A048597 -id:A048597 - cp's OEIS frontend

A076367 Primes with subscripts from the Bonse sequence.

2, 3, 3, 5, 5, 5, 7, 7, 7, 11, 11, 11, 11, 11, 13, 13, 13, 17, 17, 17, 17, 17, 19, 19, 19, 23, 23, 23, 23, 23, 29, 29, 29, 29, 29, 29, 29, 31, 31, 31, 37, 37, 37, 37, 37, 37, 37, 41, 41, 41, 41, 41, 43, 43, 43, 47, 47, 47, 47, 47, 53, 53, 53, 53, 53, 53, 53, 59, 59, 59, 59, 59

Offset: 1

Labos Elemer, Oct 14 2002

nonn

This and sequence A060646 was used to prove that 30 is the largest number whose RRS does not contain composite numbers. See A048597, A060646 and corresponding References.

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

Cf. A048597, A060646, A076368. See also A076366.

c[x_, j_] := x+1-(j+Prime[j]); c[x, 0]=x; a=1000; t=Table[0, {a}]; t1=Table[0, {a}]; Table[fl=1; (*Print["% ", u, " #"]; *)Do[s=c[u, n]; If[Equal[fl, 1]&&Equal[Sign[s], -1], Print[n]; t[[u]]=n; t1[[u]]=Prime[n]; fl=0], {n, 1, u}], {u, 1, a}] //t (*=A060646*)//t1 (* =A076367 *)

a(n) = prime(A060646(n)).

A164297 Let S(n) be the set of all positive integers that are <= n and are coprime to n. a(n) = the number of members of S(n) that are each non-coprime with at least one other member of S(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 4, 0, 3, 2, 8, 0, 9, 2, 5, 4, 13, 0, 14, 2, 7, 6, 18, 0, 15, 7, 14, 6, 24, 0, 25, 8, 14, 10, 19, 4, 31, 11, 19, 9, 35, 2, 36, 11, 17, 14, 40, 4, 35, 10, 25, 15, 45, 5, 32, 14, 28, 20, 51, 2, 52, 20, 28, 21, 40, 7, 58, 20, 35, 13, 61, 9, 62, 24, 30, 23, 50, 8, 68, 18, 43, 27

Offset: 1

Leroy Quet, Aug 12 2009

nonn

A164296(n) + A164297(n) = phi(n) (= A000010(n) = the number of elements in S(n)).

			The positive integers that are <= 9 and are coprime to 9 are: 1,2,4,5, 7,8. 1 is coprime to each other member in S(9). While 2, 4, and 8 are non-coprime to each other. 5 is coprime to each other member of S(9). And 7 is also coprime to each other member. Since there are 3 integers in S(9) that are each non-coprime with at least one other member of S(9) -- these integers being 2, 4, and 8 -- then a(9) = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A164296, A036997, A048597.

Cf. A038566, A000010.

import Data.List ((\\))
a164297 n = length [m | let ts = a038566_row n, m <- ts,
                        any ((> 1) . gcd m) (ts \\ [m])]
-- Reinhard Zumkeller, May 28 2015

Extended by Ray Chandler, Mar 16 2010

A382659 Numbers k such that k < A053669(k)^2 * A380539(k), i.e., k < A382248(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 48, 50, 54, 60, 66, 70, 72, 78, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 210, 240, 252, 270, 300, 330, 360, 390

Offset: 1

Michael De Vlieger, Apr 14 2025

Numbers k whose reduced residue system (RRS) does not intersect A126706 (i.e., the sequence of numbers that are neither squarefree nor prime powers). Alternatively, numbers k whose RRS is a subset of A303554 (i.e., the union of powers of primes and squarefree numbers).
Let p = A053669(k), and let q = A380539(k). Thus, p and q are the smallest and second smallest primes, respectively, that do not divide k. Let m = p^2 * q = A382248(k). Then this sequence is that of k such that k < m.
There are 72 terms in this sequence.
Sequences A048597 and A051250 are proper subsets of this sequence.

			Let omega = A001221.
For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = 2^2 * 3 = 12.
For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 16, 32}.
  11 is in the sequence since 11 < m, m = 2^2 * 3 = 12, but 13 is not, since 13 > 12.
  9 is in the sequence since 9 < m, m = 2^2 * 5 = 20.
  25 is not a term since 25 > 12, and 27 is not a term since 27 > 20.
For omega = 2, we have the subset {6, 10, 12, 14, 15, 18, 20, 22, 24, 26, 28, 34, 36, 38, 40, 44, 48, 50, 54, 72, 96, 108, 144, 162}.
  38 = 2*19 is a term since 38 < 45, 45 = 3^2 * 5, but 46 = 2*23 is not, since 46 > 45.
  15 = 3*5 is a term since 15 < 20, but 21 is not, since 21 > 20 and 35 is not, since 35 > 12.
  Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, 24, 36, 48, 54, 72, 96, 108, 144, 162}, since m = 5^2 * 7 = 175.
  Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, 50}, since m = 3^2 * 7 = 63.
  Intersection with A033847 = {k : rad(k) = 14} is {14, 28}, since m = 3^2 * 5 = 45, etc.
For omega = 3, we have the subset {30, 42, 60, 66, 70, 78, 84, 90, 102, 114, 120, 126, 132, 138, 150, 156, 168, 174, 180, 240, 252, 270, 300, 360, 450, 480}, of which {30, 42, 66, 70, 78, 102, 114, 138, 174} are squarefree.
  Intersection with A143207 = {k : rad(k) = 30} is {30, 60, 90, .., 480} because m = 7^2 * 11 = 539.
  Intersection with 42*A108319 = {k : rad(k) = 42} is {42, 84, 126, 168}, since m = 5^2 * 11 = 275, etc.
For omega = 4, we have the subset {210, 330, 390, 420, 510, 630, 840, 1050, 1260, 1470}, of which {210, 330, 390, 510} are squarefree.
  Intersection with A147571 = {k : rad(k) = 210} is {210, 420, 630, 840, 1050, 1260, 1470} since m = 11^2 * 13 = 1573, etc.
For omega = 5, we have 2310 = 2*3*5*7*11, a term since 2310 < 13*17 = 2873; 2730 = 2*3*5*7*13 is not a term.
There are no terms larger than 2310, since the intersection with A147572 = {2310}, 2730 is not a term, and k = Product_{i=1..j} prime(i), k > prime(j+1)^2 * prime(j+2) for j > 5. Therefore the sequence is finite like A051250.

Michael De Vlieger, Table of n, a(n) for n = 1..72

Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A126706, A303554, A380539, A382248, A382960.

Select[Range[30030], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^(2 - c)]; q = NextPrime[q]; c++]][[-1, 1]] ] ]

A382960 Numbers k such that k < A053669(k)^2 * A380539(k)^2, i.e., k < A382767(k).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84

Offset: 1

Michael De Vlieger, Apr 14 2025

Numbers k whose reduced residue system does not intersect A286708 (i.e., powerful numbers that are not prime powers).
Let p = A053669(k), and let q = A380539(k). Thus, p and q are the smallest and second smallest primes, respectively, that do not divide k. Let m = p^2 * q^2 = A382767(k). Then this sequence is that of k such that k < m.
This sequence is finite following arguments akin to those in A051250 and A382659, with 626 terms.
Sequences A048597, A051250, and A382659 are proper subsets of this sequence.

			Let omega = A001221.
For omega = 0, we have the subset {1}. 1 is in the sequence since 1 < m, m = (2*3)^2 = 36.
For omega = 1, we have the subset {2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 64, 81, 128}.
  31 is in the sequence since 31 < m, m = (2*3)^2 = 36, but 37 is not a term since 37 > 36.
  25 is in the sequence since 25 < m, m = 36.
  49 is not a term since 49 > 36, and 243 is not a term since 243 > 100, 100 = (2*5)^2, etc.
For omega = 2, we have the squarefree numbers {6, 10, 14, 15, 22, 26, 34, 35, 38, 46, 58, 62, 74, 82, 86, 94, 106, 118, 122, 134, 142, 146, 158, 166, 178, 194, 202, 206, 214, 218}.
  Intersection with A033845 = {k : rad(k) = 6} is {6, 12, 18, .., 1152}, since m = (5*7)^2 = 1225.
  Intersection with A033846 = {k : rad(k) = 10} is {10, 20, 40, ..., 400}, since m = (3*7)^2 = 441.
  Intersection with A033847 = {k : rad(k) = 14} is {14, 28, 56, ..., 224}, since m = (3*5)^2 = 225.
  Intersection with A033848 = {k : rad(k) = 15} is {15, 45, 75, 135}, since m = (2*7)^2 = 196, etc.

Michael De Vlieger, Table of n, a(n) for n = 1..626

Cf. A048597 (k such that k < p^2), A051250 (k such that k < p*q), A053669, A286708, A380539, A382659 (k such that k < p^2*q), A382767.

Select[Range[510510], Function[n, c = 0; q = 2; n < Times @@ Reap[While[c < 2, While[Divisible[n, q], q = NextPrime[q]]; Sow[q^2]; q = NextPrime[q]; c++]][[-1, 1]] ] ]

A113520 a(n) = largest nonprime nonnegative integer < n which is coprime to n.

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 6, 1, 8, 9, 10, 1, 12, 9, 14, 15, 16, 1, 18, 9, 20, 21, 22, 1, 24, 25, 26, 27, 28, 1, 30, 27, 32, 33, 34, 35, 36, 35, 38, 39, 40, 25, 42, 39, 44, 45, 46, 35, 48, 49, 50, 51, 52, 49, 54, 55, 56, 57, 58, 49, 60, 57, 62, 63, 64, 65, 66, 65, 68, 69, 70, 65, 72, 69, 74

Offset: 1

Leroy Quet, Jan 12 2006

nonn

a(1) = 0 because 0 is coprime with 1, for our purposes and is < 1.

The numbers n with a(n) = 1 are given in A048597. - Stefan Steinerberger, Oct 16 2007

			a(14) = 9 because 9 is the largest nonprime integer < 14 which is coprime to 14 (since the other nonprime integers > 9 and < 14 {10 and 12} aren't coprime with 14).

Diana Mecum, Table of n, a(n) for n = 1..1163 [From _Diana L. Mecum_, Aug 05 2008]

a = {0}; For[n = 2, n < 70, n++, i = n - 1; While[PrimeQ[i] || GCD[n, i] > 1, i-- ]; AppendTo[a, i]]; a (* Stefan Steinerberger, Oct 16 2007 *)
lnp[n_]:=Module[{k=n-1},While[PrimeQ[k]||!CoprimeQ[k,n],k--];k]; Array[ lnp,80] (* Harvey P. Dale, May 12 2019 *)

More terms from Stefan Steinerberger, Oct 16 2007

Terms 70 through 1163 from Diana L. Mecum, Aug 05 2008

A285788 Irregular triangle T(n,m): nonprime 1 <= k <= n such that n and k are coprime.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 4, 6, 1, 1, 4, 8, 1, 9, 1, 4, 6, 8, 9, 10, 1, 1, 4, 6, 8, 9, 10, 12, 1, 9, 1, 4, 8, 14, 1, 9, 15, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 1, 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 1, 9, 1, 4, 8, 10, 16, 20, 1, 9, 15, 21, 1, 4, 6, 8, 9, 10

Offset: 1

Michael De Vlieger, Apr 26 2017

Row n is a subset of A038566(n) such that the union of a(n) and A112484(n) = A038566(n).
Row lengths are A048864(n) = A000010(n)-(A000720(n)-A001221(n)), i.e., phi(n)-(pi(n)-omega(n)).
1 appears in every row since 1 is not prime and coprime to all n.
4 is the smallest composite and appears first in row 5 since 4 divides 4.
Rows that contain the single term 1 are in A048597; the largest n = 30 such that the only term is 1.
For prime p, row p contains 1 and all composites k < p, since 1 < m < p are coprime to p.

			Triangle begins:
  n\m  1  2   3   4  5   6   7
   1:  1
   2:  1
   3:  1
   4:  1
   5:  1  4
   6:  1
   7:  1  4   6
   8:  1
   9:  1  4   8
  10:  1  9
  11:  1  4   6   8  9  10
  12:  1
  13:  1  4   6   8  9  10  12
  14:  1  9
  15:  1  4   8  14
  16:  1  9  15
  ...

Michael De Vlieger, Table of n, a(n) for n = 1..11055 (rows 1 <= n <= 240)

Cf. A038566, A048597, A048864, A112484.

Table[Select[Range@ n, And[! PrimeQ@ #, CoprimeQ[#, n]] &], {n, 23}] // Flatten

from sympy import gcd, isprime
def a(n): return list(filter(lambda k: isprime(k)==0 and gcd(k, n)==1, range(1, n + 1)))
for n in range(1, 21): print(a(n)) # Indranil Ghosh, Apr 26 2017

A289172 Irregular triangle read by rows: row n lists terms m of A038566(n) such that A001221(m) = A051265(n), with a(1) = 1.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 4, 5, 6, 3, 5, 7, 2, 4, 5, 7, 8, 3, 7, 9, 6, 10, 5, 7, 11, 6, 10, 12, 3, 5, 9, 11, 13, 14, 15, 6, 10, 12, 14, 15, 5, 7, 11, 13, 17, 6, 10, 12, 14, 15, 18, 3, 7, 9, 11, 13, 17, 19, 10, 20, 15, 21, 6, 10, 12, 14, 15, 18, 20, 21, 22, 5, 7, 11

Offset: 1

Michael De Vlieger, Aug 11 2017

Consider A051265(n), the largest value of A001221(m) for 1 <= m <= n such that gcd(m, n) = 1 (i.e., m is in the reduced residue system or RRS of n, or m is a totative of n). Row n of this sequence consists of m in RRS(n) such that omega(m) = A051265(n).

			Triangle begins:
   n    T(n,m)                        A051265(n)
   1:   1                                     0
   2:   1                                     0
   3:   2                                     1
   4:   3                                     1
   5:   2    3    4                           1
   6:   5                                     1
   7:   6                                     2
   8:   3    5    7                           1
   9:   2    4    5    7    8                 1
  10:   3    7    9                           1
  11:   6   10                                2
  12:   5    7   11                           1
  13:   6   10   12                           2
  14:   3    5    9   11   13                 1
  15:  14                                     2
  16:  15                                     2
  17:   6   10   12   14   15                 2
  18:   5    7   11   13   17                 1
  19:   6   10   12   14   15   18            2
  20:   3    7    9   11   13   17   19       1

Michael De Vlieger, Table of n, a(n) for the first 1000 rows, flattened

Cf. A001221, A002110, A038566, A048597, A051250, A051265, A051266, A051267, A051268.

Table[MaximalBy[#, Last][[All, 1]] &@ Map[{#, PrimeNu@ #} &, Cases[Range[n - 1], k_ /; CoprimeQ[n, k]]] /. {} -> {1}, {n, 30}] // Flatten (* Michael De Vlieger, Aug 11 2017 *)

A383081 a(n) = A003557(A089016(n)).

Original entry on oeis.org

1, 1, 6, 2, 4, 2, 4, 1, 1, 4, 15, 1, 4, 10, 15, 1, 5, 24, 52, 1, 2, 5, 1, 3, 5, 31, 45, 2, 4, 14, 41, 1, 2, 5, 7, 2, 4, 5, 11, 30, 43, 2, 6, 18, 26, 71, 139, 3, 36, 69, 96, 5, 14, 69, 95, 1, 4, 11, 30, 57, 1, 2, 8, 39, 54, 255, 2, 3, 9, 32, 60, 82, 2, 7, 13, 45

Offset: 0

Michael De Vlieger, Apr 18 2025

nonn

Largest n-round number A089016(n) = k*P. Then a(n) = k = rad(k*P), where P is in A002110 and rad = A007947.

Michael De Vlieger, Table of n, a(n) for n = 0..5000

Cf. A002110, A003557, A007947, A048597, A089016, A122936, A122937.

Table[k = P = 1;
  While[P *= Prime[k]; P < Prime[k + 1]^(n + 1), k++];
  P /= Prime[k]; Floor[Prime[k]^(n + 1)/P], {n, 0, 75}]

A048980 Difference between number of nonprimes and primes in reduced residue system of primorial numbers.

Original entry on oeis.org

1, 1, 0, -6, -36, -196, -724, 7512, 366838, 11928316, 421130508, 14598816402, 584642184936, 25314953837836, 1128885572358548, 54492272309366314, 2950485568862138250, 213151926413154110951

Offset: 0

Labos Elemer

sign

			n=4, Q(4)=2*3*5*7=210, reduced residue system includes 48 terms:42 primes and 6 composites and 1: a(4)=6-42=-36.

Cf. A048868, A048867, A048597, A002110, A048862, A048863, A048866.

Table[Function[P, EulerPhi@ P - 2 # &[PrimePi@ P - n]]@ Product[Prime@ i, {i, n}], {n, 0, 12}] (* Michael De Vlieger, May 08 2017 *)

a(n) = A048863(n) - A048862(n) = A048866(A002110(n)).

a(n) = A005867(n) - 2*A000849(n) + 2*n.

Corrected and extended by Max Alekseyev, Feb 22 2016

A048982 Number of numbers which have a "prime-rich" reduced residue system (RRS) and binary order n.

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 15, 22, 32, 50, 85, 80, 98, 84, 59, 37, 10, 2, 0, 0, 0

Offset: 0

Labos Elemer

It is remarkable that in exponentially increasing ranges these occurrences increase to n=13 and thereafter decline to zero. So A048868 is believed to be finite.

			In binary order (A029837) zone of 7, i.e., in [65,128], 22 numbers belong to A048868: 66, 68, 70, 72, 74, 76, 78, 80, 84, 88, 90, 96, 98, 100, 102, 104, 108, 110, 112, 114, 120, and 126. The largest term is 90090. The largest 4 are divisible by 2310, the largest 28 by 210, and the largest 103 by 30.

Cf. A048862, A048863, A048864, A048865, A048866, A048867, A048868, A048869.

Cf. A002110, A029837, A048597.

cp's OEIS Frontend

A076367 Primes with subscripts from the Bonse sequence.

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A164297 Let S(n) be the set of all positive integers that are <= n and are coprime to n. a(n) = the number of members of S(n) that are each non-coprime with at least one other member of S(n).

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A382659 Numbers k such that k < A053669(k)^2 * A380539(k), i.e., k < A382248(k).

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A382960 Numbers k such that k < A053669(k)^2 * A380539(k)^2, i.e., k < A382767(k).

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A113520 a(n) = largest nonprime nonnegative integer < n which is coprime to n.

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A285788 Irregular triangle T(n,m): nonprime 1 <= k <= n such that n and k are coprime.

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A289172 Irregular triangle read by rows: row n lists terms m of A038566(n) such that A001221(m) = A051265(n), with a(1) = 1.

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A383081 a(n) = A003557(A089016(n)).

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