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 1-5 of 5 results.

A306802 Position of highly composite numbers in the sequence of products of primorials.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 12, 13, 17, 20, 24, 27, 34, 36, 43, 47, 55, 67, 77, 84, 95, 102, 107, 112, 129, 133, 138, 154, 166, 183, 198, 211, 220, 245, 252, 261, 264, 294, 314, 348, 369, 390, 406, 446, 457, 476, 500, 533, 555, 582, 634, 652, 676, 726, 756, 822
Offset: 1

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Author

Michael De Vlieger, Mar 12 2019

Keywords

Comments

Indices of A002182 in A025487. All terms in A002182 are products of terms in A002110; A025487 lists products of terms in A002110.
The first 28 terms of this sequence and those of A293635 are identical since the smallest 28 terms of A002182 and A004394 are the same.

Examples

			The number 120 is 10th in the sequence of highly composite numbers, since it sets a record for the divisor counting function. The index of this number in A025487 is 17.

Links

Crossrefs

Cf. A002182, A025487, A098719, A293635.

Programs

  • Mathematica
    Block[{P = Product[Prime@ i, {i, 8}], s, t, u}, s = Array[DivisorSigma[0, #] &, P]; t = Array[If[# == 1, {0}, Sort[FactorInteger[#][[All, -1]], Greater]] &, P]; u = Values[PositionIndex@ t][[All, 1]]; Map[FirstPosition[u, #][[1]] &, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]] ]

A332035 Indices of A004394(n) in A055932.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 13, 15, 21, 26, 30, 36, 49, 53, 63, 72, 86, 114, 149, 175, 212, 221, 285, 367, 424, 505, 541, 643, 687, 703, 886, 1015, 1198, 1272, 1496, 1586, 1704, 2116, 2491, 2912, 3076, 3587, 3791, 4052, 4971, 5999, 6087, 6845, 6955, 7330, 8481, 8933
Offset: 1

Views

Author

Michael De Vlieger, Feb 05 2020

Keywords

Comments

A055932 lists numbers m whose prime divisors p are consecutive primes starting with 2, while A004394 is a subset of A025487, the latter lists numbers m that are products of primorials. With both, we find a range of indices of primes 1, 2, ..., k that divide m. While A055932 admits any multiplicity for primes regardless of their index, the latter only admits decreasing multiplicities as prime index k increases. A004394 is a subset of A025487, which is in turn a subset of A055932.

Links

Crossrefs

Cf. A004394, A025487, A055932, A293635 (Indices of A004394(n) in A025487), A332034.

Programs

  • Mathematica
    Block[{s = TakeWhile[Import["https://oeis.org/A055932/b055932.txt", "Data"], Length@ # > 0 &][[All, -1]], t = Join @@ {Map[ToExpression@ Last@ StringSplit@ # &, #1][[All, -1]], Map[Times @@ Flatten@ {Complement[#1, Union[#2, #3]], Product[Prime@ i, {i, PrimePi@ #}] & /@ #2, Factorial /@ #3} & @@ ToExpression@ {StringSplit[#, _?(! DigitQ@ # &)], StringCases[#, (x : DigitCharacter ..) ~~ "#" :> x], StringCases[#, (x : DigitCharacter ..) ~~ "!" :> x]} &@ Last@ StringSplit[First[#]] &, TakeWhile[#2, Length@ # > 0 &]]} & @@ TakeDrop[Drop[#, 3] &@ Import["https://oeis.org/A004394/b004394.txt", "Data"], 2000] }, Reap[Do[Which[Length@ t == 0, Break[], First[t] == s[[i]] , t = Rest@ t; Sow[i]], {i, Length@ s}]][[-1, -1]]]

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

Views

Author

Amiram Eldar, Jul 02 2021

Keywords

Examples

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

Crossrefs

Cf. A025487, A138534.
Similar sequences: A098718, A098719, A293635, A306802.

Programs

  • Mathematica
    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
  • PARI
    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

Formula

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

Extensions

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

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

Views

Author

Amiram Eldar, Jul 15 2021

Keywords

Comments

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

Examples

			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.

Crossrefs

Cf. A003418, A051451, A025487.
Similar sequences: A098718, A098719, A293635, A306802, A346043.

Programs

  • Mathematica
    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

Formula

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

Views

Author

Amiram Eldar, Jun 03 2023

Keywords

Comments

Indices of records in A363455.

Examples

			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.

Crossrefs

Cf. A006939, A025487, A363455.
Similar sequences: A098718, A098719, A293635, A306802, A346043, A346407.

Programs

  • Mathematica
    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

Formula

A025487(a(n)) = A006939(n).
A363455(a(n)) = n.
Showing 1-5 of 5 results.

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