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 1-8 of 8 results.

A129912 Numbers that are products of distinct primorial numbers (see A002110).

Original entry on oeis.org

1, 2, 6, 12, 30, 60, 180, 210, 360, 420, 1260, 2310, 2520, 4620, 6300, 12600, 13860, 27720, 30030, 37800, 60060, 69300, 75600, 138600, 180180, 360360, 415800, 485100, 510510, 831600, 900900, 970200, 1021020, 1801800, 2910600, 3063060, 5405400
Offset: 1

Views

Author

Bill McEachen, Jun 05 2007, Jun 06 2007, Jul 06 2007, Aug 07 2007

Keywords

Comments

Conjecture: every odd prime p is either adjacent to a term of A129912 or a prime distance q from some term of A129912, where q < p. - Bill McEachen, Jun 03 2010, edited for clarity in Feb 26 2019
The first 2^20 terms k > 2 of A283477 all satisfy also the condition that the differences k-A151799(k) and A151800(k)-k are always either 1 or prime, like is also conjectured to hold for A002182 (cf. also the conjecture given in A117825). However, for A025487, which is a supersequence of both sequences, this is not always true: 512 is a member of A025487, but A151800(512) = 521, with 521 - 512 = 9, which is a composite number. - Antti Karttunen, Feb 26 2019

Examples

			For s = 4 there are 8 (generally 2^(s-1)) such numbers: 210 = 2*3*5*7, 420 = 2^2*3*5*7 = (2*3*5*7)*2, 1260 = 2^2*3^2*5*7 = (2*3*5*7)*(2*3), 6300 = 2^2*3^2*5^2*7 = (2*3*5*7)*(2*3*5), 2520 = 2^3*3^2*5*7 = (2*3*5*7)*(2*3)*2, 12600 = 2^3*3^2*5^2*7 = (2*3*5*7)*(2*3*5)*2, 37800 = 2^3*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3), 75600 = 2^4*3^3*5^2*7 = (2*3*5*7)*(2*3*5)*(2*3)*2.

References

  • CRC Standard Mathematical Tables, 28th Ed., CRC Press

Links

Crossrefs

Subsequence of A025487. Sequence A283477 sorted into ascending order.
Cf. A002110, A117825, A151799, A151800.

Programs

  • Mathematica
    Clear[f]; f[m_] := f[m] = Union[Times @@@ Subsets[FoldList[Times, 1, Prime[Range[m]]]]][[1 ;; 100]]; f[10]; f[m = 11]; While[f[m] != f[m-1], m++]; f[m] (* Jean-François Alcover, Mar 03 2014 *) (* or *)
pr[n_] := Product[Prime[n + 1 - i]^i, {i, n}]; upto[mx_] := Block[{ric, j = 1}, ric[n_, ip_, ex_] := If[n < mx, Block[{p = Prime[ip + 1]}, If[ex == 1, Sow@ n]; ric[n p^ex, ip + 1, ex]; If[ex > 1, ric[n p^(ex - 1), ip + 1, ex - 1]]]]; Sort@ Reap[ Sow[1]; While[pr[j] < mx, ric[2^j, 1, j]; j++]][[2, 1]]];
upto[10^30] (* faster, Giovanni Resta, Apr 02 2017 *)
  • PARI
    is(n)=my(o=valuation(n,2),t); if(o<1||n<2, return(n==1)); n>>=o; forprime(p=3,, t=valuation(n,p); n/=p^t; if(t>o || tCharles R Greathouse IV, Oct 22 2015

Formula

Apart from 1 and 2, numbers of the form 2^k(1)*3^k(2)*5^k(3)*...*p(s)^k(s), where p(s) is s-th prime, k(i)>0 for i=1..s, k(i)-k(i-1) = 0 or 1 for i=2..s and |{k(1),k(2),..,k(s)}|=k(1). - Vladeta Jovovic, Jun 14 2007
Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/A002110(n)) = 1.8177952875... . - Amiram Eldar, Jun 03 2023

Extensions

Edited by N. J. A. Sloane, Jun 09 2007, Aug 08 2007
I corrected the Potter link to reflect its relocation. - Bill McEachen, Sep 12 2009
I added link to Wikicommons image. - Bill McEachen, Sep 16 2009
I again corrected the Potter link for its relocation - Bill McEachen, May 30 2013

A068507 Highly composite numbers sandwiched between twin primes.

Original entry on oeis.org

4, 6, 12, 60, 180, 240, 7560, 55440, 110880, 73329656400, 18632716502400, 130429015516800, 48519593772249600, 149602080797769600, 74377068101903920953600, 927967188666725711881005276648000, 241271469053348685089061371928480000
Offset: 1

Views

Author

Lekraj Beedassy, Mar 25 2002

Keywords

Comments

Intersection of (A072826 - 1) and (A072828 + 1). - Lekraj Beedassy, Nov 27 2003
The next term, a(18), is A002182(1002), it has 77 digits. - M. F. Hasler, Jun 23 2019
a(22) > 10^17030, if it exists. - Amiram Eldar, Dec 03 2020

Examples

			60 is between 59 and 61.

Links

Crossrefs

Cf. A002182, A072826, A072828, A117825, A321995.
This is also the intersection of A002182 and A014574.

Formula

a(n) = A002182(A321995(n)). - Amiram Eldar, Dec 03 2020

Extensions

Corrected and extended by Lior Manor, Jun 03 2002
More terms from Bill McEachen, May 24 2006
a(18)-a(20) from M. F. Hasler, Jun 23 2019

A141345 Distance from the n-th highly composite number, A002182(n), to the next prime.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 5, 1, 7, 1, 1, 7, 7, 13, 17, 13, 1, 11, 1, 11, 1, 1, 19, 13, 1, 11, 1, 17, 1, 29, 13, 13, 1, 1, 17, 13, 23, 17, 19, 17, 17, 19, 1, 19, 23, 37, 53, 1, 17, 29, 43, 29, 1, 19, 19, 1, 23, 23, 1, 41, 41, 1, 53, 29, 19, 19, 23, 23, 47, 29, 23, 37, 1, 59, 71, 41, 1, 29, 37
Offset: 1

Views

Author

T. D. Noe, Jun 26 2008

Keywords

Comments

It appears that (1) every term is either 1 or a prime and (2) every prime greater than 3 appears. Note that a prime can occur only a finite number of times. Similar to Fortune's conjecture (A005235) and McEachen's conjecture (A117825).
The arithmetic mean of a(n)/log(A002182(n)) for the terms 3..10000 is 1.513, i.e., a rough approximation is given by a(n) ~ log(A002182(n)^(3/2)). - A.H.M. Smeets, Dec 02 2020

Links

Programs

  • Mathematica
    With[{s = Array[DivisorSigma[0, #] &, 10^6]}, Map[NextPrime[#] - # &@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* or *)
Map[NextPrime[#] - # &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[1 ;; 80, -1]] ] (* Michael De Vlieger, Dec 11 2020 *)

A324385 Distance from the n-th highly composite number, A002182(n), from the largest prime <= A002182(n).

Original entry on oeis.org

0, 1, 1, 1, 1, 5, 1, 1, 7, 1, 1, 1, 1, 1, 1, 11, 17, 1, 1, 1, 13, 11, 11, 19, 17, 13, 1, 23, 1, 1, 13, 17, 17, 13, 17, 1, 17, 1, 1, 23, 17, 17, 17, 1, 19, 83, 37, 23, 17, 23, 1, 43, 19, 1, 19, 43, 19, 31, 23, 19, 31, 19, 19, 1, 1, 1, 1, 47, 1, 31, 47, 23, 53, 23, 83, 37, 31, 1, 31, 1, 23, 61, 1, 41, 47, 61, 41, 29, 41, 29, 43, 73, 29, 47, 31, 31
Offset: 2

Views

Author

Antti Karttunen, Feb 26 2019

Keywords

Comments

Like in A141345 it appears (or is conjectured) that no composite numbers ever occur here. Taken together, this leads to McEachen's conjecture given in A117825. Here in range 2..10000 term 1 occurs for 313 times.
The arithmetic mean of a(n)/log(A002182(n)) for the terms 3..10000 is 1.513, i.e., a rough approximation is given by a(n) ~ log(A002182(n)^(3/2)). - A.H.M. Smeets, Dec 02 2020

Examples

			A002182(2) = 2, the largest prime <= 2 is 2 itself, thus a(2) = 2-2 = 0.
A002182(7) = 36, the largest prime <= 36 is 31, thus a(7) = 36-31 = 5.

Links

Crossrefs

Cf. A002182, A007917, A060270, A117825, A141345.

Programs

  • Mathematica
    With[{s = Array[DivisorSigma[0, #] &, 10^6]}, {0}~Join~Map[# - NextPrime[#, -1] &@ FirstPosition[s, #][[1]] &, Drop[Union@ FoldList[Max, s], 2]]] (* or *)
{0}~Join~Map[# - NextPrime[#, -1] &, Import["https://oeis.org/A002182/b002182.txt", "Data"][[3 ;; 97, -1]] ] (* Michael De Vlieger, Dec 11 2020 *)
  • PARI
    A324385(n) = (A002182(n)-precprime(A002182(n)));

Formula

a(n) = A002182(n) - A007917(A002182(n)).

A140645 Digit sums of highly composite numbers.

Original entry on oeis.org

1, 2, 4, 6, 3, 6, 9, 12, 6, 3, 9, 6, 9, 9, 12, 9, 15, 9, 9, 18, 9, 9, 9, 9, 18, 18, 9, 18, 18, 18, 18, 18, 18, 18, 36, 18, 27, 18, 18, 18, 18
Offset: 1

Views

Author

Parthasarathy Nambi, Jul 08 2008

Keywords

Comments

Except for the first three terms, all others are multiples of 3.

Examples

			2162160 is a highly composite number whose digit sum is 18.

Links

Crossrefs

Cf. A002182, A002183, A007953, A117825, A127390.

Programs

  • Mathematica
    HCN=NestList[Function[last,Module[{d = DivisorSigma[0, last]},NestWhile[# + 1 &, last, DivisorSigma[0, #] <= d &]]], 1,40]; DigitSum/@HCN (* James C. McMahon, Jul 12 2025 *)

Formula

a(n) = A007953(A002182(n)). - Amiram Eldar, Nov 13 2019

A228945 Number of ways to write highly composite numbers (A002182(n)) as the difference of two primes, both <= 2*A002182(n).

Original entry on oeis.org

0, 0, 1, 1, 3, 6, 6, 7, 11, 19, 25, 28, 40, 61, 87, 109, 138, 184, 326, 437, 550, 721, 935, 1103, 1326, 1792, 1903, 2351, 3261, 4119, 5773, 7386, 8736, 10307, 14404, 15953, 18290, 21480, 30294, 38516, 54874, 70132, 85419, 99583, 142053, 155243, 182169, 220996
Offset: 1

Views

Author

Jaycob Coleman, Sep 08 2013

Keywords

Comments

Conjectures: (a) This sequence is strictly increasing beginning with n=7. (b) If p is the smallest prime with p > A002182(n)+1, then p-A002182(n) is prime. This is a strengthening of a conjecture regarding A117825.

Examples

			a(5) = 3, since A002182(5) = 12 = 23-11 = 19-7 = 17-5.

Links

Crossrefs

Cf. A002182, A117825, A202472.

Formula

a(n) = A202472(A002182(n)/2) for n>1.

Extensions

More terms from Amiram Eldar, Nov 03 2024

A228943 Number of decompositions of highly composite numbers (A002182) into unordered sums of two primes.

Original entry on oeis.org

0, 0, 1, 1, 1, 3, 4, 5, 6, 12, 14, 18, 22, 39, 51, 68, 83, 112, 184, 251, 315, 431, 527, 652, 768, 1011, 1128, 1305, 1836, 2344, 3240, 4082, 4955, 5725, 8023, 8723, 10260, 11945, 16771, 21466, 30280, 38583, 46645, 54789, 77430, 85067, 99199, 120742, 154753
Offset: 1

Views

Author

Jaycob Coleman, Sep 08 2013

Keywords

Comments

a(n) = A045917(A002182(n)/2) for n>1.
Conjecture: (a) This sequence is strictly increasing beginning with n=5. (b) For all n>2, if p is the greatest prime with p<A002182(n)-1, then A002182(n)-p is prime. This is a strengthening of a conjecture regarding A117825. - Jaycob Coleman, Sep 08 2013

Examples

			a(6)=3, since 24=5+19=7+17=11+13.

Crossrefs

Cf. A116979, A117825.

Programs

  • PARI
    nbd(n) = my(s); forprime(p=2, n\2, s+=isprime(n-p)); s;
lista(nn) = {last = 1; print1(nbd(last), ", "); forstep(n=2, nn, 2, if(numdiv(n)> last, last=numdiv(n); print1(nbd(n), ", ")););} \\ Michel Marcus, Sep 10 2013

Extensions

More terms from Michel Marcus, Sep 10 2013

A272817 Distance from n-th highly composite number (cf. A002182) to nearest prime or square.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 13, 1, 11, 1, 17, 1, 1, 13, 13, 1, 1, 17, 1, 17, 1, 1, 17, 17, 17, 1, 1, 19, 37, 37, 1, 1, 23, 1, 29, 1, 1, 19, 1, 19, 23, 1, 19, 31, 1, 19, 1, 1, 1, 1, 23, 1, 29, 23, 23, 1, 23, 71, 37
Offset: 1

Views

Author

Vladimir Shevelev, May 07 2016

Keywords

Comments

Conjecture: Terms are either 1 or prime, n>7. - Bill McEachen, Jun 11 2025

Links

Crossrefs

Cf. A002182, A005235, A117825, A141345.

Extensions

a(25)-a(77) from Giovanni Resta, May 07 2016
Showing 1-8 of 8 results.

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

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