A084969 -id:A084969 - cp's OEIS frontend

A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

2, 4, 3, 6, 9, 5, 8, 15, 25, 7, 10, 21, 35, 49, 11, 12, 27, 55, 77, 121, 13, 14, 33, 65, 91, 143, 169, 17, 16, 39, 85, 119, 187, 221, 289, 19, 18, 45, 95, 133, 209, 247, 323, 361, 23, 20, 51, 115, 161, 253, 299, 391, 437, 529, 29, 22, 57, 125, 203, 319, 377, 493, 551, 667

Offset: 2

Yasutoshi Kohmoto, Jun 05 2003

This is permutation of natural numbers larger than 1.
From Antti Karttunen, Dec 19 2014: (Start)
If we assume here that a(1) = 1 (but which is not explicitly included because outside of the array), then A252460 gives an inverse permutation. See also A249741.
For navigating in this array:
A055396(n) gives the row number of row where n occurs, and A078898(n) gives its column number, both starting their indexing from 1.
A250469(n) gives the number immediately below n, and when n is an odd number >= 3, A250470(n) gives the number immediately above n. If n is a composite, A249744(n) gives the number immediately left of n.
First cube of each row, which is {the initial prime of the row}^3 and also the first number neither a prime or semiprime, occurs on row n at position A250474(n).
(End)
The n-th row contains the numbers whose least prime factor is the n-th prime: A020639(T(n,k)) = A000040(n). - Franklin T. Adams-Watters, Aug 07 2015

			The top left corner of the array:
   2,   4,   6,    8,   10,   12,   14,   16,   18,   20,   22,   24,   26
   3,   9,  15,   21,   27,   33,   39,   45,   51,   57,   63,   69,   75
   5,  25,  35,   55,   65,   85,   95,  115,  125,  145,  155,  175,  185
   7,  49,  77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329
  11, 121, 143,  187,  209,  253,  319,  341,  407,  451,  473,  517,  583
  13, 169, 221,  247,  299,  377,  403,  481,  533,  559,  611,  689,  767
  17, 289, 323,  391,  493,  527,  629,  697,  731,  799,  901, 1003, 1037
  19, 361, 437,  551,  589,  703,  779,  817,  893, 1007, 1121, 1159, 1273
  23, 529, 667,  713,  851,  943,  989, 1081, 1219, 1357, 1403, 1541, 1633
  29, 841, 899, 1073, 1189, 1247, 1363, 1537, 1711, 1769, 1943, 2059, 2117
  ...

Transpose of A083140.
One more than A249741.
Inverse permutation: A252460.
Column 1: A000040, Column 2: A001248.
Row 1: A005843, Row 2: A016945, Row 3: A084967, Row 4: A084968, Row 5: A084969, Row 6: A084970.
Main diagonal: A083141.
First semiprime in each column occurs at A251717; A251718 & A251719 with additional criteria. A251724 gives the corresponding semiprimes for the latter. See also A251728.
Permutations based on mapping numbers between this array and A246278: A249817, A249818, A250244, A250245, A250247, A250249. See also: A249811, A249814, A249815.
Also used in the definition of the following arrays of permutations: A249821, A251721, A251722.
Cf. A002260, A004736, A004280, A020639, A038179, A055396, A078898, A138511, A249820, A249730, A249735, A249744, A250469, A250470, A250472, A250474.

lim = 11; a = Table[Take[Prime[n] Select[Range[lim^2], GCD[# Prime@ n, Product[Prime@ i, {i, 1, n - 1}]] == 1 &], lim], {n, lim}]; Flatten[Table[a[[i, n - i + 1]], {n, lim}, {i, n}]] (* Michael De Vlieger, Jan 04 2016, after Yasutoshi Kohmoto at A083140 *)

More terms from Hugo Pfoertner, Jun 13 2003

A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

Original entry on oeis.org

2, 3, 4, 5, 9, 6, 7, 25, 15, 8, 11, 49, 35, 21, 10, 13, 121, 77, 55, 27, 12, 17, 169, 143, 91, 65, 33, 14, 19, 289, 221, 187, 119, 85, 39, 16, 23, 361, 323, 247, 209, 133, 95, 45, 18, 29, 529, 437, 391, 299, 253, 161, 115, 51, 20, 31, 841, 667, 551, 493, 377, 319, 203, 125, 57, 22

Offset: 2

Yasutoshi Kohmoto, Jun 05 2003

A permutation of natural numbers >= 2.
The proportion of integers in the n-th row of the array is given by A005867(n-1)/A002110(n) = A038110(n)/A038111(n). - Peter Kagey, Jun 03 2019, based on comments by Jamie Morken and discussion with Tom Hanlon.
The proportion of the integers after the n-th row of the array is given by A005867(n)/A002110(n). - Tom Hanlon, Jun 08 2019

			Array begins:
   2   4   6   8  10  12  14  16  18  20  22  24 .... (A005843 \ {0})
   3   9  15  21  27  33  39  45  51  57  63  69 .... (A016945)
   5  25  35  55  65  85  95 115 125 145 155 175 .... (A084967)
   7  49  77  91 119 133 161 203 217 259 287 301 .... (A084968)
  11 121 143 187 209 253 319 341 407 451 473 517 .... (A084969)
  13 169 221 247 299 377 403 481 533 559 611 689 .... (A084970)

Cf. A083141 (main diagonal), A083221 (transpose), A004280, A038179, A084967, A084968, A084969, A084970, A084971.
Arrays of integers grouped into rows by various criteria:
by greatest prime factor: A125624,
by lowest prime factor: this sequence (upward antidiagonals), A083221 (downward antidiagonals),
by number of distinct prime factors: A125666,
by number of prime factors counted with multiplicity: A078840,
by prime signature: A095904,
by ordered prime signature: A096153,
by number of divisors: A119586,
by number of 1's in binary expansion: A066884 (upward), A067576 (downward),
by distance to next prime: A192179.
Cf. A002110, A005867, A038110, A038111.

a = Join[ {Table[2n, {n, 1, 12}]}, Table[ Take[ Prime[n]*Select[ Range[100], GCD[ Prime[n] #, Product[ Prime[i], {i, 1, n - 1}]] == 1 &], 12], {n, 2, 12}]]; Flatten[ Table[ a[[i, n - i]], {n, 2, 12}, {i, n - 1, 1, -1}]]
(* second program: *)
rows = 12; Clear[T]; Do[For[m = p = Prime[n]; k = 1, k <= rows, m += p, If[ FactorInteger[m][[1, 1]] == p, T[n, k++] = m]], {n, rows}]; Table[T[n - k + 1, k], {n, rows}, {k, n}] // Flatten (* Jean-François Alcover, Mar 08 2016 *)

More terms from Hugo Pfoertner and Robert G. Wilson v, Jun 13 2003

A065642 a(1) = 1; for n > 1, a(n) = Min {m > n | m has same prime factors as n ignoring multiplicity}.

Original entry on oeis.org

1, 4, 9, 8, 25, 12, 49, 16, 27, 20, 121, 18, 169, 28, 45, 32, 289, 24, 361, 40, 63, 44, 529, 36, 125, 52, 81, 56, 841, 60, 961, 64, 99, 68, 175, 48, 1369, 76, 117, 50, 1681, 84, 1849, 88, 75, 92, 2209, 54, 343, 80, 153, 104, 2809, 72, 275, 98, 171, 116, 3481, 90, 3721

Offset: 1

Reinhard Zumkeller, Dec 03 2001

After the initial 1, a permutation of the nonsquarefree numbers A013929. The array A284457 is obtained as a dispersion of this sequence. - Antti Karttunen, Apr 17 2017

Numbers such that a(n)/n is not an integer are listed in A284342.

			a(10) = a(2 * 5) = 2 * 2 * 5 = 20; a(12) = a(2^2 * 3) = 2 * 3^2 = 18.

Reinhard Zumkeller (terms 1..1000) & Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A005117, A007947, A013929, A020639, A000040, A001221, A081382.
Cf. A285328 (a left inverse).
Cf. also arrays A284457 & A284311, A285321 and permutations A284572, A285112, A285332.
Cf. A084968, A084969, A084970, A284342.

a065642 1 = 1
a065642 n = head [x | let rad = a007947 n, x <- [n+1..], a007947 x == rad]
-- Reinhard Zumkeller, Jun 12 2015, Jul 27 2011

ffi[x_]:= Flatten[FactorInteger[x]]; lf[x_]:= Length[FactorInteger[x]]; ba[x_]:= Table[Part[ffi[x], 2*w-1], {w, 1, lf[x]}]; cor[x_]:= Apply[Times, ba[x]]; Join[{1}, Table[Min[Flatten[Position[Table[cor[w], {w, n+1, n^2}]-cor[n], 0]]+n], {n, 2, 100}]] (* This code is suitable since prime factor set is invariant iff squarefree kernel is invariant. *) (* G. C. Greubel, Oct 31 2018 *)
Array[If[# == 1, 1, Function[{n, c}, SelectFirst[Range[n + 1, n^2], Times @@ FactorInteger[#][[All, 1]] == c &]] @@ {#, Times @@ FactorInteger[#][[All, 1]]}] &, 61] (* Michael De Vlieger, Oct 31 2018 *)

A065642(n)={ my(r=A007947(n)); if(1==n,n, n += r; while(A007947(n) <> r, n += r); n)} \\ Antti Karttunen, Apr 17 2017

a(n)=if(n<2, return(1)); my(f=factor(n),r,mx,mn,t); if(#f~==1, return(f[1,1]^(f[1,2]+1))); f=f[,1]; r=factorback(f); mn=mx=n*f[1]; forvec(v=vector(#f,i,[1,logint(mx/r,f[i])+1]), t=prod(i=1,#f, f[i]^v[i]); if(tn, mn=t)); mn \\ Charles R Greathouse IV, Oct 18 2017

from sympy import primefactors, prod
def a007947(n): return 1 if n < 2 else prod(primefactors(n))
def a(n):
    if n==1: return 1
    r=a007947(n)
    n += r
    while a007947(n)!=r:
        n+=r
    return n
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 17 2017

(define (A065642 n) (if (= 1 n) n (let ((k (A007947 n))) (let loop ((n (+ n k))) (if (= (A007947 n) k) n (loop (+ n k))))))) ;; (Semi-naive implementation) - Antti Karttunen, Apr 17 2017

A007947(a(n)) = A007947(n); a(A007947(n)) = A007947(n) * A020639(n), where A007947 is the squarefree kernel (radical), A020639 is the least prime factor (lpf).
a(A000040(n)^k) = A000040(n)^(k+1); A001221(a(n)) = A001221(n).
A285328(a(n)) = n. - Antti Karttunen, Apr 17 2017
n < a(n) <= n*lpf(n) <= n^2. - Charles R Greathouse IV, Oct 18 2017

A084968 Multiples of 7 coprime to 30.

Original entry on oeis.org

7, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 343, 371, 413, 427, 469, 497, 511, 539, 553, 581, 623, 637, 679, 707, 721, 749, 763, 791, 833, 847, 889, 917, 931, 959, 973, 1001, 1043, 1057, 1099, 1127, 1141, 1169, 1183, 1211, 1253, 1267, 1309

Offset: 1

Robert G. Wilson v, Jun 15 2003

Numbers 7*k such that gcd(k,30) = 1.

Numbers congruent to 7, 49, 77, 91, 119, 133, 161, 203 modulo 210. - Jianing Song, Nov 18 2022

			77 is in the sequence because gcd(77, 30) = 1.
84 is not in the sequence because gcd(84, 3) = 6.
91 is in the sequence because gcd(91, 30) = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,1,-1).

Subsequence of A008589.
Fourth row of A083140.
Cf. A084967 (5), A084969 (11), A084970 (13), A332799 (17), A332798 (19), A332797 (23), A007775 (7-rough numbers).

q:= k-> igcd(k, 30)=1:
select(q, [7*i$i=1..300])[];  # Alois P. Heinz, Feb 25 2020

7Select[ Range[190], GCD[ #, 2*3*5] == 1 & ]

is(n)=gcd(210,n)==7 \\ Charles R Greathouse IV, Aug 05 2013

G.f.: 7*x*(x^8 + 6*x^7 + 4*x^6 + 2*x^5 + 4*x^4 + 2*x^3 + 4*x^2 + 6*x + 1) / ((x-1)^2*(x+1)*(x^2+1)*(x^4+1)). - Colin Barker, Feb 24 2013
Lim_{n->oo} a(n)/n = A038111(4)/A038110(4) = 105/4. - Vladimir Shevelev, Jan 20 2015
a(n) = 7*A007775(n).
a(n+8) = a(n) + 210. - Jianing Song, Nov 18 2022
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(23 + sqrt(5) - sqrt(6*(5 + sqrt(5))))*Pi/105. - Amiram Eldar, Jul 15 2023

A084970 Numbers whose smallest prime factor is 13.

Original entry on oeis.org

13, 169, 221, 247, 299, 377, 403, 481, 533, 559, 611, 689, 767, 793, 871, 923, 949, 1027, 1079, 1157, 1261, 1313, 1339, 1391, 1417, 1469, 1651, 1703, 1781, 1807, 1937, 1963, 2041, 2119, 2171, 2197, 2249, 2327, 2353, 2483, 2509, 2561, 2587, 2743, 2873

Offset: 1

Robert G. Wilson v, Jun 15 2003

			a(2) = 13*13, a(3) = 13*17.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, order 481.

Sixth row of A083140.

Cf. A084967 (5), A084968 (7), A084969 (11), A332799 (17), A332798 (19), A332797 (23), A008365 (13-rough numbers).

Select[ 13Range@ 225, GCD[#, 2310] == 1 &] (* Robert G. Wilson v, Dec 18 2014 *)

is(n)=n%13==0 && gcd(n,2310)==1 \\ Charles R Greathouse IV, Nov 19 2014

a(n) = a(n-480) + 30030 = a(n-1) + a(n-480) - a(n-481). - Charles R Greathouse IV, Nov 19 2014
Lim_{n->infinity} a(n)/n = A038111(6)/A038110(6) = 1001/16 = 62.5625. - Vladimir Shevelev, Jan 20 2015
a(n) = 13*A008365(n).

More terms from David Wasserman, Oct 19 2004

A332797 Numbers whose smallest prime factor is 23.

Original entry on oeis.org

23, 529, 667, 713, 851, 943, 989, 1081, 1219, 1357, 1403, 1541, 1633, 1679, 1817, 1909, 2047, 2231, 2323, 2369, 2461, 2507, 2599, 2921, 3013, 3151, 3197, 3427, 3473, 3611, 3749, 3841, 3979, 4117, 4163, 4393, 4439, 4531, 4577, 4853, 5129, 5221, 5267, 5359, 5497

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 55296/7436429. - Amiram Eldar, Dec 06 2020

			a(2) = 23*23, a(3) = 23*29.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

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

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332799 (17), A332798 (19), A166063 (23-rough numbers).

23 * Select[Range[240], CoprimeQ[#, 9699690] &] (* Amiram Eldar, Feb 24 2020 *)
Select[Range[6000],FactorInteger[#][[1,1]]==23&] (* Harvey P. Dale, Aug 29 2025 *)

P = 23         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 23*A166063(n).

A332798 Numbers whose smallest prime factor is 19.

Original entry on oeis.org

19, 361, 437, 551, 589, 703, 779, 817, 893, 1007, 1121, 1159, 1273, 1349, 1387, 1501, 1577, 1691, 1843, 1919, 1957, 2033, 2071, 2147, 2413, 2489, 2603, 2641, 2831, 2869, 2983, 3097, 3173, 3287, 3401, 3439, 3629, 3667, 3743, 3781, 4009, 4237, 4313, 4351, 4427

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 3072/323323. - Amiram Eldar, Dec 06 2020

			a(2) = 19*19, a(3) = 19*23.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

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

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332799 (17), A332797 (23), A166061 (19-rough numbers).

19 * Select[Range[235], CoprimeQ[#, 510510] &] (* Amiram Eldar, Feb 24 2020 *)

P = 19         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 19*A166061(n).

A332799 Numbers whose smallest prime factor is 17.

Original entry on oeis.org

17, 289, 323, 391, 493, 527, 629, 697, 731, 799, 901, 1003, 1037, 1139, 1207, 1241, 1343, 1411, 1513, 1649, 1717, 1751, 1819, 1853, 1921, 2159, 2227, 2329, 2363, 2533, 2567, 2669, 2771, 2839, 2941, 3043, 3077, 3247, 3281, 3349, 3383, 3587, 3791, 3859, 3893

Offset: 1

Frank Ellermann, Feb 24 2020

The asymptotic density of this sequence is 192/17017. - Amiram Eldar, Dec 06 2020

			a(2) = 17*17, a(3) = 17*19.

Emmanuel Desurvire, Classical and Quantum Information Theory: An Introduction for the Telecom Scientist, Cambridge University Press, 2009, table 20.5 p. 421.

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

Cf. A084967 (5), A084968 (7), A084969 (11), A084970 (13), A332798 (19), A332797 (23), A008366 (17-rough numbers).

17 * Select[Range[230], CoprimeQ[#, 30030] &] (* Amiram Eldar, Feb 24 2020 *)

P = 17         ;  S = P
do N = P by 2 while length( S ) < 255
   do I = 1 until P = X
      X = PRIME( I )
      if P = X       then  leave I
      if N // X = 0  then  iterate N
   end I
   S = S || ',' P*N
end N
say S          ;  return S

a(n) = 17*A008366(n).

A255415 Row 5 of Ludic array A255127.

Original entry on oeis.org

11, 55, 103, 151, 203, 251, 299, 343, 391, 443, 491, 539, 587, 631, 683, 731, 779, 827, 877, 923, 971, 1019, 1067, 1117, 1165, 1211, 1259, 1307, 1357, 1405, 1453, 1499, 1547, 1597, 1645, 1693, 1741, 1787, 1837, 1885, 1933, 1981, 2033, 2077, 2125, 2173, 2221, 2273, 2321, 2365, 2413, 2461, 2513, 2561, 2609, 2653, 2701, 2753, 2801, 2849, 2897, 2941, 2993, 3041

Offset: 1

Antti Karttunen, Feb 22 2015

nonn

First differences are periodic with period length 48, cf. formulas. - M. F. Hasler, Nov 17 2024

Antti Karttunen, Table of n, a(n) for n = 1..10001

Row 5 of A255127. See A255414 for row 4 and A255416 for row 6.

Cf. A084969, A255407.

L255415=[n*337\14*2+7|n<-[0..47]]+digits(54129937554927109457534, 3)*2
apply( A255415(n)=n--\48*2310+L255415[n%48+1], [1..66]) \\ M. F. Hasler, Nov 10 2024

(define (A255415 n) (A255127bi 5 n)) ;; Code for A255127bi given in A255127.

a(n) = A255407(A084969(n)).

a(n) = a(n-48) + 2310 = a((n-1)%48 + 1) + [(n-1)/48]*2310, where % = mod = remainder operator, and [.] = floor. - M. F. Hasler, Nov 10 2024

A008367 Composite but smallest prime factor >= 17.

Original entry on oeis.org

289, 323, 361, 391, 437, 493, 527, 529, 551, 589, 629, 667, 697, 703, 713, 731, 779, 799, 817, 841, 851, 893, 899, 901, 943, 961, 989, 1003, 1007, 1037, 1073, 1081, 1121, 1139, 1147, 1159, 1189, 1207, 1219, 1241, 1247, 1271, 1273, 1333, 1343, 1349, 1357, 1363, 1369, 1387, 1403, 1411, 1457

Offset: 1

N. J. A. Sloane

nonn

Composite numbers k such that k^720 mod 30030 = 1. - Gary Detlefs, May 02 2012

The asymptotic density of this sequence is 192/1001. - Amiram Eldar, Mar 22 2021

T. D. Noe, Table of n, a(n) for n = 1..10000

Intersection of A002808 and A008366.
Cf. A038511, A084969, A084970.
Cf. A287391.

Filtered([17..1500],n->PowerMod(n,720,30030)=1 and not IsPrime(n)); # Muniru A Asiru, Nov 24 2018

for i from 1 to 2000 do if gcd(i,30030) = 1 and not isprime(i) then print(i); fi; od;

Select[ Range[ 1500 ], (GCD[ #1, 30030 ]==1&&!PrimeQ[ #1 ])& ]
Select[Range[2000], ! PrimeQ[#] && FactorInteger[#][[1, 1]] >= 17 &] (* T. D. Noe, Mar 16 2013 *)

is(n)={gcd(n,30030)==1 && !ispseudoprime(n)} \\ M. F. Hasler, Oct 04 2018

For 1 <= n < 107, a(n) = A287391(n+2); then a(107) = 2329, a(108) = 2363 are not in A287391, but again a(n) = A287391(n) for 108 < n < 120. - M. F. Hasler, Oct 04 2018

cp's OEIS Frontend

A083221 Sieve of Eratosthenes arranged as an array and read by antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), ...

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A083140 Sieve of Eratosthenes arranged as an array and read by antidiagonals in the up direction; n-th row has property that smallest prime factor is prime(n).

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A065642 a(1) = 1; for n > 1, a(n) = Min {m > n | m has same prime factors as n ignoring multiplicity}.

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A084968 Multiples of 7 coprime to 30.

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A084970 Numbers whose smallest prime factor is 13.

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A332797 Numbers whose smallest prime factor is 23.

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A332798 Numbers whose smallest prime factor is 19.

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