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.

A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520
Offset: 1

Views

Author

Paul Boddington, Jul 21 2005

Keywords

Comments

This sequence is a permutation of A025487.
And thus also a permutation of A181812, see the formula section. - Antti Karttunen, Jul 21 2014
A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015

Examples

			a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).

Links

Crossrefs

Cf. A319626, A329900 (left inverses).
Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.

Programs

  • Mathematica
    a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)
  • PARI
    primorial(n)=prod(i=1,primepi(n),prime(i))
a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015
  • Python
    from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017
  • Sage
    def sharp_primorial(n): return sloane.A002110(prime_pi(n))
def p(f):
    return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1,51)]
# Giuseppe Coppoletta, Feb 07 2015

Formula

Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
Completely multiplicative with a(p_i) = A002110(i) = prime(i)#. [Franklin T. Adams-Watters, Jun 24 2009; typos corrected by Antti Karttunen, Jul 21 2014]
From Antti Karttunen, Jul 21 2014: (Start)
a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).
a(n) = n * A181811(n).
a(n) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]
a(n) = A181812(A048673(n)).
Other identities:
A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
A071178(a(n)) = A071178(n). [And also its exponent.]
a(2^n) = 2^n. [Fixes the powers of two.]
A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]
(End)
From Antti Karttunen, Nov 19 2019: (Start)
Further identities:
a(A307035(n)) = A000142(n).
a(A003418(n)) = A181814(n).
a(A025487(n)) = A181817(n).
a(A181820(n)) = A181822(n).
a(A019565(n)) = A283477(n).
A001221(a(n)) = A061395(n).
A001222(a(n)) = A056239(n).
A181819(a(n)) = A122111(n).
A124859(a(n)) = A181821(n).
A085082(a(n)) = A238690(n).
A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)
A005361(a(n)) = A329382(n). (product of exponents of prime factors)
A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
A000005(a(n)) = A329605(n). (number of divisors)
A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
A324580(a(n)) = A324887(n).
A276150(a(n)) = A324888(n). (digit sum in primorial base)
A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
A243055(a(n)) = A329343(n).
A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
A328114(a(n)) = A329344(n). (maximal digit in primorial base)
A062977(a(n)) = A325226(n).
A097248(a(n)) = A283478(n).
A324895(a(n)) = A324896(n).
A324655(a(n)) = A329046(n).
A327860(a(n)) = A329047(n).
A329601(a(n)) = A329607(n).
(End)
a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - Antti Karttunen, Dec 29 2019
From Antti Karttunen, Jul 09 2021: (Start)
a(n) = A346092(n) + A346093(n).
a(n) = A346108(n) - A346109(n).
a(A342012(n)) = A004490(n).
a(A337478(n)) = A336389(n).
A336835(a(n)) = A337474(n).
A342002(a(n)) = A342920(n).
A328571(a(n)) = A346091(n).
A328572(a(n)) = A344592(n).
(End)
Sum_{n>=1} 1/a(n) = A161360. - Amiram Eldar, Aug 04 2022

Extensions

More terms computed by Antti Karttunen, Jul 21 2014
The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015
Name "Primorial inflation" (coined by Matthew Vandermast in A181815) prefixed to the name by Antti Karttunen, Jan 14 2020

A325241 Numbers > 1 whose maximum prime exponent is one greater than their minimum.

Original entry on oeis.org

12, 18, 20, 28, 44, 45, 50, 52, 60, 63, 68, 72, 75, 76, 84, 90, 92, 98, 99, 108, 116, 117, 124, 126, 132, 140, 147, 148, 150, 153, 156, 164, 171, 172, 175, 180, 188, 198, 200, 204, 207, 212, 220, 228, 234, 236, 242, 244, 245, 252, 260, 261, 268, 275, 276, 279
Offset: 1

Views

Author

Gus Wiseman, Apr 15 2019

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose maximum multiplicity is one greater than their minimum (counted by A325279).
The asymptotic density of this sequence is 1/zeta(3) - 1/zeta(2) = A088453 - A059956 = 0.22398... . - Amiram Eldar, Jan 30 2023

Examples

			The sequence of terms together with their prime indices begins:
  12: {1,1,2}
  18: {1,2,2}
  20: {1,1,3}
  28: {1,1,4}
  44: {1,1,5}
  45: {2,2,3}
  50: {1,3,3}
  52: {1,1,6}
  60: {1,1,2,3}
  63: {2,2,4}
  68: {1,1,7}
  72: {1,1,1,2,2}
  75: {2,3,3}
  76: {1,1,8}
  84: {1,1,2,4}
  90: {1,2,2,3}
  92: {1,1,9}
  98: {1,4,4}
  99: {2,2,5}

Links

Crossrefs

Positions of 1's in A062977. Supersequence of A054753, A096156.
Cf. A001221, A001222, A001694, A051903, A051904, A052485, A056239, A112798, A118914, A325240, A325259, A325270, A325279.
Cf. A059956, A088453.

Programs

  • Mathematica
    Select[Range[100],Max@@FactorInteger[#][[All,2]]-Min@@FactorInteger[#][[All,2]]==1&]
Select[Range[300],  Min[e = FactorInteger[#][[;; , 2]]] +1 == Max[e] &] (* Amiram Eldar, Jan 30 2023 *)
  • PARI
    is(n)={my(e=factor(n)[,2]); n>1 && vecmin(e) + 1 == vecmax(e); } \\ Amiram Eldar, Jan 30 2023
  • Python
    from sympy import factorint
def ok(n):
    e = sorted(factorint(n).values())
    return n > 1 and max(e) == 1 + min(e)
print([k for k in range(280) if ok(k)]) # Michael S. Branicky, Dec 18 2021

Formula

A051903(a(n)) - A051904(a(n)) = 1.

A325226 Number of prime factors of n that are less than the largest, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 3, 0, 2, 0, 2, 2, 1, 0, 4, 0, 1, 1, 2, 0, 1, 1, 3, 1, 1, 0, 3, 0, 1, 2, 0, 1, 2, 0, 2, 1, 2, 0, 3, 0, 1, 1, 2, 1, 2, 0, 4, 0, 1, 0, 3, 1, 1, 1, 3, 0, 3, 1, 2, 1, 1, 1, 5, 0, 1, 2, 2, 0, 2, 0, 3, 2
Offset: 1

Views

Author

Gus Wiseman, Apr 12 2019

Keywords

Examples

			The prime factors of 300 are {2,2,3,5,5} of which {2,2,3} are less than the largest, so a(300) = 3.

Links

Crossrefs

Positions of 0's are A000961. Positions of 1's are A325230. Positions of terms > 1 are A307517.
Cf. A000094, A000245, A001222, A052126, A053585, A061395, A062977, A064989, A071178, A105441, A108951, A256617.

Programs

  • Mathematica
    Table[PrimeOmega[n/Power@@FactorInteger[n][[-1]]],{n,100}]
  • PARI
    A071178(n) = if(1==n, 0, factor(n)[omega(n), 2]);
A325226(n) = (bigomega(n) - A071178(n)); \\ Antti Karttunen, Nov 17 2019

Formula

a(n) = A001222(n/A053585(n)).
a(n) = A001222(n) - A071178(n) = A062977(A108951(n)). - Antti Karttunen, Nov 17 2019

Extensions

Data section extended up to term a(105) by Antti Karttunen, Nov 17 2019

A376514 Number of divisors of n that are neither squarefree nor prime powers.

Original entry on oeis.org

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

Views

Author

Michael De Vlieger, Sep 25 2024

Keywords

Comments

This sequence is distinct from both A062977 and A335460.
A062977(36) = 0 while a(36) = 3 = card({12, 18, 36}).
A335460(60) = 6 while a(60) = 3 = card({12, 20, 60}).

Links

Crossrefs

Cf. A000005, A001221, A001222, A027750, A046660, A126706, A376504.

Programs

  • Mathematica
    {0}~Join~Table[DivisorSigma[0, n] - Total@ #1 - 2^#2 + #2 & @@ {#, Length[#]} &[FactorInteger[n][[All, -1]] ], {n, 2, 120}] (* or *)
Table[DivisorSum[n, 1 &, Nor[PrimePowerQ[#], SquareFreeQ[#]] &], {n, 120}]

Formula

a(n) = tau(n) - 2^omega(n) - bigomega(n) + omega(n).
a(n) = A000005(n) - 2^A001221(n) - A001222(n) + A001221(n).
a(n) = A000005(n) - 2^A001221(n) - A046660(n).
Intersection of row n of A027750 and A126706.
a(n) > 0 for n in A126706.

A382219 Product of the largest and smallest exponents in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 9, 4, 1, 1, 2, 1, 1, 1, 16, 1, 2, 1, 2, 1, 1, 1, 3, 4, 1, 9, 2, 1, 1, 1, 25, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 4, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 36, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 16, 1, 1, 2, 1, 1, 1, 3, 1, 2
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 19 2025

Keywords

Comments

The asymptotic density of the occurrences of k = 1, 2, ... in this sequence is 1/zeta(2) for k = 1 and 1/zeta(k+1) - 1/zeta(k) for k >= 2, and the asymptotic mean of this sequence is A033150, the same densities and mean as in A051903, since a(n) = A051903(n) for nonpowerful numbers n (A052485) whose asymptotic density is 1. - Amiram Eldar, Mar 28 2025

Links

Crossrefs

Cf. A005361, A033150, A051903, A051904, A052485, A062977, A066048, A304233, A333352.

Programs

  • Mathematica
    Table[Max @@ (#[[2]] & /@ FactorInteger[n]) Min @@ (#[[2]] & /@ FactorInteger[n]), {n, 90}]
  • PARI
    a(n) = if(n == 1, 1, my(e = factor(n)[,2]); vecmin(e) * vecmax(e)); \\ Amiram Eldar, Mar 28 2025

Formula

If n = Product (p_j^k_j) then a(n) = min{k_j} * max{k_j}.
a(n) = A051903(n) * A051904(n) for n > 1.
Showing 1-5 of 5 results.

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