A006094 -id:A006094 - cp's OEIS frontend

A340537 Primes that are sums of a sequence of consecutive terms of A006094.

127, 491, 1201, 1427, 2003, 2713, 2767, 5431, 7229, 7639, 13001, 17231, 18061, 20753, 24509, 37337, 37589, 38149, 38261, 44563, 44839, 50969, 51517, 53609, 55201, 60859, 76519, 77191, 80239, 80783, 81703, 90823, 91583, 96493, 103079, 103687, 110573, 126713, 130411, 134093, 137777, 139199, 139663

Offset: 1

J. M. Bergot and Robert Israel, Jan 10 2021

nonn

Each term is the sum of at least three consecutive terms of A006094.

A number that can be expressed as such a sum in more than one way is only listed once. The first such number is 50911291 = 547*557+...+1051*1061 = 1423*1427+...+1559*1567.

			a(1) = 5*7+7*11+11*13 = 127.
a(2) = 5*7+7*11+11*13+13*17+17*19 = 491.
a(3) = 11*13+13*17+17*19+19*23+23*29 = 1201.
a(4) = 19*23+23*29+29*31 = 1427.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A006094, A340465. Includes A287653.

SP:= [seq(ithprime(i)*ithprime(i+1),i=1..100)]:
SSP:= ListTools:-PartialSums([0,op(SP)]):
select(t -> t <= SP[-1] and isprime(t),
  {seq(seq(SSP[j]-SSP[i],i=1..j-3),j=4..nops(SSP))});

A073491 Numbers having no prime gaps in their factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 19, 23, 24, 25, 27, 29, 30, 31, 32, 35, 36, 37, 41, 43, 45, 47, 48, 49, 53, 54, 59, 60, 61, 64, 67, 71, 72, 73, 75, 77, 79, 81, 83, 89, 90, 96, 97, 101, 103, 105, 107, 108, 109, 113, 120, 121, 125, 127, 128, 131, 135

Offset: 1

Reinhard Zumkeller, Aug 03 2002

A073490(a(n)) = 0; subsequences are: A000040, A000961, A006094, A002110, A000142, A073485.
A137721(n) = number of terms not greater than n; A137794(a(n))=1; complement of A073492. - Reinhard Zumkeller, Feb 11 2008
Essentially the same as A066311. - R. J. Mathar, Sep 23 2008
The Heinz numbers of the partitions that have no gaps. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product_{j=1..r} (p_j-th prime) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: (i) 18 (= 2*3*3) is in the sequence because it is the Heinz number of the partition [1,2,2]; (ii) 10 (= 2*5) is not in the sequence because it is the Heinz number of the partition [1,3]. - Emeric Deutsch, Oct 02 2015

			360 is a term, as 360 = 2*2*2*3*3*5 with consecutive prime factors.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A137791, A137792, A137793, A137895.

a073491 n = a073491_list !! (n-1)
a073491_list = filter ((== 0) . a073490) [1..]
-- Reinhard Zumkeller, Dec 20 2013

ok[n_] := (p = FactorInteger[n][[All, 1]]; PrimePi[Last@p] - PrimePi[First@p] == Length[p] - 1); Select[Range[135], ok] (* Jean-François Alcover, Apr 29 2011 *)
npgQ[n_]:=Module[{f=Transpose[FactorInteger[n]][[1]]},f==Prime[Range[ PrimePi[ f[[1]]], PrimePi[f[[-1]]]]]]; Join[{1},Select[Range[2,200],npgQ]] (* Harvey P. Dale, Apr 12 2013 *)

is(n)=my(f=factor(n)[,1]); for(i=2,#f,if(precprime(f[i]-1)>f[i-1], return(0))); 1 \\ Charles R Greathouse IV, Apr 28 2015

A078898 Number of times the smallest prime factor of n is the smallest prime factor for numbers <= n; a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 4, 11, 1, 12, 2, 13, 5, 14, 1, 15, 1, 16, 6, 17, 3, 18, 1, 19, 7, 20, 1, 21, 1, 22, 8, 23, 1, 24, 2, 25, 9, 26, 1, 27, 4, 28, 10, 29, 1, 30, 1, 31, 11, 32, 5, 33, 1, 34, 12, 35, 1, 36, 1, 37, 13, 38, 3, 39, 1, 40, 14, 41, 1, 42, 6, 43

Offset: 0

Reinhard Zumkeller, Dec 12 2002

nonn

From Antti Karttunen, Dec 06 2014: (Start)
For n >= 2, a(n) tells in which column of the sieve of Eratosthenes (see A083140, A083221) n occurs in. A055396 gives the corresponding row index.
(End)

Harvey P. Dale (terms 1 - 1000) & Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A002110, A008683, A008836, A020639, A032742, A054272, A055396, A078899, A078896, A083140, A083221, A243055, A246277, A249738, A249744, A249808, A249809, A249810, A249820, A249818, A250470, A250474, A250477, A250478, A251719, A251724, A251728.

Cf. A001248, A006094, A038110, A038111, A090076.

import Data.IntMap (empty, findWithDefault, insert)
a078898 n = a078898_list !! n
a078898_list = 0 : 1 : f empty 2 where
   f m x = y : f (insert p y m) (x + 1) where
           y = findWithDefault 0 p m + 1
           p = a020639 x
-- Reinhard Zumkeller, Apr 06 2015

N:= 1000: # to get a(0) to a(N)
Primes:= select(isprime, [2,seq(2*i+1,i=1..floor((N-1)/2))]):
A:= Vector(N):
for p in Primes do
  t:= 1:
  A[p]:= 1:
  for n from p^2 to N by p do
    if A[n] = 0 then
       t:= t+1:
       A[n]:= t
    fi
  od
od:
0,1,seq(A[i],i=2..N); # Robert Israel, Jan 04 2015

Module[{nn=90,spfs},spfs=Table[FactorInteger[n][[1,1]],{n,nn}];Table[ Count[ Take[spfs,i],spfs[[i]]],{i,nn}]] (* Harvey P. Dale, Sep 01 2014 *)

\\ Not practical for computing, but demonstrates the sum moebius formula:
A020639(n) = { if(1==n,n,vecmin(factor(n)[, 1])); };
A055396(n) = { if(1==n,0,primepi(A020639(n))); };
A002110(n) = prod(i=1, n, prime(i));
A078898(n) = { my(k,p); if(1==n, n, k = A002110(A055396(n)-1); p = A020639(n); sumdiv(k, d, moebius(d)*(n\(p*d)))); };
\\ Antti Karttunen, Dec 05 2014

;; With memoizing definec-macro.
(definec (A078898 n) (if (< n 2) n (+ 1 (A078898 (A249744 n)))))
;; Much better for computing. Needs also code from A249738 and A249744. - Antti Karttunen, Dec 06 2014

Ordinal transform of A020639 (Lpf). - Franklin T. Adams-Watters, Aug 28 2006
From Antti Karttunen, Dec 05-08 2014: (Start)
a(0) = 0, a(1) = 1, a(n) = 1 + a(A249744(n)).
a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(n / (A020639(n)*d)).
a(0) = 0, a(1) = 1, a(n) = sum_{d | A002110(A055396(n)-1)} moebius(d) * floor(A032742(n) / d).
[Instead of Moebius mu (A008683) one could use Liouville's lambda (A008836) in the above formulas, because all primorials (A002110) are squarefree. A020639(n) gives the smallest prime dividing n, and A055396 gives its index].
a(0) = 0, a(1) = 1, a(2n) = n, a(2n+1) = a(A250470(2n+1)). [After a similar recursive formula for A246277. However, this cannot be used for computing the sequence, unless a definition for A250470(n) is found which doesn't require computing the value of A078898(n).]
For n > 1: a(n) = A249810(n) - A249820(n).
(End)
Other identities:
a(2*n) = n.
For n > 1: a(n)=1 if and only if n is prime.
For n > 1: a(n) = A249808(n, A055396(n)) = A249809(n, A055396(n)).
For n > 1: a(n) = A246277(A249818(n)).
From Antti Karttunen, Jan 04 2015: (Start)
a(n) = 2 if and only if n is a square of a prime.
For all n >= 1: a(A251728(n)) = A243055(A251728(n)) + 2. That is, if n is a semiprime of the form prime(i)*prime(j), prime(i) <= prime(j) < prime(i)^2, then a(n) = (j-i)+2.
(End)
a(A000040(n)^2) = 2; a(A000040(n)*A000040(n+1)) = 3. - Reinhard Zumkeller, Apr 06 2015
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Sum_{k>=1} (A038110(k)/A038111(k))^2 = 0.2847976823663... . - Amiram Eldar, Oct 26 2024

a(0) = 0 prepended for recurrence's sake by Antti Karttunen, Dec 06 2014

A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 6, 1, 7, 3, 8, 1, 9, 1, 10, 5, 11, 1, 12, 2, 13, 4, 14, 1, 15, 1, 16, 7, 17, 3, 18, 1, 19, 11, 20, 1, 21, 1, 22, 6, 23, 1, 24, 2, 25, 13, 26, 1, 27, 5, 28, 17, 29, 1, 30, 1, 31, 10, 32, 7, 33, 1, 34, 19, 35, 1, 36, 1, 37, 9, 38, 3, 39, 1, 40, 8, 41, 1, 42

Offset: 1

Antti Karttunen, Aug 21 2014

nonn

If n >= 2, n occurs in column a(n) of A246278.

By convention, a(1) = 0 because 1 does not occur in A246278.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from indices in prime factorization

Terms of A348717 halved. A305897 is the restricted growth sequence transform.
Positions of terms 1 .. 8 in this sequence are given by the following sequences: A000040, A001248, A006094, A030078, A090076, A251720, A090090, A030514.
Cf. A078898 (has the same role with array A083221 as this sequence has with A246278).
Cf. A000040, A001222, A001358, A003961, A055396, A064989, A064216, A243055, A246272, A249810, A249820, A249735, A252463.
This sequence is also used in the definition of the following permutations: A246274, A246276, A246675, A246677, A246683, A249815, A249817 (A249818), A249823, A249825, A250244, A250245, A250247, A250249.
Also in the definition of arrays A249821, A251721, A251722.
Sum of prime indices of a(n) is A359358(n) + A001222(n) - 1, cf. A326844.
A112798 lists prime indices, length A001222, sum A056239.
Cf. A005940, A241916, A253565, A356958, A358195.

a246277[n_Integer] := Module[{f, p, a064989, a},
  f[x_] := Transpose@FactorInteger[x];
  p[x_] := Which[
    x == 1, 1,
    x == 2, 1,
    True, NextPrime[x, -1]];
  a064989[x_] := Times @@ Power[p /@ First[f[x]], Last[f[x]]];
  a[1] = 0;
  a[x_] := If[EvenQ[x], x/2, NestWhile[a064989, x, OddQ]/2];
a/@Range[n]]; a246277[84] (* Michael De Vlieger, Dec 19 2014 *)

A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A246277(n) = { if(1==n, 0, while((n%2), n = A064989(n)); (n/2)); };

A246277(n) = if(1==n, 0, my(f = factor(n), k = primepi(f[1,1])-1); for (i=1, #f~, f[i,1] = prime(primepi(f[i,1])-k)); factorback(f)/2); \\ Antti Karttunen, Apr 30 2022

from sympy import factorint, prevprime
from operator import mul
from functools import reduce
def a064989(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==2 else prevprime(i)**f[i] for i in f])
def a(n): return 0 if n==1 else n//2 if n%2==0 else a(a064989(n))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 15 2017

;; two different variants, the second one employing memoizing definec-macro)
(define (A246277 n) (if (= 1 n) 0 (let loop ((n n)) (if (even? n) (/ n 2) (loop (A064989 n))))))
(definec (A246277 n) (cond ((= 1 n) 0) ((even? n) (/ n 2)) (else (A246277 (A064989 n)))))

a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)) = a(A064216(n+1)). [Cf. the formula for A252463.]
Instead of the equation for a(2n+1) above, we may write a(A003961(n)) = a(n). - Peter Munn, May 21 2022
Other identities. For all n >= 1, the following holds:
For all w >= 0, a(p_{i} * p_{j} * ... * p_{k}) = a(p_{i+w} * p_{j+w} * ... * p_{k+w}).
For all n >= 2, A001222(a(n)) = A001222(n)-1. [a(n) has one less prime factor than n. Thus each semiprime (A001358) is mapped to some prime (A000040), etc.]
For all n >= 2, a(n) = A078898(A249817(n)).
For semiprimes n = p_i * p_j, j >= i, a(n) = A000040(1+A243055(n)) = p_{1+j-i}.
a(n) = floor(A348717(n)/2). - Antti Karttunen, Apr 30 2022
If n has prime factorization Product_{i=1..k} prime(x_i), then a(n) = Product_{i=2..k} prime(x_i-x_1+1). The opposite version is A358195, prime indices A358172, even bisection A241916. - Gus Wiseman, Dec 29 2022

A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

Original entry on oeis.org

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 27, 35, 49, 11, 12, 21, 125, 77, 121, 13, 14, 45, 55, 343, 143, 169, 17, 16, 33, 175, 91, 1331, 221, 289, 19, 18, 81, 65, 539, 187, 2197, 323, 361, 23, 20, 75, 625, 119, 1573, 247, 4913, 437, 529, 29, 22, 63, 245, 2401, 209, 2873, 391, 6859, 667, 841, 31

Offset: 2

Antti Karttunen, Aug 21 2014

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
This array can be obtained by taking every second column from array A242378, starting from its column 2.
Permutation of natural numbers larger than 1.
The terms on row n are all divisible by n-th prime, A000040(n).
Each column is strictly growing, and the terms in the same column have the same prime signature.
A055396(n) gives the row number of row where n occurs,
and A246277(n) gives its column number, both starting from 1.
From Antti Karttunen, Jan 03 2015: (Start)
A252759(n) gives their sum minus one, i.e. the Manhattan distance of n from the top left corner.
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252752 gives the inverse permutation. See also A246276.
(End)

			The top left corner of the array:
   2,     4,     6,     8,    10,    12,    14,    16,    18, ...
   3,     9,    15,    27,    21,    45,    33,    81,    75, ...
   5,    25,    35,   125,    55,   175,    65,   625,   245, ...
   7,    49,    77,   343,    91,   539,   119,  2401,   847, ...
  11,   121,   143,  1331,   187,  1573,   209, 14641,  1859, ...
  13,   169,   221,  2197,   247,  2873,   299, 28561,  3757, ...

Antti Karttunen, Table of n, a(n) for n = 2..1276; the first 50 antidiagonals of the array

First row: A005843 (the even numbers), from 2 onward.
Row 2: A249734, Row 3: A249827.
Column 1: A000040 (primes), Column 2: A001248 (squares of primes), Column 3: A006094 (products of two successive primes), Column 4: A030078 (cubes of primes).
Transpose: A246279.
Inverse permutation: A252752.
One more than A246275.
Cf. A005940, A242378, A246259, A000040, A002260, A004736, A003961, A055396, A083221, A114537, A246277 (terms of A348717 halved), A246675, A246684, A249818, A252759, A253515.
Arrays obtained by applying a particular function (given in parentheses) to the entries of this array. Cases where the columns grow monotonically are indicated with *: A249822 (A078898), A253551 (* A156552), A253561 (* A122111), A341605 (A017665), A341606 (A017666), A341607 (A006530 o A017666), A341608 (A341524), A341626 (A341526), A341627 (A341527), A341628 (A006530 o A341527), A342674 (A341530), A344027 (* A003415, arithmetic derivative), A355924 (A342671), A355925 (A009194), A355926 (A355442), A355927 (* sigma), A356155 (* A258851), A372562 (A252748), A372563 (A286385), A378979 (* deficiency, A033879), A379008 (* (probably), A294898), A379010 (* A000010, Euler phi), A379011 (* A083254).
Cf. A329050 (subtable).

f[p_?PrimeQ] := f[p] = Prime[PrimePi@ p + 1]; f[1] = 1; f[n_] := f[n] = Times @@ (f[First@ #]^Last@ # &) /@ FactorInteger@ n; Block[{lim = 12}, Table[#[[n - k, k]], {n, 2, lim}, {k, n - 1, 1, -1}] &@ NestList[Map[f, #] &, Table[2 k, {k, lim}], lim]] // Flatten (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003961 *)

(define (A246278 n) (if (<= n 1) n (A246278bi (A002260 (- n 1)) (A004736 (- n 1))))) ;; Square array starts with offset=2, and we have also tacitly defined a(1) = 1 here.
(define (A246278bi row col) (if (= 1 row) (* 2 col) (A003961 (A246278bi (- row 1) col))))

A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).
As a composition of other similar sequences:
a(n) = A122111(A253561(n)).
a(n) = A249818(A083221(n)).
For all n >= 1, a(n+1) = A005940(1+A253551(n)).
A(n, k) = A341606(n, k) * A355925(n, k). - Antti Karttunen, Jul 22 2022

Starting offset of the linear sequence changed from 1 to 2, without affecting the column and row indices by Antti Karttunen, Jan 03 2015

A003277 Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k, i.e., A000001(k) = 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, 141, 143, 145, 149, 151, 157, 159, 161, 163, 167, 173

Offset: 1

N. J. A. Sloane and Richard Stanley

Except for a(2)=2, all the terms in the sequence are odd. This is because of the existence of a non-cyclic dihedral group of order 2n for each n>1. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 09 2001
Also gcd(n, A051953(n)) = 1. - Labos Elemer
n such that x^n == 1 (mod n) has no solution 2 <= x <= n. - Benoit Cloitre, May 10 2002
There is only one group (the cyclic group of order n) whose order is n. - Gerard P. Michon, Jan 08 2008  [This is a 1947 result of Tibor Szele. - Charles R Greathouse IV, Nov 23 2011]
Any divisor of a Carmichael number (A002997) must be odd and cyclic. Conversely, G. P. Michon conjectured (c. 1980) that any odd cyclic number has at least one Carmichael multiple (if the conjecture is true, each of them has infinitely many such multiples). In 2007, Michon & Crump produced explicit Carmichael multiples of all odd cyclic numbers below 10000 (see link, cf. A253595). - Gerard P. Michon, Jan 08 2008
Numbers n such that phi(n)^phi(n) == 1 (mod n). - Michel Lagneau, Nov 18 2012
Contains A000040, and all members of A006094 except 6. - Robert Israel, Jul 08 2015
Number m such that n^n == r (mod m) is solvable for any r. - David W. Wilson, Oct 01 2015
Numbers m such that A074792(m) = m + 1. - Thomas Ordowski, Jul 16 2017
Squarefree terms of A056867 (see McCarthy link p. 592 and similar comment with "cubefree" in A051532). - Bernard Schott, Mar 24 2022

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
J. S. Rose, A Course on Group Theory, Camb. Univ. Press, 1978, see p. 7.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Max Alekseyev, Michon's conjecture (Open Problem Garden, Aug. 2007).
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025. See pp. 1, 2, 30.
Keith Conrad, When are all groups of order n cyclic?, University of Connecticut, 2019.
Peter J. Dukes and Joanna Niezen, Pairwise balanced designs of dimension three, Australasian Journal Of Combinatorics, Volume 61(1) (2015), pages 98-113.
Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.) 12 (1948), pp. 75-78.
Jose M. Grau, Manuel Rodríguez, A. Oller-Marcen, and Daniel Sadornil, Fermat test with gaussian base and Gaussian pseudoprimes, arXiv preprint arXiv:1401.4708 [math.NT], 2014.
Donald J. McCarthy, A survey of partial converses to Lagrange's theorem on finite groups, Transactions of the New York Academy of Sciences, Vol. 33, No. 6, Series II (1971), pp. 586-594. See page 592.
Gérard P. Michon, Carmichael Divisors
Gérard P. Michon and J. K. Crump, Carmichael Multiples of Odd Cyclic Numbers (up to 10000)
Jonathan Pakianathan and Krishnan Shankar, Nilpotent Numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
R. P. Stanley, Letter to N. J. A. Sloane, c. 1991
T. Szele, Über die endlichen Ordnungszahlen, zu denen nur eine Gruppe gehört, Commentarii Mathematici Helvetici 20 (1947), pp. 265-267.
Index entries for sequences related to groups

Subsequence of A051532. Intersection of A056867 and A005117.
Cf. A000010, A008966, A009195, A050384 (the same sequence but with the primes removed). Also A000001(a(n)) = 1.
Cf. A002997, A006094, A054395, A055561, A054396, A054397, A056867, A135850, A249550, A249551, A249552, A249553, A249554, A249555, A036537, A051953, A253595.

import Data.List (elemIndices)
a003277 n = a003277_list !! (n-1)
a003277_list = map (+ 1) $ elemIndices 1 a009195_list
-- Reinhard Zumkeller, Feb 27 2012

[n: n in [1..200] | Gcd(n, EulerPhi(n)) eq 1]; // Vincenzo Librandi, Jul 09 2015

select(t -> igcd(t, numtheory:-phi(t))=1, [$1..1000]); # Robert Israel, Jul 08 2015

Select[Range[175], GCD[#, EulerPhi[#]] == 1 &] (* Jean-François Alcover, Apr 04 2011 *)
Select[Range@175, FiniteGroupCount@# == 1 &] (* Robert G. Wilson v, Feb 16 2017 *)
Select[Range[200],CoprimeQ[#,EulerPhi[#]]&] (* Harvey P. Dale, Apr 10 2022 *)

isA003277(n) = gcd(n,eulerphi(n))==1 \\ Michael B. Porter, Feb 21 2010

# Compare A050384.
def isPrimeTo(n, m): return gcd(n, m) == 1
def isCyclic(n): return isPrimeTo(n, euler_phi(n))
[n for n in (1..173) if isCyclic(n)] # Peter Luschny, Nov 14 2018

n = p_1*p_2*...*p_k (for some k >= 0), where the p_i are distinct primes and no p_j-1 is divisible by any p_i.
A000001(a(n)) = 1.
Erdős proved that a(n) ~ e^gamma n log log log n, where e^gamma is A073004. - Charles R Greathouse IV, Nov 23 2011
A000005(a(n)) = 2^k. - Carlos Eduardo Olivieri, Jul 07 2015
A008966(a(n)) = 1. - Bernard Schott, Mar 24 2022

More terms from Christian G. Bower

A030229 Numbers that are the product of an even number of distinct primes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202, 203, 205, 206, 209, 210, 213, 214

Offset: 1

David W. Wilson

These are the positive integers k with moebius(k) = 1 (cf. A008683). - N. J. A. Sloane, May 18 2021
From Enrique Pérez Herrero, Jul 06 2012: (Start)
This sequence and A030059 form a partition of the squarefree numbers set: A005117.
Also solutions to equation mu(n)=1.
Sum_{n>=1} 1/a(n)^s = (Zeta(s)^2 + Zeta(2*s))/(2*Zeta(s)*Zeta(2*s)).
(End)
A008683(a(n)) = 1; a(A220969(n)) mod 2 = 0; a(A220968(n)) mod 2 = 1. - Reinhard Zumkeller, Dec 27 2012
Characteristic function for values of a(n) = (mu(n)+1)! - 1, where mu(n) is the Mobius function (A008683). - Wesley Ivan Hurt, Oct 11 2013
Conjecture: For the matrix M(i,j) = 1 if j|i and 0 otherwise, Inverse(M)(a,1) = -1, for any a in this sequence. - Benedict W. J. Irwin, Jul 26 2016
Solutions to the equation Sum_{d|n} mu(d)*d = Sum_{d|n} mu(n/d)*d. - Torlach Rush, Jan 13 2018
Solutions to the equation Sum_{d|n} mu(d)*sigma(d) = n, where sigma(n) is the sum of divisors function (A000203). - Robert D. Rosales, May 20 2024
From Peter Munn, Oct 04 2019: (Start)
Numbers n such that omega(n) = bigomega(n) = 2*k for some integer k.
The squarefree numbers in A000379.
The squarefree numbers in A028260.
This sequence is closed with respect to the commutative binary operation A059897(.,.), thus it forms a subgroup of the positive integers under A059897(.,.). A006094 lists a minimal set of generators for this subgroup. The lexicographically earliest ordered minimal set of generators is A100484 with its initial 4 removed.
(End)
The asymptotic density of this sequence is 3/Pi^2 (cf. A104141). - Amiram Eldar, May 22 2020

			(empty product), 2*3, 2*5, 2*7, 3*5, 3*7, 2*11, 2*13, 3*11, 2*17, 5*7, 2*19, 3*13, 2*23,...

B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary, see p. 23; AMS Providence RI 1995
S. Ramanujan, Collected Papers, pp. xxiv, 21.

T. D. Noe, Table of n, a(n) for n = 1..1000
Debmalya Basak, Nicolas Robles, and Alexandru Zaharescu, Exponential sums over Möbius convolutions with applications to partitions, arXiv:2312.17435 [math.NT], 2023. Mentions this sequence.
S. Ramanujan, Irregular numbers, J. Indian Math. Soc. 5 (1913) 105-106.
Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Moebius Function
Eric Weisstein's World of Mathematics, Prime Sums
H. S. Wilf, A Greeting; and a view of Riemann's Hypothesis, Amer. Math. Monthly, 94:1 (1987), 3-6.

A006881, A046386, A067885, A123322, A281222 are subsequences.

Cf. A000379, A005117, A008683, A028260, A030059, A104141, A151797, A245630.

import Data.List (elemIndices)
a030229 n = a030229_list !! (n-1)
a030229_list = map (+ 1) $ elemIndices 1 a008683_list
-- Reinhard Zumkeller, Dec 27 2012

a := n -> `if`(numtheory[mobius](n)=1,n,NULL); seq(a(i),i=1..214); # Peter Luschny, May 04 2009
with(numtheory); t := [ ]: f := [ ]: for n from 1 to 250 do if mobius(n) = 1 then t := [ op(t), n ] else f := [ op(f), n ]; fi; od: t; # Wesley Ivan Hurt, Oct 11 2013
# alternative
A030229 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if numtheory[mobius](a) = 1 then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A030229(n),n=1..40) ; # R. J. Mathar, Sep 22 2020

Select[Range[214], MoebiusMu[#] == 1 &] (* Jean-François Alcover, Oct 04 2011 *)

isA030229(n)= #(n=factor(n)[,2]) % 2 == 0 && (!n || vecmax(n)==1 )

is(n)=moebius(n)==1 \\ Charles R Greathouse IV, Jan 31 2017
for(n=1,500, isA030229(n)&print1(n",")) \\ M. F. Hasler

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A030229(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n-1+x-sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,x.bit_length(),2)))
    kmin, kmax = 0,1
    while f(kmax) > kmax:
        kmax <<= 1
    while kmax-kmin > 1:
        kmid = kmax+kmin>>1
        if f(kmid) <= kmid:
            kmax = kmid
        else:
            kmin = kmid
    return kmax # Chai Wah Wu, Aug 29 2024

a(n) < n*Pi^2/3 infinitely often; a(n) > n*Pi^2/3 infinitely often. - Charles R Greathouse IV, Oct 04 2011; corrected Sep 07 2017

{a(n)} = {m : m = A059897(A030059(k), p), k >= 1} for prime p, where {a(n)} denotes the set of integers in the sequence. - Peter Munn, Oct 04 2019

A348717 a(n) is the least k such that A003961^i(k) = n for some i >= 0 (where A003961^i denotes the i-th iterate of A003961).

Original entry on oeis.org

1, 2, 2, 4, 2, 6, 2, 8, 4, 10, 2, 12, 2, 14, 6, 16, 2, 18, 2, 20, 10, 22, 2, 24, 4, 26, 8, 28, 2, 30, 2, 32, 14, 34, 6, 36, 2, 38, 22, 40, 2, 42, 2, 44, 12, 46, 2, 48, 4, 50, 26, 52, 2, 54, 10, 56, 34, 58, 2, 60, 2, 62, 20, 64, 14, 66, 2, 68, 38, 70, 2, 72, 2

Offset: 1

Rémy Sigrist, Oct 31 2021

nonn

All terms except a(1) = 1 are even.
To compute a(n) for n > 1:
- if n = Product_{j = 1..o} prime(p_j)^e_j (where prime(i) denotes the i-th prime number, p_1 < ... < p_o and e_1 > 0)
- then a(n) = Product_{j = 1..o} prime(p_j + 1 - p_1)^e_j.
This sequence has similarities with A304776: here we shift down prime indexes, there prime exponents.
Smallest number generated by uniformly decrementing the indices of the prime factors of n. Thus, for n > 1, the smallest m > 1 such that the first differences of the indices of the ordered prime factors (including repetitions) are the same for m and n. As a function, a(.) preserves properties such as prime signature. - Peter Munn, May 12 2022

Positions of particular values (see formula section): A000040, A001248, A006094, A030078, A030514, A046301, A050997, A090076, A090090, A166329, A251720.
Also see formula section for the relationship to: A000265, A003961, A004277, A005940, A020639, A046523, A055396, A071364, A122111, A156552, A243055, A243074, A297845, A322993.
Sequences with comparable definitions: A304776, A316437.
Cf. A246277 (terms halved), A305897 (restricted growth sequence transform), A354185 (Möbius transform), A354186 (Dirichlet inverse), A354187 (sum with it).

a[1] = 1; a[n_] := Module[{f = FactorInteger[n], d}, d = PrimePi[f[[1, 1]]] - 1; Times @@ ((Prime[PrimePi[#[[1]]] - d]^#[[2]]) & /@ f)]; Array[a, 100] (* Amiram Eldar, Oct 31 2021 *)

a(n) = { my (f=factor(n)); if (#f~>0, my (pi1=primepi(f[1,1])); for (k=1, #f~, f[k,1] = prime(primepi(f[k,1])-pi1+1))); factorback(f) }

a(n) = n iff n belongs to A004277.
A003961^(A055396(n)-1)(a(n)) = n for any n > 1.
a(n) = 2 iff n belongs to A000040 (prime numbers).
a(n) = 4 iff n belongs to A001248 (squares of prime numbers).
a(n) = 6 iff n belongs to A006094 (products of two successive prime numbers).
a(n) = 8 iff n belongs to A030078 (cubes of prime numbers).
a(n) = 10 iff n belongs to A090076.
a(n) = 12 iff n belongs to A251720.
a(n) = 14 iff n belongs to A090090.
a(n) = 16 iff n belongs to A030514.
a(n) = 30 iff n belongs to A046301.
a(n) = 32 iff n belongs to A050997.
a(n) = 36 iff n belongs to A166329.
a(1) = 1, for n > 1, a(n) = 2*A246277(n). - Antti Karttunen, Feb 23 2022
a(n) = A122111(A243074(A122111(n))). - Peter Munn, Feb 23 2022
From Peter Munn and Antti Karttunen, May 12 2022: (Start)
a(1) = 1; a(2n) = 2n; a(A003961(n)) = a(n). [complete definition]
a(n) = A005940(1+A322993(n)) = A005940(1+A000265(A156552(n))).
Equivalently, A156552(a(n)) = A000265(A156552(n)).
A297845(a(n), A020639(n)) = n.
A046523(a(n)) = A046523(n).
A071364(a(n)) = A071364(n).
a(n) >= A071364(n).
A243055(a(n)) = A243055(n).
(End)

A325698 Numbers with as many even as odd prime indices, counted with multiplicity.

Original entry on oeis.org

1, 6, 14, 15, 26, 33, 35, 36, 38, 51, 58, 65, 69, 74, 77, 84, 86, 90, 93, 95, 106, 119, 122, 123, 141, 142, 143, 145, 156, 158, 161, 177, 178, 185, 196, 198, 201, 202, 209, 210, 214, 215, 216, 217, 219, 221, 225, 226, 228, 249, 262, 265, 278, 287, 291, 299

Offset: 1

Gus Wiseman, May 17 2019

nonn

These are Heinz numbers of the integer partitions counted by A045931.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The integers in the multiplicative subgroup of positive rational numbers generated by the products of two consecutive primes (A006094). The sequence is closed under multiplication, prime shift (A003961), and - where the result is an integer - under division. Using these closures, all the terms can be derived from the presence of 6. For example, A003961(6) = 15, A003961(15) = 35, 6 * 35 = 210, 210/15 = 14. Closed also under A297845, since A297845 can be defined using squaring, prime shift and multiplication. - Peter Munn, Oct 05 2020

			The sequence of terms together with their prime indices begins:
    1: {}
    6: {1,2}
   14: {1,4}
   15: {2,3}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   51: {2,7}
   58: {1,10}
   65: {3,6}
   69: {2,9}
   74: {1,12}
   77: {4,5}
   84: {1,1,2,4}
   86: {1,14}
   90: {1,2,2,3}
   93: {2,11}
   95: {3,8}

David A. Corneth, Table of n, a(n) for n = 1..10000

Positions of 0's in A195017.
A257992(n) = A257991(n).
Cf. A000712, A001222, A001405, A006094, A026010, A045931, A063886, A097613, A112798, A130780, A171966, A239241, A241638, A325700.
Closed under: A003961, A003991, A297845.
Subsequence of A028260, A332820.

Select[Range[100],Total[Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>k*(-1)^PrimePi[p]]]==0&]

is(n) = {my(v = vector(2), f = factor(n));for(i = 1, #f~,v[1 + primepi(f[i, 1])%2]+=f[i, 2]);v[1] == v[2]} \\ David A. Corneth, Oct 06 2020

from sympy import factorint, primepi
def ok(n):
    v = [0, 0]
    for p, e in factorint(n).items(): v[primepi(p)%2] += e
    return v[0] == v[1]
print([k for k in range(300) if ok(k)]) # Michael S. Branicky, Apr 16 2022 after David A. Corneth

A319630 Positive numbers that are not divisible by two consecutive prime numbers.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 76, 79, 80, 81, 82, 83, 85, 86, 87

Offset: 1

Rémy Sigrist, Sep 25 2018

nonn

This sequence is the complement of A104210.
Equivalently, this sequence corresponds to the positive numbers k such that:
- A300820(k) <= 1,
- A087207(k) is a Fibbinary number (A003714).
For any n > 0 and k >= 0, a(n)^k belongs to the sequence.
The numbers of terms not exceeding 10^k, for k=1,2,..., are 9, 78, 758, 7544, 75368, 753586, 7535728, 75356719, 753566574, ... Apparently, the asymptotic density of this sequence is 0.75356... - Amiram Eldar, Apr 10 2021
Numbers not divisible by any term of A006094. - Antti Karttunen, Jul 29 2022

			The number 10 is only divisible by 2 and 5, hence 10 appears in the sequence.
The number 42 is divisible by 2 and 3, hence 42 does not appear in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003714, A006094, A087207, A104210, A300820, A356171 (odd terms only).

Positions of 1's in A322361 and in A356173 (characteristic function).

N:= 1000: # for terms <= N
R:= {}:
p:= 2:
do
  q:= p; p:= nextprime(p);
  if p*q > N then break fi;
  R:= R union {seq(i,i=p*q..N,p*q)}
od:
sort(convert({$1..N} minus R,list)); # Robert Israel, Apr 13 2020

q[n_] := SequenceCount[FactorInteger[n][[;; , 1]], {p1_, p2_} /; p2 == NextPrime[p1]] ==  0; Select[Range[100], q] (* Amiram Eldar, Apr 10 2021 *)

is(n) = my (f=factor(n)); for (i=1, #f~-1, if (nextprime(f[i,1]+1)==f[i+1,1], return (0))); return (1)

A300820(a(n)) <= 1.

cp's OEIS Frontend

A340537 Primes that are sums of a sequence of consecutive terms of A006094.

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A073491 Numbers having no prime gaps in their factorization.

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A078898 Number of times the smallest prime factor of n is the smallest prime factor for numbers <= n; a(0)=0, a(1)=1.

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A246277 Column index of n in A246278: a(1) = 0, a(2n) = n, a(2n+1) = a(A064989(2n+1)).

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A246278 Prime shift array: Square array read by antidiagonals: A(1,col) = 2*col, and for row > 1, A(row,col) = A003961(A(row-1,col)).

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A003277 Cyclic numbers: k such that k and phi(k) are relatively prime; also k such that there is just one group of order k, i.e., A000001(k) = 1.

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