, 32, 1094, 113, 18, 608, 6, 63, 95, 3281, 338, 53, 1823, 17, 188, 284, 9842, 1013, 158, 5468, 50, 563, 25, 851, 29525, 88, 3038, 28, 313, 473, 16403, 149, 1688, 74, 2552, 88574, 263, 9113, 7, 83, 938, 1418, 49208, 446, 5063, 221, 7655, 265721, 788, 27338, 20

Offset: 1

Antti Karttunen and Michael De Vlieger, Feb 10 2021

nonn

Maxima are in A007051 and appear at n in A025488, which are the indices of 2^k in A025487. 2^k is idempotent via A181815 but transformed by A003961 to 3^n, which are rendered by A048673 to (3^n + 1)/2.

Local minima are in A111333 and appear at n in A098719, which are the indices of P(k) = A002110(k) in A025487. P(k) is transformed by A181815 to p_k = A000040(k), which become p_(k+1) under A003961. Therefore these become (p_(k+1)+1)/2 via A048673.

Michael De Vlieger, Annotated logarithmic plot of a(n) for 1 <= n <= A098719(8) accentuating records in red and local minima in blue.
Michael De Vlieger, log-log plot of a(n) for 1 <= n <= A098719(16).
Michael De Vlieger, Notes on this sequence.
Index entries for sequences that are permutations of the natural numbers

Cf. A002110, A025487, A025488, A048673, A098719, A108951, A111333, A181812, A181815.
Cf. A341352 (inverse).
Cf. A007051 (record values).

a025487[n_] := {{1}}~Join~Block[{lim = Product[Prime@ i, {i, n}], ww = NestList[Append[#, 1] &, {1}, n - 1]}, Map[Block[{w = #, k = 1}, Sort@ Prepend[If[Length@ # == 0, #, #[[1]]], Product[Prime@ i, {i, Length@ w}]] &@ Reap[Do[If[# < lim, Sow[#]; k = 1, If[k >= Length@ w, Break[], k++]] &@ Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, #]] &@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1]]], {i, Infinity}]][[-1]] ] &, ww]]; Map[(1 + If[# == 1, 1, Apply[Times, NextPrime[#1]^#2 & @@@ FactorInteger[#]]])/2 &@ Apply[Times, Prime@ Table[LengthWhile[#1, # >= j &], {j, #2}] & @@ {#, Max[#]} &@ If[# == 1, {0}, Function[f, ReplacePart[ConstantArray[0, PrimePi@ f[[-1, 1]] ], #] &@ Map[PrimePi@ First@ # -> Last@ # &, f]]@ FactorInteger@ #]] &, Union@ Flatten@ a025487@ 5] (* Michael De Vlieger, Feb 11 2021 *)

PARI
```
A341351(n) = A048673(A181815(n));
```

a(n) = A048673(A181815(n)).

For all n >= 1, A181812(a(n)) = A025487(n).

A346043 a(n) is the position of A138534(n) in A025487.

Original entry on oeis.org

1, 2, 6, 17, 67, 166, 676, 1373, 4475, 10446, 30036, 51032, 196386, 315302, 737515, 1654229, 4227565, 6301902, 17975187, 26010425, 70085244, 133337963

Offset: 0

Amiram Eldar, Jul 02 2021

			A138534(2) = A025487(6) = 12, so a(2) = 6.

Cf. A025487, A138534.

Similar sequences: A098718, A098719, A293635, A306802.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; s = {}; Do[p = Position[lps, Product[Prime[k]^Floor[n/k], {k, 1, n}]]; If[p == {}, Break[]]; AppendTo[s, p[[1, 1]]], {n, 0, 20}]; s

f(m) = my(c=1, p, q=2, v=vector(logint(m, 2), i, 2^i), w); while(#v, c+=#v; p=q; q=nextprime(q+1); w=List([]); for(i=1, #v, for(j=1, min(valuation(v[i], p), logint(m\v[i], q)), listput(w, v[i]*q^j))); v=w); c;
a(n) = f(prod(k=1, n, prime(k)^(n\k))); \\ Jinyuan Wang, Jul 08 2021

A025487(a(n)) = A138534(n).

a(20)-a(21) from Jinyuan Wang, Jul 08 2021

A307133 T(n,m) = number of k <= A002110(n) such that A001221(k) = m, where k is a term in A025487.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 9, 4, 1, 1, 11, 21, 15, 5, 1, 1, 14, 38, 36, 18, 6, 1, 1, 18, 64, 79, 53, 23, 7, 1, 1, 23, 97, 148, 122, 63, 26, 7, 1, 1, 27, 140, 258, 251, 157, 76, 30, 7, 1, 1, 32, 196, 425, 480, 349, 195, 89, 33, 8, 1, 1, 37, 261, 655, 853

Offset: 0

Michael De Vlieger, Mar 26 2019

Terms m in A025487 are products of p_i# in A002110.

The primorial A002110(n) is the smallest number k that is the product of the n smallest primes (i.e., A001221(k) = n) and is a subset of A025487.

			Row 3 = {1,4,3,1}. The terms k in A025487 such that k <= A002110(3) are {1, 2, 4, 6, 8, 12, 16, 24, 30}. Of these, 1 has 0 distinct prime divisors, 4 {2,4,8,16} have 1 distinct prime divisor, 3 {6,12,24} have 2 distinct prime divisors, and 1 {30} has 3 distinct prime divisors.
Triangle begins:
   0: 1
   1: 1   1
   2: 1   2    1
   3: 1   4    3    1
   4: 1   7    9    4     1
   5: 1  11   21   15     5     1
   6: 1  14   38   36    18     6    1
   7: 1  18   64   79    53    23    7    1
   8: 1  23   97  148   122    63   26    7    1
   9: 1  27  140  258   251   157   76   30    7    1
  10: 1  32  196  425   480   349  195   89   33    8   1
  11: 1  37  261  655   853   700  443  228  102   37   9   1
  12: 1  42  340  975  1438  1323  928  533  268  119  41  11   1
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..560 (rows 0 <= n <= 1000).

Cf. A000012, A001221, A002110, A025487, A054850, A098719.

Block[{nn = 12, f, w}, f[n_] := {{1}}~Join~Block[{lim = Product[Prime@ i, {i, n}], ww = NestList[Append[#, 1] &, {1}, n - 1], g}, g[x_] := Apply[Times, MapIndexed[Prime[First@ #2]^#1 &, x]]; Map[Block[{w = #, k = 1}, Sort@ Prepend[If[Length@ # == 0, #, #[[1]]], Product[Prime@ i, {i, Length@ w}]] &@ Reap[Do[If[# < lim, Sow[#]; k = 1, If[k >= Length@ w, Break[], k++]] &@ g@ Set[w, If[k == 1, MapAt[# + 1 &, w, k], PadLeft[#, Length@ w, First@ #] &@ Drop[MapAt[# + Boole[i > 1] &, w, k], k - 1]]], {i, Infinity}]][[-1]]] &, ww]]; s = MapAt[Flatten, f@ nn, 1]; Array[Function[P, TakeWhile[Map[Count[#, _?(# <= P &)] &, s, {1}], # > 0 &]]@ Product[Prime@ i, {i, #}] &, nn + 1, 0]] // Flatten

T(n,0) = T(n,n) = A000012(n).
T(n,1) = A054850(n).
A098719(n) = sum of row n.

A346407 a(n) is the position of A051451(n) in A025487.

Original entry on oeis.org

1, 2, 4, 6, 13, 29, 36, 55, 112, 223, 264, 514, 956, 1749, 2345, 2847, 5005, 8567, 9507, 16073, 26792, 43730, 70482, 88969, 140871, 221370, 342958, 368588, 565510, 859401, 1290994, 1927925, 2128165, 3142980, 4616207, 6754033, 9810997, 14133201, 20230329, 28744301

Offset: 1

Amiram Eldar, Jul 15 2021

nonn

Equivalently, the positions of the distinct terms of A003418 in A025487.

			A138534(1) = A025487(1) = 1, so a(1) = 1.
A138534(2) = A025487(2) = 2, so a(2) = 2.
A138534(3) = A025487(4) = 6, so a(3) = 4.

Cf. A003418, A051451, A025487.

Similar sequences: A098718, A098719, A293635, A306802, A346043.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]]; s = {}; lcms = Union @ Table[LCM @@ Range[n], {n, 1, 31}]; Do[p = Position[lps, lcms[[n]]]; If[p == {}, Break[]]; AppendTo[s, p[[1, 1]]], {n, 1, Length[lcms]}]; s

A025487(a(n)) = A003418(n).

A363456 Positions of the terms of the Chernoff sequence (A006939) in A025487.

Original entry on oeis.org

1, 2, 6, 27, 150, 900, 5697, 37226, 246280, 1648592, 11204274

Offset: 0

Amiram Eldar, Jun 03 2023

Indices of records in A363455.

			A006939(0) = A025487(1) = 1, so a(0) = 1.
A006939(1) = A025487(2) = 2, so a(1) = 2.
A006939(2) = A025487(6) = 12, so a(2) = 6.

Cf. A006939, A025487, A363455.

Similar sequences: A098718, A098719, A293635, A306802, A346043, A346407.

lps = Cases[Import["https://oeis.org/A025487/b025487.txt", "Table"], {, }][[;; , 2]];
cher = Table[Product[Prime[k]^(n - k + 1), {k, 1, n}], {n, 0, 8}]
Position[lps, #] & /@ cher // Flatten

A025487(a(n)) = A006939(n).

A363455(a(n)) = n.

cp's OEIS Frontend

A341351 a(n) = A048673(A181815(n)).

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A346043 a(n) is the position of A138534(n) in A025487.

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A307133 T(n,m) = number of k <= A002110(n) such that A001221(k) = m, where k is a term in A025487.

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A346407 a(n) is the position of A051451(n) in A025487.

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A363456 Positions of the terms of the Chernoff sequence (A006939) in A025487.

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Examples

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Programs

Mathematica

Formula