A048597 -id:A048597 - cp's OEIS frontend

A051266 Numbers n such that maximal value of prime divisors of reduced residue system for n is 2.

7, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 50, 51, 52, 54, 56, 57, 58, 62, 63, 64, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126

Offset: 1

Labos Elemer

nonn

Largest value of A001221(k) = 2 for 1 <= k <= n such that gcd (k, n) = 1, i.e., k in row n of A038566. - Michael De Vlieger, Aug 10 2017

			n = 29 is here because for terms of RRS(29) = {1, 2, ..., 27, 28} the number of prime divisors is 0(for 1), 1(for prime powers) or 2 (for 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28).

Michael De Vlieger, Table of n, a(n) for n = 1..161

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

Block[{n = 2, P}, P = Product[Prime@ i, {i, n}]; P + Position[#, n][[All, 1]] &@ Array[Max@ Map[PrimeNu, Cases[Range@ #, k_ /; CoprimeQ[#, k]]] &, 120, P + 1]] (* Michael De Vlieger, Aug 10 2017 *)

A051267 Numbers n such that maximal value of prime divisors of reduced residue system for n is 3.

Original entry on oeis.org

31, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 106, 107, 109, 111, 113, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 127, 128, 129, 131, 133, 134, 136, 137, 139, 141, 142, 143, 145, 146, 147

Offset: 1

Labos Elemer

nonn

Largest value of A001221(k) = 3 for 1 <= k <= n such that gcd(k, n) = 1, i.e., k in row n of A038566. - Michael De Vlieger, Aug 10 2017

Michael De Vlieger, Table of n, a(n) for n = 1..2446

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

Block[{n = 3, P}, P = Product[Prime@ i, {i, n}]; P + Position[#, n][[All, 1]] &@ Array[Max@ Map[PrimeNu, Cases[Range@ #, k_ /; CoprimeQ[#, k]]] &, 117, P + 1]] (* Michael De Vlieger, Aug 10 2017 *)

A051268 Numbers n such that maximal value of prime divisors of reduced residue system for n is 4.

Original entry on oeis.org

211, 221, 223, 227, 229, 233, 239, 241, 247, 251, 253, 257, 263, 269, 271, 277, 281, 283, 289, 293, 299, 307, 311, 313, 317, 319, 323, 331, 337, 341, 343, 347, 349, 353, 359, 361, 367, 371, 373, 377, 379, 383, 389, 391, 397, 401, 403, 407, 409, 413

Offset: 1

Labos Elemer

nonn

Largest value of A001221(k) = 4 for 1 <= k <= n such that gcd(k, n) = 1, i.e., k in row n of A038566. - Michael De Vlieger, Aug 10 2017

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A002110, A006862, A038566, A048597, A051250, A051265, A051266, A051267.

Block[{n = 4, P}, P = Product[Prime@ i, {i, n}]; P + Position[#, n][[All, 1]] &@ Array[Max@ Map[PrimeNu, Cases[Range@ #, k_ /; CoprimeQ[#, k]]] &, 175, P + 1]] (* Michael De Vlieger, Aug 10 2017 *)

More terms from Michael De Vlieger, Aug 10 2017

A076366 Number of numbers for which the count of nonprimes (i.e., 1 and composites) in their reduced residue set equals n.

Original entry on oeis.org

10, 6, 6, 4, 4, 7, 3, 4, 3, 7, 4, 4, 0, 6, 5, 1, 4, 3, 7, 4, 7, 2, 3, 3, 2, 2, 6, 5, 2, 2, 0, 6, 4, 3, 5, 4, 5, 3, 1, 3, 3, 4, 4, 6, 2, 3, 1, 6, 1, 6, 3, 6, 1, 4, 4, 4, 1, 1, 3, 6, 3, 2, 4, 4, 1, 1, 2, 4, 6, 0, 3, 4, 3, 5, 4, 1, 2, 8, 2, 5, 6, 2, 2, 5, 1, 4, 2, 4, 7, 2, 1, 2, 6, 1, 3, 5, 2, 3, 5, 3

Offset: 1

Labos Elemer, Oct 10 2002

nonn

			A048864(x) = 13: S = {},                                a(13) =  0;
A048864(x) = 16: S = {144},                             a(16) =  1;
A048864(x) = 22: S = {57,92},                           a(22) =  2;
A048864(x) = 7:  S = {13,34,50},                        a(7)  =  3;
A048864(x) = 4:  S = {15,22,54,84},                     a(4)  =  4;
A048864(x) = 15: S = {35,64,68,156,240},                a(15) =  5;
A048864(x) = 2:  S = {5,10,14,20,42,60},                a(2)  =  6;
A048864(x) = 6:  S = {11,21,32,40,72,78,210},           a(6)  =  7;
A048864(x) = 78: S = {133,177,268,440,490,552,870,990}, a(78) =  8;
A048864(x) = 1:  S = {1,2,3,4,6,8,12,18,24,30},         a(1)  = 10; See A048597.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A000010, A000720, A001221, A048597, A048864, A048865, A070971, A072022, A072023, A074915.

listn(nn) = {my(v = vector(10^5, n, eulerphi(n) - (primepi(n) - omega(n)))); vector(nn, k, if (#(w=Vec(select(x->(x==k), v, 1))) == 0, 0, #w));} \\ Michel Marcus, Feb 23 2020

a(n) = Card{x; A048864(x) = n}; a(n)=0 if supposedly no such number exists (see A072023).

A089016 Largest n-round number.

Original entry on oeis.org

2, 30, 1260, 60060, 2042040, 446185740, 25878772920, 7420738134810, 304250263527210, 52331045326680120, 9223346738827371150, 1922760350154212639070, 469153525437627883933080

Offset: 0

Paul Boddington, Nov 04 2003

A positive integer m is said to be n-round if it is divisible by all primes p satisfying p^(n+1) < m, or equivalently if all positive integers t < m satisfying GCD(t,m)=1 are divisible by at most n primes (counting multiplicities). Using the fact that p_(t+1)<2*p_t (p_t the (t)th prime) it is easy to prove that there are only finitely many n-round numbers for each n. 1-round numbers are usually called very round (A048597).

			a(4)=2042040 as follows. Certainly it is 4-round since it is <= 19^5 and divisible by all primes < 19. Also it is > 17^5, hence the largest 4-round number must be a multiple of 510510 = 2.3.5.7.11.13.17. But no 4-round number can be > 19^5 (since it is easy to prove that if p is a prime >= 19 and q is the next prime after p then 2.3.5....p > q^5 ). Thus 2042040, being the largest multiple of 510510 which is <= 19^5, must be the largest 4-round number.

T. D. Noe, Table of n, a(n) for n=0..100

Cf. A048597, A122936 (2-round numbers), A122937 (3-round numbers).

Table[k=1; While[prod=Times@@Prime[Range[k]]; prodT. D. Noe, Sep 21 2006 *)

More terms from T. D. Noe, Sep 21 2006

A122937 3-Round numbers: numbers n such that every number less than n and relatively prime to n has at most three prime factors (counting multiplicities).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 84, 90, 96, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 192

Offset: 1

T. D. Noe, Sep 21 2006

This sequence, for r=3 prime factors, is finite. Maillet proved that such sequences are finite for any fixed r. The case r=1 is A048597; case r=2 is A122936.

Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York, 1952, p. 134.

T. D. Noe, Table of n, a(n) for n=1..265 (complete sequence)

Cf. A048597 (very round numbers), A051250, A089016 (largest n-round number).

Omega[n_] := If[n==1, 0, Plus@@(Transpose[FactorInteger[n]][[2]])]; nn=60060; r=3; moreThanR=Select[Range[nn], Omega[ # ]>r&]; lst={1}; Do[s=Select[Range[n],GCD[n,# ]==1&]; If[Intersection[s,moreThanR]=={}, AppendTo[lst,n]], {n,2,nn}]; lst

A144740 Partial totient function phi(c, n) for c = 2: number of semiprimes less than and coprime to n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 2, 0, 1, 1, 4, 0, 4, 1, 2, 2, 6, 0, 6, 1, 2, 3, 8, 0, 6, 4, 6, 3, 10, 0, 10, 4, 5, 5, 7, 2, 13, 6, 8, 4, 15, 1, 15, 6, 6, 7, 16, 2, 13, 5, 10, 8, 18, 3, 12, 7, 11, 11, 21, 1, 21, 11, 11, 11, 15, 4, 23, 11, 14, 6, 24, 5, 24, 13, 11, 12, 18, 5, 26, 9, 17, 14, 27, 3, 19, 15, 19

Offset: 1

Reikku Kulon, Sep 20 2008

phi(c, n) = 0 iff n is in A048597.

			phi(2, 7) = 2: the two semiprimes less than 7 are 4 and 6.
phi(2, 15) = 2: there are five semiprimes less than 15 (4, 6, 9, 10, 14), but only 4 and 14 are relatively prime to 15.

Reikku Kulon, Table of n, phi(2, n) for n = 1..10000

Cf. A048597.

Cf. A036997 (phi(n) - max(phi(c, n)) over all nonnegative integers c).

A122936 2-Round numbers: numbers n such that every number less than n and relatively prime to n has at most two prime factors (counting multiplicities).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102, 108, 114, 120, 150, 180, 210, 240, 270, 300, 330, 420, 630, 840, 1050, 1260

Offset: 1

T. D. Noe, Sep 21 2006

This sequence, for r=2 prime factors, is finite. Maillet proved that such sequences are finite for any fixed r. The case r=1 is A048597; case r=3 is A122937.

Dickson, History of the Theory of Numbers, Vol. I, Chelsea, New York, 1952, p. 134.

H. Bonse, Über eine bekannte Eigenshaft der Zahl 30 und ihre Verallgemeinerung, Archiv d. Math. u. Physik (3) vol. 12 (1907) 292-295. See page 295.

Cf. A048597 (very round numbers), A051250, A089016 (largest n-round number).

Omega[n_] := If[n==1, 0, Plus@@(Transpose[FactorInteger[n]][[2]])]; nn=1260; r=2; moreThanR=Select[Range[nn], Omega[ # ]>r&]; lst={1}; Do[s=Select[Range[n],GCD[n,# ]==1&]; If[Intersection[s,moreThanR]=={}, AppendTo[lst,n]], {n,2,nn}]; lst
tpfQ[n_] :=Max[PrimeOmega /@ Select[Range[n - 1], CoprimeQ[#, n] &]] < 3; Select[Range[1300],tpfQ] (* Harvey P. Dale, Mar 16 2016 *)

A141341 Totally Goldbach numbers: Positive integers k such that for all primes p < k-1 with p not dividing k, k-p is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 18, 24, 30

Offset: 1

Rick L. Shepherd, Jun 25 2008

As Browers et al. point out, A141340 = A141341 union {7,14,16,36,42,48,60,90,210}, A020490 = A141341\{5} and A048597 = A141341\{5,10}. The authors show that the first strategy of Deshouillers et al. to establish a bound (of 10^520) for A141340 is sufficient for then determining the totally Goldbach numbers and "leads us naturally to interesting questions concerning primes in a fixed residue class".

J-M. Deshouillers, A. Granville, W. Narkiewicz and C. Pomerance, An upper bound in Goldbach's problem, Math. Comp. 61 (1993), 209-213.
David van Golstein Brouwers, John Bamberg and Grant Cairns, Totally Goldbach numbers and related conjectures, The Australian Mathematical Society, Gazette, Volume 31 Number 4, September 2004.
Index entries for sequences related to Goldbach conjecture

Cf. A020490, A048597, A141340.

q[k_]:=AllTrue[k-Select[Prime[Range[PrimePi[k-2]]],!Divisible[k,#]&],PrimeQ];Select[Range[30],q[#]&] (* James C. McMahon, Jul 21 2025 *)

A048869 Numbers for which reduced residue system contains as many primes as nonprimes.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 10, 15, 21, 45, 58, 82, 86, 92, 105, 116, 196, 238, 308, 310, 320, 380, 972, 978, 996, 1068, 1368, 5640, 10890

Offset: 1

Labos Elemer

This sequence is finite, since the number of primes < n is ~ n/log(n), but liminf phi(n) / ( n*log(log(n)) ) = exp(-gamma), a consequence of Mertens's theorem (see Hardy and Wright's Theory of Numbers). Also, if there exists a further element, it is >700000 (as verified with the enclosed Mathematica code). (Question: is it possible to show that there are no further such elements by using explicit bounds in the Prime Number Theorem and in Mertens's theorem?) - Reiner Martin, Jan 16 2002

There are no terms larger than 10890; it suffices to check to 52024. [Charles R Greathouse IV, Dec 19 2011]

			n=45, phi(45)=24 and the reduced residue system mod 45 contains 12 primes {2,7,11,13,17,19,23,29,31,37,41,43} and 12 nonprimes {1,4,8,14,16,22,26,28,32,34,38,44}.

A000720(n)-A001221(n) = A000010(n) - [ A000720(n)-A001221(n) ].

Cf. A048597-A002110.

Select[Range[700000], 2(PrimePi[ # ] - Length[FactorInteger[ # ]]) == EulerPhi[ # ]&]
For[i = 1, i < 20000, i++, If[2(PrimePi[i] - Length[FactorInteger[i]]) == EulerPhi[i], Print[i]]]; (* Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 29 2006 *)

p=0;for(n=1,6e4,if(isprime(n),p++);if(p==eulerphi(n)/2+omega(n),print1(n", "))) \\ Charles R Greathouse IV, Dec 19 2011

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 29 2006

cp's OEIS Frontend

A051266 Numbers n such that maximal value of prime divisors of reduced residue system for n is 2.

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A051267 Numbers n such that maximal value of prime divisors of reduced residue system for n is 3.

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A051268 Numbers n such that maximal value of prime divisors of reduced residue system for n is 4.

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A076366 Number of numbers for which the count of nonprimes (i.e., 1 and composites) in their reduced residue set equals n.

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A089016 Largest n-round number.

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A122937 3-Round numbers: numbers n such that every number less than n and relatively prime to n has at most three prime factors (counting multiplicities).

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A144740 Partial totient function phi(c, n) for c = 2: number of semiprimes less than and coprime to n.

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A122936 2-Round numbers: numbers n such that every number less than n and relatively prime to n has at most two prime factors (counting multiplicities).

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A141341 Totally Goldbach numbers: Positive integers k such that for all primes p < k-1 with p not dividing k, k-p is prime.