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.

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A137701 a(n) = semiprime(n)^prime(n).

Original entry on oeis.org

16, 216, 59049, 10000000, 4049565169664, 1946195068359375, 30041942495081691894741, 32064977213018365645815808, 142108547152020037174224853515625, 108199957741720996894562347292921981566976, 118558347188026655500106547231096910504441858017
Offset: 1

Views

Author

Jonathan Vos Post, Apr 27 2008

Keywords

Examples

			a(1) = Semiprime(1)^prime(1) = 4^2 = 16.
a(2) = Semiprime(2)^prime(2) = 6^3 = 216.
a(3) = Semiprime(3)^prime(3) = 9^5 = 59049.
a(4) = Semiprime(4)^prime(4) = 10^7 = 10000000.

Crossrefs

Cf. A000040, A001358, A051674, A114850, A111096.

Formula

a(n) = A001358(n)^A000040(n).

A279686 Numbers that are the least integer of a prime tower factorization equivalence class (see Comments for details).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 30, 36, 40, 48, 60, 64, 72, 81, 90, 108, 144, 162, 180, 192, 200, 210, 225, 240, 256, 280, 320, 324, 360, 405, 420, 432, 450, 500, 512, 540, 576, 600, 630, 648, 720, 768, 810, 900, 960, 1260, 1280, 1296, 1350, 1400, 1536, 1575, 1600
Offset: 1

Views

Author

Rémy Sigrist, Dec 16 2016

Keywords

Comments

The prime tower factorization of a number is defined in A182318.
We say that two numbers, say n and m, belong to the same prime tower factorization equivalence class iff there is a permutation of the prime numbers, say f, such that replacing each prime p by f(p) in the prime tower factorization of n leads to m.
The notion of prime tower factorization equivalence class can be seen as a generalization of the notion of prime signature; thereby, this sequence can be seen as an equivalent of A025487.
This sequence contains all primorial numbers (A002110).
This sequence contains A260548.
This sequence contains the terms > 0 in A014221.
If n appears in the sequence, then 2^n appears in the sequence.
If n appears in the sequence and k>=0, then A002110(k)^n appears in the sequence.
With the exception of term 1, this sequence contains no term from A182318.
Odd numbers appearing in this sequence: 1, 81, 225, 405, 1575, 2025, 2835, 6125, 10125, 11025, 14175, 15625, 16875, 17325, 31185, 33075, 50625, 67375, 70875, 99225, ...
Here are some prime tower factorization equivalence classes:
- Class 1: the number one (the only finite equivalence class),
- Class p: the prime numbers (A000040),
- Class p*q: the squarefree semiprimes (A006881),
- Class p^p: the numbers of the form p^p with p prime (A051674),
- Class p^q: the numbers of the form p^q with p and q distinct primes,
- Class p*q*r: the sphenic numbers (A007304),
- Class p*q*r*s: the products of four distinct primes (A046386),
- Class p*q*r*s*t: the products of five distinct primes (A046387),
- Class p*q*r*s*t*u: the products of six distinct primes (A067885).

Examples

			2 is the least number of the form p with p prime, hence 2 appears in the sequence.
6 is the least number of the form p*q with p and q distinct primes, hence 6 appears in the sequence.
72 is the least number of the form p^q*q^p with p and q distinct primes, hence 72 appears in the sequence.
36000 is the least number of the form p^q*q^r*r^p with p, q and r distinct primes, hence 36000 appears in the sequence.

Links

Crossrefs

Cf. A000040, A002110, A006881, A007304, A014221, A025487, A046386, A046387, A051674, A067885, A182318, A260548, A279690.

A300332 Integers of the form Sum_{j in 0:p-1} x^j*y^(p-j-1) where x and y are positive integers with max(x, y) >= 2 and p is some prime.

Original entry on oeis.org

3, 4, 7, 12, 13, 19, 21, 27, 28, 31, 37, 39, 43, 48, 49, 52, 57, 61, 63, 67, 73, 75, 76, 79, 80, 84, 91, 93, 97, 103, 108, 109, 111, 112, 117, 121, 124, 127, 129, 133, 139, 147, 148, 151, 156, 157, 163, 169, 171, 172, 175, 181, 183, 189, 192, 193, 196, 199
Offset: 1

Views

Author

Peter Luschny, Mar 03 2018

Keywords

Comments

Equivalently these are the integers represented by a cyclotomic binary form Phi_p(x,y) where p is prime and x and y are positive integers with max(x,y) >= 2. A cyclotomic binary form (over Z) is a homogeneous polynomial in two variables of the form f(x, y) = y^phi(k)*Phi(k, x/y) where Phi(k, z) is a cyclotomic polynomial of index k and phi is Euler's totient function.
An efficient and safe calculation of this sequence requires a precise knowledge of the range of possible solutions of the associated Diophantine equations. The bounds used in the Julia program below were specified by Fouvry, Levesque and Waldschmidt.

Examples

			Let p denote an odd prime. Subsequences are numbers of the form
2^p - 1,         (A001348) (x = 1, y = 2) (Mersenne numbers),
p*2^(p - 1),     (A299795) (x = 2, y = 2),
(3^p - 1)/2,     (A003462) (x = 1, y = 3),
3^p - 2^p,       (A135171) (x = 2, y = 3),
p*3^(p - 1),     (A027471) (x = 3, y = 3),
(4^p - 1)/3,     (A002450) (x = 1, y = 4),
2^(p-1)*(2^p-1), (A006516) (x = 2, y = 4),
4^p - 3^p,       (A005061) (x = 3, y = 4),
p*4^(p - 1),     (A002697) (x = 4, y = 4),
(p^p-1)/(p-1),   (A023037),
p^p,             (A000312, A051674).
.
The generalized cuban primes A007645 are a subsequence, as are the quintan primes A002649, the septan primes and so on.
All primes in this sequence less than 1031 are generalized cuban primes. 1031 is an element because 1031 = f(5,2) where f(x,y) = x^4 + y*x^3 + y^2*x^2 + y^3*x + y^4, however 1031 is not a cuban prime because 1030 is not divisible by 6.

Links

Crossrefs

Indices of the nonzero values of A300333.
Cf. A001348, A299795, A003462, A135171, A027471, A002450, A006516, A005061, A002697, A000312, A051674, A023037, A007645.

Programs

  • Julia
    using Primes
function isA300332(n)
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    k = 2
    while k <= K
        if k == 7
            K = Int(floor(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
        for y in 2:M, x in 1:y
            r = x == y ? k*y^(k - 1) : div(x^k - y^k, x - y)
            n == r && return true
        end
        k = nextprime(k+1)
    end
    return false
end
A300332list(upto) = [n for n in 1:upto if isA300332(n)]
println(A300332list(200))

A328383 a(n) gives the number of iterations of x -> A003415(x) needed to reach the first number which is either a divisor or multiple of n, but not both at the same time. If no such number can ever be reached, a(n) is 0 (when either n is of the form p^p, or if the iteration would never stop). When the number reached is a divisor of n, a(n) is -1 * iteration count.

Original entry on oeis.org

-1, -1, 0, -1, -2, -1, 2, -3, -2, -1, 9, -1, -4, 23, 1, -1, -4, -1, 5, -2, -2, -1, 2, -3, 24, 0, 18, -1, -2, -1, 6, -5, -2, 85, 7, -1, -4, 21, 10, -1, -2, -1, 35, 53, -4, -1, 2, -5, 44, 18, 34, -1, 2, 21, 4, -3, -2, -1, 16, -1, -6, 21, 1, -5, -2, -1, 7, 85, -2, -1, 4, -1, 23, 55, 5, -4, -2, -1, 4, 9, -2, -1, 42, -3, 42
Offset: 2

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Comments

The absolute value of a(n) tells how many columns right from the leftmost column in array A258651 one needs to go at row n, before one (again) finds either a divisor or a multiple of n, with 0's reserved for cases like 4 and 27 where the same value continues forever. If one finds a divisor before a multiple, then the value of a(n) will be negative, otherwise it will be positive.
Question: What is the value of a(91) ?

Examples

			For n = 6, its arithmetic derivative A003415(6) = 5 is neither its divisor nor its multiple, but the second arithmetic derivative A003415(5) = 1 is its divisor, thus a(6) = -2.
For n = 8, its arithmetic derivative A003415(8) = 12 is neither its divisor nor its multiple, but the second arithmetic derivative A003415(12) = 16 is its multiple, thus a(8) = +2.
Numbers reached for n=2..28 (with positions of the form p^p are filled with the same p^p): 1, 1, 4, 1, 1, 1, 16, 1, 1, 1, 8592, 1, 1, 410267320320, 32, 1, 1, 1, 240, 7, 1, 1, 48, 1, 410267320320, 27, 9541095424. For example, we have a(12) = 9 and the 9th arithmetic derivative of 12 is A003415^(9)(12) = 8592 = 716*12.

Links

Crossrefs

Cf. A003415, A258651, A328235, A328236.
Cf. A051674 (indices of zeros provided for all n >= 2 either a divisor or multiple can be found).
Cf. A256750, A328248, A328384 for similar counts.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A328383(n) = { my(u=A003415(n),k=1); if(u==n,return(0)); while((n%u) && (u%n), k++; u = A003415(u)); if(u%n,-k,k); };

Formula

a(A000040(n)) = -1.
a(A051674(n)) = 0.

A328384 If n is of the form p^p, a(n) = 0, otherwise a(n) gives the number of iterations of x -> A003415(x) needed to reach the first number different from n which is either a prime, or whose degree (A051903) differs from the degree of n. If the degree of the final number is <= that of n, then a(n) = -1 * iteration count.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Oct 15 2019

Keywords

Comments

The records -1, 0, 1, 2, 3, 5, 8, 10, 11, 13, ... occur at n = 1, 4, 12, 26, 36, 80, 108, 4887, 18688, 22384, ...

Examples

			For n = 21 = 3*7, A051903(21) = 1, A003415(21) = 10 = 2*5, is of the same degree as A051903(10) = 1, but then A003415(10) = 7, which is a prime, having degree <= of the starting value (as we have A051903(7) <= A051903(21)), thus a(21) = -1 * 2 = -2.
For n = 33 = 3*11, A051903(33) = 1, A003415(33) = 14 = 2*7, is of the same degree, but on the second iteration, A003415(14) = 9 = 3^2, with A051903(9) = 2, which is larger than the initial degree, thus a(33) = +2.

Links

Crossrefs

Cf. A003415, A051903, A051674, A157037, A258651, A328252, A328310, A328320.
Cf. A328385 (the number found in the iteration).
Cf. A256750, A328248, A328383 for similar counts.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((n<=1),n-1,vecmax(factor(n)[, 2]));
A328384(n) = { my(d=A051903(n), u=A003415(n), k=1); if(u==n,return(0)); while(u && (u!=n) && !isprime(u) && A051903(u)==d, k++; n = u; u = A003415(u)); if(A051903(u)<=d,-k,k); };

Formula

a(1) = -1 as 0 is here considered having a smaller degree than 1.
a(p) = -1 for all primes.
a(A051674(n)) = 0.
a(A157037(n)) = -1.
a(A328252(n)) = -1.
a(A328320(n)) = -1.

A343222 Number of iterations of x -> A003961(x) needed until A003415(x) <= x, when starting from x=n, where A003415(x) gives the arithmetic derivative of x, and A003961 shifts its prime factorization one step towards the larger primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0
Offset: 1

Views

Author

Antti Karttunen, Apr 08 2021

Keywords

Links

Crossrefs

Cf. A003415, A003961.
Positions of zeros: Union of A051674 and A083347.
Cf. also A343221, A344027.

Programs

  • PARI
    A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A343222(n) = if(A003415(n)<=n,0,1+A343222(A003961(n)));

A383299 Numbers k such that A276086(k) is a multiple of A276086(A003415(k)), where A003415 is the arithmetic derivative, and A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 27, 29, 31, 37, 41, 43, 45, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 97, 101, 103, 107, 109, 113, 117, 119, 127, 131, 137, 139, 141, 147, 149, 151, 157, 161, 163, 165, 167, 171, 173, 177, 179, 181, 191, 193, 197, 199, 203, 207, 209, 211, 223, 227, 229, 233, 239
Offset: 1

Views

Author

Antti Karttunen, May 15 2025

Keywords

Comments

The sequence contains the intersection of A048103, A369650, and A328387. That is, {1, 15, 5005}, at least.

Examples

			5 is a term as A003415(5) = 1, and A276086(5) = 18 is a multiple of A276086(1) = 2, and ditto for all odd primes.
9 is a term as A003415(9) = 6, and A276086(9) = 30 is a multiple of A276086(6) = 5.
15 is a term as A003415(15) = 8, and A276086(15) = 150 is a multiple of A276086(8) = 15.
5005 is a term as A003415(5005) = 2556, and A276086(5005) = 39055266250 = 7803250 * A276086(2556) = 7803250 * 5005. See also A369650.
See also examples in A383300.

Links

Crossrefs

Cf. A003415, A048103, A276086, A327859, A328387, A383298 (characteristic function).
Cf. A006005, A051674, A383300, A383301 (subsequences).
Cf. also A369650.

Programs

A058067 Number of polynomial functions from Z to Z/nZ.

Original entry on oeis.org

1, 1, 4, 27, 64, 3125, 108, 823543, 1024, 19683, 12500, 285311670611, 1728, 302875106592253, 3294172, 84375, 65536, 827240261886336764177, 78732, 1978419655660313589123979, 200000, 22235661, 1141246682444, 20880467999847912034355032910567
Offset: 0

Views

Author

N. J. A. Sloane, Nov 24 2000

Keywords

Comments

The first formula for a(n) is due to Kempner (1921). - Jonathan Sondow, Nov 05 2017

Links

Crossrefs

Cf. A051674, A090622, A240098.

Programs

  • Maple
    A058067 := n->mul(n/gcd(n,k!),k=0..n-1);
  • Mathematica
    a[0] = 1; a[n_] := Product[n/GCD[n, k!], {k, 0, n - 1}]; Array[a, 24, 0] (* Amiram Eldar, Sep 29 2020 *)
  • PARI
    a(n) = prod(k=0, n-1, n/gcd(n, k!)); \\ Michel Marcus, Nov 06 2017

Formula

a(n) = Product_{k=0..n-1} n/gcd(n, k!).
Multiplicative with a(p^e) = p^t_p(e). - David W. Wilson, Aug 14 2005 [t_p(e) = Sum_{k>=0: e > A090622(k, p)} (e - A090622(k, p)) = p * Sum_{k = 1..e} max(0, k - A090622(e-k, p)). In particular, t_p(e) = p*e*(e+1)/2 for e <= p. - Andrey Zabolotskiy, Nov 09 2017 and Sep 29 2020]
a(prime(n)) = A051674(n). - R. J. Mathar, Apr 01 2014 [Edited by Andrey Zabolotskiy, Nov 08 2017]
a(n) = n^n / A240098(n). - Jonathan Sondow, Nov 10 2017

A074107 a(n) = Product of (prime + 1) for first n primes - primorial (n); Sum of proper divisors of the n-th primorial.

Original entry on oeis.org

0, 1, 6, 42, 366, 4602, 66738, 1231314, 25136790, 612982650, 18612572370, 602072009070, 23079296834790, 976751205195990, 43281303292150770, 2090585319354906990, 113506497027753468870, 6842978980142398176930, 426187457118982899608730, 29098035465450244144376910, 2102916875063497845451016610, 156173789584825539524342644530
Offset: 0

Views

Author

Amarnath Murthy, Aug 22 2002

Keywords

Examples

			a(3) = (2+1)*(3+1)*(5+1) - 2*3*5 = 72 - 30 = 42.

Links

Crossrefs

Cf. A001065, A002110, A003415, A024451, A051674, A054640, A070826, A348507.

Programs

  • Maple
    for n from 1 to 25 do a[n] := product(ithprime(i)+1,i=1..n)-product(ithprime(i),i=1..n): od:seq(a[j],j=1..25);
  • Mathematica
    Module[{nn=20,p,pr,pr1},p=Prime[Range[nn]];pr=FoldList[Times,1,p];pr1= FoldList[Times,1,p+1];#[[2]]-#[[1]]&/@Rest[Thread[{pr,pr1}]]](* Harvey P. Dale, Feb 07 2015 *)
  • PARI
    A074107(n) = (prod(i=1,n,1+prime(i))-prod(i=1,n,prime(i))); \\ Antti Karttunen, Nov 19 2024

Formula

From Antti Karttunen, Nov 19 2024: (Start)
a(n) = A348507(A002110(n)) = A054640(n) - A002110(n) = A001065(A002110(n)).
a(n) >= A024451(n), because A348507(n) >= A003415(n).
For n >= 1, a(n) <= A070826(1+n) [= A002110(1+n)/2] < A051674(n).
(End)

Extensions

More terms from Sascha Kurz, Feb 01 2003
Term a(0)=0 prepended, data section further extended, and secondary definition added by Antti Karttunen, Nov 19 2024

A101340 a(n) = p^p + p, with p = prime(n).

Original entry on oeis.org

6, 30, 3130, 823550, 285311670622, 302875106592266, 827240261886336764194, 1978419655660313589123998, 20880467999847912034355032910590, 2567686153161211134561828214731016126483498
Offset: 1

Views

Author

Jorge Coveiro, Dec 24 2004

Keywords

Links

Programs

Formula

a(n) = A051674(n) + A000040(n). - R. J. Mathar, Apr 26 2007
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