A088860 -id:A088860 - cp's OEIS frontend

A381032 The radix prime of the least significant digit > 1 in the primorial base expansion of n, or 1 if there is no such digit.

1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 5, 5, 5, 5, 3, 3, 5, 5, 5, 5, 3, 3, 5, 5, 5, 5, 3, 3, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 3, 3, 5, 5, 5, 5, 3, 3, 5, 5, 5, 5, 3, 3, 5, 5, 5, 5, 3, 3, 7, 7, 7, 7, 3, 3, 7, 7, 7, 7, 3, 3, 5, 5, 5, 5, 3, 3, 5, 5, 5, 5, 3, 3, 5, 5, 5, 5, 3, 3, 7, 7, 7, 7, 3, 3, 7, 7, 7, 7, 3, 3, 5, 5, 5, 5, 3, 3, 5

Offset: 0

Antti Karttunen, Feb 13 2025

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to primorial base.

Cf. A008578, A020639, A249739, A276086, A327860, A328572, A328828, A381033, A381034.
Cf. A088860 (positions of records for values > 1), A276156 (positions of 1's).
Cf. also A053669, A351566.

A381032(n) = { my(p=2); while(n, if((n%p)>1, return(p)); n = n\p; p = nextprime(1+p)); (1); };

a(n) = A008578(1+A328828(n)).
a(n) = A020639(A328572(n)) = A249739(A276086(n)).
For all n, a(n) divides A327860(n).

A370134 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 1 <= k <= n; sums of two primorials > 1, not necessarily distinct.

Original entry on oeis.org

4, 8, 12, 32, 36, 60, 212, 216, 240, 420, 2312, 2316, 2340, 2520, 4620, 30032, 30036, 30060, 30240, 32340, 60060, 510512, 510516, 510540, 510720, 512820, 540540, 1021020, 9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380, 223092872, 223092876, 223092900, 223093080, 223095180, 223122900, 223603380

Offset: 1

Antti Karttunen, Mar 07 2024

			Triangle begins as:
        4;
        8,      12;
       32,      36,      60;
      212,     216,     240,     420;
     2312,    2316,    2340,    2520,    4620;
    30032,   30036,   30060,   30240,   32340,   60060;
   510512,  510516,  510540,  510720,  512820,  540540, 1021020;
  9699692, 9699696, 9699720, 9699900, 9702000, 9729720, 10210200, 19399380;

A370121 without its leftmost column. Subsequence of  A370132.
Cf. A002110, A049345, A087112, A276150, A370135.
Cf. A088860 (right edge).

nn = 20; MapIndexed[Set[P[First[#2] - 1], #1] &, FoldList[Times, 1, Prime@ Range[nn + 1]]]; Table[(P[n] + P[k]), {n, nn}, {k, n}] (* Michael De Vlieger, Mar 08 2024 *)

A002110(n) = prod(i=1,n,prime(i));
A370134(n) = { n--; my(c = (sqrtint(8*n + 1) - 1) \ 2); (A002110(1+c) + A002110(1+n - binomial(c + 1, 2))); };

For n >= 1, A276150(a(n)) = 2.

A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

Original entry on oeis.org

1, 2, 12, 120, 1680, 36960, 960960, 32672640, 1241560320, 57111774720, 3312482933760, 205373941893120, 15197671700090880, 1246209079407452160, 107173980829040885760, 10074354197929843261440, 1067881544980563385712640, 126010022307706479514091520, 15373222721540190500719165440

Offset: 0

Ilya Gutkovskiy, Apr 26 2018

nonn

For n > 0, a(n) is the smallest number m having exactly n distinct prime divisors and exactly 2*n - 1 prime divisors counted with multiplicity.

			a(1) = 2^1;
a(2) = 2^2*3;
a(3) = 2^3*3*5;
a(4) = 2^4*3*5*7;
a(5) = 2^5*3*5*7*11, etc.

Central terms of triangle A303555 (for n > 0).

Cf. A000079, A002110, A003680, A005179, A011782, A038547, A061283, A070175, A088860, A102476.

Join[{1}, Table[2^(n - 1) Product[Prime[j], {j, n}], {n, 18}]]

a(n) = A011782(n)*A002110(n).

A378133 Irregular triangle T(n,k) = P(n)*2^k, n >= 0, k = 0..floor(log_2 prime(k+1)), where P = A002110.

Original entry on oeis.org

1, 2, 4, 6, 12, 24, 30, 60, 120, 210, 420, 840, 1680, 2310, 4620, 9240, 18480, 30030, 60060, 120120, 240240, 480480, 510510, 1021020, 2042040, 4084080, 8168160, 9699690, 19399380, 38798760, 77597520, 155195040, 223092870, 446185740, 892371480, 1784742960, 3569485920

Offset: 0

Michael De Vlieger, Nov 17 2024

Subset of A060735.

a(n) = min(2*a(n-1), f(a(n-1))), where f(k) is the smallest primorial (A002110) greater than k, for n > 0. - Hal M. Switkay, Mar 19 2025

			Rows n = 0..9:
n\k |         0          1          2           3           4
-------------------------------------------------------------
  0 |         1          .          .           .           .
  1 |         2          4          .           .           .
  2 |         6         12         24           .           .
  3 |        30         60        120           .           .
  4 |       210        420        840        1680           .
  5 |      2310       4620       9240       18480           .
  6 |     30030      60060     120120      240240      480480
  7 |    510510    1021020    2042040     4084080     8168160
  8 |   9699690   19399380   38798760    77597520   155195040
  9 | 223092870  446185740  892371480  1784742960  3569485920

Michael De Vlieger, Table of n, a(n) for n = 0..3494 (rows n = 0..349, flattened).

Cf. A000079, A002110, A060735, A088860, A098388, A102476, A378144.

nn = 16;
MapIndexed[Set[P[First[#2] - 1], #1] &,
  FoldList[Times, 1, Prime@ Range[nn + 1] ] ];
Union@ Flatten@
  Table[P[i]*2^Range[0, Floor[Log2[Prime[i + 1] ] ] ], {i, 0, nn}]

T(n,k) = A002110(n)*A000079(k), n >= 0, k = 0..A098388(k+1).
T(n,0) = A002110(n).
T(n,1) = A088860(n), n >= 1.
T(n,2) = A102476(n), n >= 2.
T(n,A098388(k+1)) = A378144(n).
Let S(n,j) = A002110(n)*j, n >= 0, j = 0..A006093(n+1) = P(n)*j, n >= 0, j = 0..prime(n+1)-1. Then T(n,k) = S(n, 2^k).

A384656 a(n) = Sum_{k=1..n} A051903(ugcd(n,k)), where ugcd(n,k) is the greatest divisor of k that is a unitary divisor of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 4, 1, 3, 2, 6, 1, 9, 1, 8, 7, 4, 1, 12, 1, 13, 9, 12, 1, 16, 2, 14, 3, 17, 1, 22, 1, 5, 13, 18, 11, 24, 1, 20, 15, 22, 1, 30, 1, 25, 18, 24, 1, 27, 2, 28, 19, 29, 1, 32, 15, 28, 21, 30, 1, 51, 1, 32, 22, 6, 17, 46, 1, 37, 25, 46, 1, 41, 1, 38, 30

Offset: 1

Amiram Eldar, Jun 06 2025

nonn

The terms of this sequence can be calculated efficiently using the 1st formula. The value of the function f(n, k) is equal to the number of integers i from 1 to n such that the greatest divisor of k that is a unitary divisor of n is is 1 if k = 1, or k-free if k >= 2 (k-free numbers are numbers that are not divisible by a k-th power other than 1). E.g., f(n, 1) = A047994(n), f(n, 2) = A384048(n), and f(n, 3) = A384049(n).

The record values of a(n)/n are 1, 2, 6, 12, 60, 420, ..., i.e, 1, 2, 6, followed by twice the primorials (A088860, A097250) starting from 2*primorial(2) = 2*A002110(2) = 12. The record values of a(n)/n converge to 5/4.

			a(4) = A051903(ugcd(4,1)) + A051903(ugcd(4,2)) + A051903(ugcd(4,3)) + A051903(ugcd(4,4)) = A051903(1) + A051903(1) + A051903(1) + A051903(4) = 0 + 0 + 0 + 2 = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Unitary analog of A384655.

Cf. A000961, A002110, A005117, A047994, A051953, A088860, A097250, A384046, A384048, A384049, A383159.

f[p_, e_, k_] := p^e - If[e < k, 0, 1]; a[n_] := Module[{fct = FactorInteger[n], emax, s}, emax = Max[fct[[;; , 2]]]; s = emax * n; Do[s -= Times @@ (f[#1, #2, k] & @@@ fct), {k, 1, emax}]; s]; a[1] = 0; Array[a, 100]

a(n) = if(n == 1, 0, my(f = factor(n), p = f[,1], e = f[,2], emax = vecmax(e), s = emax*n); for(k = 1, emax, s -= prod(i = 1, #p, p[i]^e[i] - if(e[i] < k, 0, 1))); s);

a(n) = Sum_{k=1..A051903(n)} (n - f(n, k)) = A051903(n) * n  - Sum_{k=1..A051903(n)} f(n, k), where f(n, k) is multiplicative for a given k, with f(p^e, k) = p^e - 1 if e >= k and f(p^e, k) = p^e if e < k.
a(n) = 1 if and only if n is prime.
a(n) >= 2 if and only if n is composite.
a(n) >= n - A047994(n) with equality if and only if n is squarefree (A005117).
a(n) >= 2*n - A047994(n) - A384048(n) with equality if and only if n is cubefree that is not squarefree (i.e., n in A067259, or equivalently, A051903(n) = 2).
a(n) <= A384655(n) with equality if and only if n is squarefree (A005117).
a(n) < 5*n/4 and lim sun_{n->oo} a(n)/n = 5/4.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum{k>=1} (1 - Product_{p prime} (1 - 1/(p^(2*k-1)*(p+1)))) = 0.36292303251495264373... .

A084737 Beginning with 1, numbers such that (a(n+2)-a(n+1))/(a(n+1)-a(n)) = prime(n).

Original entry on oeis.org

1, 2, 4, 10, 40, 250, 2560, 32590, 543100, 10242790, 233335660, 6703028890, 207263519020, 7628001653830, 311878265181040, 13394639596851070, 628284422185342480, 33217442899375387210, 1955977793053588026280

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 14 2003

nonn

Successive differences are primorials.

			a(3) = 4, a(4) = 10 and a(5) = 40 and (40-10)/(10-4) = 5 = prime(3).

Index entries for sequences related to primorial numbers

Cf. A002110, A060389, A088860, A143293, A276085.

Join[{1},Accumulate[FoldList[Times,1,Prime[Range[20]]]]+1] (* Harvey P. Dale, Dec 14 2011 *)

From Antti Karttunen, Feb 06 2024: (Start)
For n >= 1, a(n) = A276085(2*A002110(n-1)).
For n >= 2, a(n) = 1 + A143293(n-2).
For n >= 3, a(n) = 2 + A060389(n-2).
(End)

More terms from Vladeta Jovovic, Jun 17 2003

A309308 Least number k with A309004(k) = n.

Original entry on oeis.org

1, 12, 60, 420, 4620, 360, 1021020, 19399380, 446185740, 13860, 401120980260, 2520, 608500527054420, 26165522663340060, 180180, 65178316954380089460, 3845520700308425278140, 234576762718813941966540, 15716643102160534111758180, 27720, 3063060, 6435289534681345815798169108260

Offset: 1

Amiram Eldar, Jul 22 2019

nonn

			a(3) = 60 since 60 is the least number with 3 numbers having the same prime signature and set of distinct prime factor as 60: 60 = 2^2 * 3 * 5, 90 = 3^2 * 2 * 5, and 150 = 5^2 * 2 * 3.

Cf. A088860, A111470, A309004.

Subsequence of A025487.

a[n_] := Multinomial @@ Tally[FactorInteger[n][[;;,2]]][[;;,2]]; m = 6; s = Table[0, {m}]; c = 0; n = 1; While[c < m, i = a[n]; If[i <= m && s[[i]] == 0, s[[i]] = n; c++]; n++]; s

a(k) <= A088860(k).

A364878 Triangle read by rows: T(n,k), 0 <= k <= n, is the smallest number that has n distinct prime factors, k of which are unique.

Original entry on oeis.org

1, 4, 2, 36, 12, 6, 900, 180, 60, 30, 44100, 6300, 1260, 420, 210, 5336100, 485100, 69300, 13860, 4620, 2310, 901800900, 69369300, 6306300, 900900, 180180, 60060, 30030, 260620460100, 15330615300, 1179278100, 107207100, 15315300, 3063060, 1021020, 510510

Offset: 0

Jon E. Schoenfield, Aug 11 2023

If a number's prime factorization is written as the product of the number's prime factors without using exponents (e.g., 1260 = 2*2*3*3*5*7), a prime factor is unique (i.e., one of a kind) if it appears only once; prime factors that have one or more duplicates are not unique. E.g., the distinct prime factors of 1260 are 2, 3, 5, and 7, but the only unique prime factors of 1260 are 5 and 7.
From Michael De Vlieger, Aug 12 2023: (Start)
T(n,k) is in A025487 by definition.
The number 2 is the only prime term and 4 the only composite prime power term.
T(n,k), k < n, n > 2 is neither squarefree nor a prime power and is in A126706.
No prime power divisor p^e | T(n,k) is such that e > 2. (End)

			T(2,0) = 36: 36 = 2*2*3*3, so 36 has 2 distinct prime factors (2 and 3) but no unique prime factors (each prime factor has a duplicate), and 36 is the smallest number with this property.
T(2,2) = 6: 6 = 2*3, so 6 has 2 distinct prime factors (2 and 3), and each of those is a unique prime factor (having no duplicates), and 6 is the smallest number with this property.
T(3,2) = 60: 60 = 2*2*3*5, so 60 has 3 distinct prime factors (2, 3, and 5), but only 2 unique prime factors (3 and 5, since the factor 2 is duplicated), and 60 is the smallest number having this property.
Table begins:
  n\k|         0         1        2       3       4      5      6
  ---+-----------------------------------------------------------
  0  |         1;
  1  |         4,        2;
  2  |        36,       12,       6;
  3  |       900,      180,      60,     30;
  4  |     44100,     6300,    1260,    420,    210;
  5  |   5336100,   485100,   69300,  13860,   4620,  2310;
  6  | 901800900, 69369300, 6306300, 900900, 180180, 60060, 30030;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0..150, flattened)

Cf. A002110, A025487, A061742, A088860, A126706, A228593.

T[n_,k_]:=Module[{primes=Array[Prime,n],primeProducts},primeProducts=Table[If[j>n-k,primes[[j]],primes[[j]]^2],{j,1,n}];Times@@primeProducts];lst[rows_]:=Table[T[n,k],{n,0,rows},{k,0,n}]//Flatten;lst[7] (* Robert P. P. McKone, Aug 12 2023 *)

T(n,k) = Product_{j=1..n} prime(j)^m where m = 1 if j > n - k, m = 2 otherwise.
T(n,k) = A002110(n)*A002110(n-k).
T(n,n) = A002110(n).
T(n,0) = A002110(n)^2.

A387153 Squarefree 3-abundant numbers: squarefree numbers k such that A000203(k) > 3*k.

Original entry on oeis.org

30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 79170, 82110, 84630, 85470, 91770, 94710, 99330, 101010, 103530, 108570, 111930, 117390, 122430, 128310, 136290, 140910, 144690, 154770, 161070, 164010, 166530, 168630, 182490, 191730, 205590

Offset: 1

Amiram Eldar, Aug 19 2025

nonn

First differs from A067885 at n = 11: A067885(11) = 72930 is not a term of this sequence. a(59) = 510510 is the least term of this sequence that is not in A067885.
Subsequence of A285615 and first differs from it at n = 51: A285615(51) = 390390 is not a term of this sequence.
This sequence is not the same as the sequence of numbers k such that A048250(k) > 3*k which includes all the terms of this sequence but also nonsquarefree numbers, the least of them is 2*A002110(52) = A088860(52) = 2.1248...*10^96.
The least odd term is A002110(17)/2 = 961380175077106319535, the least term that is not divisible by 3 is a(5607800) = 66853496710, and the least term that is coprime to 6 is A002110(52)/6 = 1.7706...*10^95.
If k is a term and m is a squarefree number coprime to k, then k*m is also a term.
The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 17, 95, 795, 8162, 86331, 854164, 8372782, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00008... .

			30030 = 2 * 3 * 5 * 7 * 11 * 13 is a term since it is squarefree, and sigma(30030) = 96768 > 3*30030 = 90090.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005117 and A068403.
Subsequence of A087248 and A285615.
Cf. A000203, A001221, A002110, A013929, A048250, A088860.

q[k_] := Module[{f = FactorInteger[k]}, Max[f[[;;, 2]]] == 1 && Times @@ (1 + f[[;; , 1]]) > 3*k]; Select[Range[2*10^5], q]

isok(k) = {my(f = factor(k)); issquarefree(f) && vecprod(apply(x -> x+1, f[, 1])) > 3*k;}

A001221(a(n)) >= 6.

cp's OEIS Frontend

A381032 The radix prime of the least significant digit > 1 in the primorial base expansion of n, or 1 if there is no such digit.

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A370134 Triangle read by rows: T(n,k) = A002110(n) + A002110(k), 1 <= k <= n; sums of two primorials > 1, not necessarily distinct.

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A303557 a(0) = 1; a(n) = 2^(n-1)*prime(n)#, where prime(n)# is the product of first n primes.

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A378133 Irregular triangle T(n,k) = P(n)*2^k, n >= 0, k = 0..floor(log_2 prime(k+1)), where P = A002110.

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A384656 a(n) = Sum_{k=1..n} A051903(ugcd(n,k)), where ugcd(n,k) is the greatest divisor of k that is a unitary divisor of n.

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A084737 Beginning with 1, numbers such that (a(n+2)-a(n+1))/(a(n+1)-a(n)) = prime(n).

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A309308 Least number k with A309004(k) = n.

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A364878 Triangle read by rows: T(n,k), 0 <= k <= n, is the smallest number that has n distinct prime factors, k of which are unique.

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