A112774 -id:A112774 - cp's OEIS frontend

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (ink - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

3, 5, 2, 9, 3, 2, 13, 5, 3, 2, 21, 7, 4, 3, 2, 25, 11, 5, 4, 3, 2, 33, 13, 7, 5, 4, 3, 2, 37, 17, 9, 6, 5, 4, 3, 2, 45, 19, 10, 7, 6, 5, 4, 3, 2, 57, 23, 13, 9, 7, 6, 5, 4, 3, 2, 61, 29, 15, 11, 8, 7, 6, 5, 4, 3, 2, 73, 31, 17, 12, 9, 8, 7, 6, 5, 4, 3, 2

Offset: 1

David Lovler, Oct 15 2019

All the U(i;n,k) mimic the ordinary multiplication table in that they are commutative, associative, have identity element 1 and have 0. However (except when i=2) they are partially distributive, meaning that distributivity works except if an even number is partitioned into a sum of two odd numbers. Only when i=2, the odd-even-dependent A319929 term disappears and normal distributivity holds.
U(0;n,k) = A319929(n,k);
U(1;n,k) = A322630(n,k);
U(2;n,k) = n*k;
U(3;n,k) = A322744(n,k);
U(4;n,k) = A327259(n,k);
U(i;n,k) = 2i*floor(n/2)*floor(k/2) + A319929(n,k).
Row 1 is 2p-1 where p is a prime number (A076274 without 1).
Row 2 is the prime numbers.
Row 3 is A307002.
Row 4 is A327261.
The i-th row of T(n,k) consists of numbers that sieve out of the array U(i;n,k) = (i*n*k - (i-2)*A319929(n,k))/2, in numerical order.
From David Lovler, Sep 02 2020: (Start)
Row 1 has no even numbers. Row 2 has one even number. Generally, the even numbers of the i-th row start with i-1 consecutive even numbers (from 2). This is because U(i;2,2) = 2*i gives the first even number not in row i.
Row 3 seems to have even numbers that, after 2, coincide with A112774 which has an infinite number of terms. For i > 3, as i increases, row i has a denser presence of even numbers, thus each row has an infinite number of even terms.
Generalization of the twin prime conjecture: Since row 2 is the prime numbers, we can observe the twin prime conjecture that after the first three odd primes, the sprinkling of pairs of consecutive prime numbers never ends. Concerning just odd terms, a similar conjecture can be stated for rows i >= 3. Row 3 starts with four odd numbers then the sprinkling of three consecutive odd number never ends. Row 4 starts with five odd numbers then the sprinkling of four consecutive odd numbers never ends. The pattern continues as row i starts with i+1 odd numbers then the sprinkling of i consecutive odd numbers never ends. We can take this back to row 1 which starts with two odd numbers then continues with isolated odd numbers.
Studying the even terms, there is an analog to the above generalization of the twin prime conjecture. Row 3 starts with two even numbers then continues with isolated even numbers. Row 4 starts with three even numbers then the sprinkling of pairs of consecutive even numbers never ends. Row 5 starts with four even numbers then the sprinkling of three consecutive even numbers never ends. The pattern continues as row i starts with i-1 even numbers then the sprinkling of i-2 consecutive even numbers never ends.
(End)

			3  5  9  13  21  25  33  37  45  57  61  73  81  85  93 105 117 121 133 141 145 ...
2  3  5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73 ...
2  3  4   5   7   9  10  13  15  17  21  22  23  25  29  31  34  37  39  41  45 ...
2  3  4   5   6   7   9  11  12  14  15  17  19  21  22  25  27  28  29  31  35 ...
2  3  4   5   6   7   8   9  11  13  14  16  17  18  19  21  23  25  26  28  29 ...
2  3  4   5   6   7   8   9  10  11  13  15  16  18  19  20  21  22  23  25  27 ...
2  3  4   5   6   7   8   9  10  11  12  13  15  17  18  20  21  22  23  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  17  19  20  22  23  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  19  21  22  24  25 ...
2  3  4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  21  23  24 ...
...

Cf. A000040, A003991, A076274, A112774.
Cf. A319929, A322630, A322744, A327259, A345474.
Cf. A354596.

row=12;max=200;U[i_,n_,k_]:=(i*n*k-(i-2)If[OddQ@n,If[OddQ@k,n+k-1,k],If[OddQ@k,n,0]])/2;t=Table[c=Union@Flatten@Table[U[i,n,k],{n,2,max},{k,2,max}];Complement[Range[2,max],c][[;;row]],{i,row}];Flatten@Table[t[[m,k-m+1]],{k,row},{m,k}] (* Giorgos Kalogeropoulos, Jun 08 2021 *)

With one exception there are likely no formulas for the rows of T(n,k) since their creation is based on a sieving process like the familiar prime number sieve. The exception is T(1,k) = 2*T(2,k)-1.

A307002 Numbers > 1 not of the form (3n*k - A319929(n,k))/2 where n and k > 1.

Original entry on oeis.org

2, 3, 4, 5, 7, 9, 10, 13, 15, 17, 21, 22, 23, 25, 29, 31, 34, 37, 39, 41, 45, 46, 49, 53, 55, 57, 58, 63, 65, 69, 71, 73, 77, 79, 81, 82, 85, 93, 94, 95, 97, 101, 105, 106, 109, 111, 118, 119, 121, 125, 129, 133, 135, 137, 141, 142, 143, 149, 151, 153, 157

Offset: 1

David Lovler, Mar 19 2019

All even terms > 2 appear to be semiprimes of the form 6m+4 (A112774).

The subsequence of odd terms is A307001. - David Lovler, Jan 17 2022

David Lovler, Table of n, a(n) for n = 1..4399

Cf. A112774, A307001, A322630, A322744, A340746, A340747.

Third row of array A327263.

T319929(n, k) = if (n%2, if (k%2, n+k-1, k), if (k%2, n, 0));
T(n, k) = (3*n*k - T319929(n, k))/2; \\ A322744
lista(nn) = {my(list = List()); for (n=2, nn, for (k=2, nn\n, listput(list, T(n, k)););); setminus([2..nn], Set(list));} \\ Michel Marcus, Jan 24 2021

Name amended by David Lovler, Jan 25 2022

A112772 Semiprimes of the form 6n+2.

Original entry on oeis.org

14, 26, 38, 62, 74, 86, 122, 134, 146, 158, 194, 206, 218, 254, 278, 302, 314, 326, 362, 386, 398, 422, 446, 458, 482, 542, 554, 566, 614, 626, 662, 674, 698, 734, 746, 758, 794, 818, 842, 866, 878, 914, 926, 974, 998, 1046, 1082, 1094, 1142, 1154, 1202, 1214

Offset: 1

Jonathan Vos Post and Ray Chandler, Oct 15 2005

{6} + A112772 + A112774 = A100484 = 2*A000040.

Rado showed that for a given Bernoulli number B_n there exist infinitely many Bernoulli numbers B_m having the same denominator. As a special case, if n = 2p where p is an odd prime p == 1 (mod 3), then the denominator of the Bernoulli number B_n equals 6. - Bernd C. Kellner, Mar 21 2018

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. Rado, A note on the Bernoullian numbers, J. London Math. Soc. 9 (1934) 88-90.

Subsequence of A051222. - Bernd C. Kellner, Mar 21 2018

Cf. A027642.

IsSemiprime:= func; [s: n in [0..210] | IsSemiprime(s) where s is 6*n + 2]; // Vincenzo Librandi, Sep 22 2012

Select[6Range[0,300]+2,PrimeOmega[#]==2&] (* Harvey P. Dale, Oct 04 2011 *)

2*select(n->n%3==1,primes(100)) \\ Charles R Greathouse IV, Sep 22 2012

a(n) = 2 * A002476(n) = 6 * A024892(n) + 2.

denominator(Bernoulli(a(n))) = 6. - Bernd C. Kellner, Mar 21 2018

A319996 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 6), n/g (A052126), and a single bit A319988(n) telling whether the largest prime factor is unitary.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 5, 11, 7, 12, 13, 14, 5, 15, 7, 16, 17, 10, 5, 18, 19, 12, 20, 21, 5, 22, 7, 23, 13, 10, 24, 25, 7, 12, 17, 26, 5, 27, 7, 16, 28, 10, 5, 29, 30, 31, 13, 21, 5, 32, 33, 34, 17, 10, 5, 35, 7, 12, 36, 37, 24, 22, 7, 16, 13, 38, 5, 39, 7, 12, 40, 21, 41, 27, 7, 42, 43, 10, 5, 44, 33, 12, 13, 26, 5, 45, 46, 16, 17, 10, 24, 47, 7, 48, 28, 49, 5, 22, 7, 34

Offset: 1

Antti Karttunen, Oct 05 2018

nonn

Restricted growth sequence transform of triple [A010875(A006530(n)), A052126(n), A319988(n)], with a separate value allotted for a(1).
Many of the same comments as given in A319717 apply also here, except for this filter, the "blind spot" area (where only unique values are possible for a(n)) is different, and contains at least all numbers in A070003. Because presence of 2 or 3 in the prime factorization of n do not force the value of a(n) unique, this is substantially less lax (i.e., more exact) filter than A319717. Here among the first 100000 terms, only 2393 have a unique value, compared to 74355 in A319717.
For all i, j:
  a(i) = a(j) => A002324(i) = A002324(j),
  a(i) = a(j) => A067029(i) = A067029(j),
  a(i) = a(j) => A071178(i) = A071178(j),
  a(i) = a(j) => A077462(i) = A077462(j) => A101296(i) = A101296(j),
  a(i) = a(j) => A319690(i) = A319690(j).

			For n = 15 (3*5) and n = 33 (3*11), the mod 6 residue of the largest prime factor is 5, also in both cases it is unitary (A319988(n) = 1), and the quotient n/A006530(n) is equal, in this case 3. Thus a(15) and a(33) are alloted the same running count (13 in this case) by rgs-transform.
For n = 2275 (5^2 * 7 * 13), n = 3325 (5^2 * 7 * 19), 5425 (5^2 * 7 * 31) and 6475 (5^2 * 7 * 37), the largest prime factor = 1 (mod 6), and A052126(n) = 175, thus these numbers are allotted the same running count (394 in this case) by rgs-transform.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A006530, A052126, A319988, A319717.
Cf. A007528 (positions of 5's), A002476 (of 7's), A112774 (after its initial term gives the position of 10's in this sequence).
Cf. also A319994 (modulo 4 analog).

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
A052126(n) = (n/A006530(n));
A319988(n) = ((n>1)&&(factor(n)[omega(n),2]>1));
A319996aux(n) = if(1==n,0,[A006530(n)%6, A052126(n), A319988(n)]);
v319996 = rgs_transform(vector(up_to,n,A319996aux(n)));
A319996(n) = v319996[n];

A232540 Numbers n such that (n(n+1)/2) modulo sigma(n) = 1.

Original entry on oeis.org

10, 22, 34, 46, 58, 82, 94, 106, 118, 142, 166, 178, 202, 214, 226, 262, 274, 298, 334, 346, 358, 382, 385, 394, 430, 454, 466, 478, 502, 514, 526, 538, 562, 586, 622, 634, 694, 706, 718, 766, 778, 802, 838, 862, 886, 898, 922, 934, 958, 982, 1006, 1018, 1042

Offset: 1

Jaroslav Krizek, Nov 25 2013

nonn

Also numbers n such that antisigma(n) modulo sigma(n) = 1. Antisigma(n) = A024816(n) = the sum of the nondivisors of n that are between 1 and n, sigma(n) = A000203(n) = the sum of the divisors of n.
Numbers n such that A232324(n) = 1.
Number 5950 is only squareful number from first 1400 terms (< 50000) of this sequence.
Conjecture: supersequence of A112774 (semiprimes of the form 6n+4).

			106 is in sequence because antisigma(106) mod sigma(106) = 5509 mod 162 = 1.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A000203, A076617, A024816, A112774, A232324, A232538.

Select[Range[1100],Mod[(#(#+1))/2,DivisorSigma[1,#]]==1&] (* Harvey P. Dale, Sep 08 2017 *)

A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.

Original entry on oeis.org

1, 4, 7, 10, 13, 19, 22, 25, 28, 34, 40, 43, 49, 52, 55, 64, 67, 73, 82, 85, 88, 94, 97, 112, 115, 118, 124, 127, 130, 133, 139, 145, 154, 157, 172, 175, 178, 190, 193, 199, 208, 214, 220, 223, 229, 232, 238, 244, 250, 253, 259, 277, 280, 283, 292, 295, 298, 307, 319

Offset: 1

Eric Desbiaux, Feb 11 2010

With the Bachet-Bézout theorem implicating Gauss Lemma and the Fundamental Theorem of Arithmetic,
for k > 1, k = 2*a + 3*b (a and b integers)
first type
A001477 = (2*A080425) + (3*A008611)
A000040 = (2*A039701) + (3*A157966)
A024893 Numbers k such that 3*k + 2 is prime
A034936 Numbers k such that 3*k + 4 is prime
OR second type
A001477 = (2*A028242) + (3*A059841)
A000040 = (2*A067076) + (3*1)
A067076 Numbers k such that 2*k + 3 is prime
   k   a b OR a b
  --   - -    - -
   0   0 0    0 0
   1   - -    - -
   2   1 0    1 0
   3   0 1    0 1
   4   2 0    2 0
   5   1 1    1 1
   6   0 2    3 0
   7   2 1    2 1
   8   1 2    4 0
   9   0 3    3 1
  10   2 2    5 0
  11   1 3    4 1
  12   0 4    6 0
  13   2 3    5 1
  14   1 4    7 0
  15   0 5    6 1
  ...
  2* 1 + 3 OR 3* 1 + 2 =  5;
  2* 4 + 3 OR 3* 3 + 2 = 11;
  2* 7 + 3 OR 3* 5 + 2 = 17;
  2*10 + 3 OR 3* 7 + 2 = 23;
  2*13 + 3 OR 3* 9 + 2 = 29;
  2*19 + 3 OR 3*13 + 2 = 41;
  2*22 + 3 OR 3*15 + 2 = 47;
  2*25 + 3 OR 3*17 + 2 = 53;
  2*28 + 3 OR 3*19 + 2 = 59.
A024893 Numbers k such that 3k+2 is prime.
A007528 Primes of the form 6k-1.
A024898 Positive integers k such that 6k-1 is prime.
1, 4, 7, 10, 13, 19, ... = (3*(4*A024898 - A024893) - 7)/2 = (A112774 - 3*A024893 - 5)/2 = A003627 - (3*A024893 - 5)/2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Chris K. Caldwell, FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?, Frequently asked questions about primes.

Cf. A067076, A024893, A007528, A024898, A059325.

Select[Range[0, 320], PrimeQ[(p = 2*# + 3)] && Mod[p, 3] == 2 &] (* Amiram Eldar, Jul 30 2024 *)

a(n) = 3*A059325(n) + 1. - Amiram Eldar, Jul 30 2024

Data corrected and extended by Amiram Eldar, Jul 30 2024

A250398 Composite numbers whose sum of divisors divides the sum of their unrelated numbers.

Original entry on oeis.org

4, 10, 22, 34, 46, 58, 82, 94, 106, 118, 142, 166, 178, 202, 214, 226, 262, 274, 298, 334, 346, 358, 382, 394, 454, 466, 478, 493, 502, 514, 526, 538, 562, 586, 622, 634, 694, 706, 718, 766, 778, 802, 838, 862, 886, 898, 922, 934, 958, 982, 1006, 1018, 1042, 1114

Offset: 1

Paolo P. Lava, Nov 21 2014

nonn

For a definition of unrelated number see A045763.

Most of the terms belong to A112774. First difference is at a(28) = 493.

			There is no unrelated number to 4. Therefore 0 / sigma(4) = 0 / 7 = 0 is an integer.
Unrelated numbers to 10 are 4, 6, 8 and their sum is 18. Now, 18 / sigma(10) = 18 / 18 = 1, that is an integer.

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Cf. A000203, A045763, A066760, A112774, A250399.

with(numtheory); P:=proc(q) local n;
for n from 1 to q do if not isprime(n) then
if type((1+n*(n+1)/2-sigma(n)-n*phi(n)/2)/sigma(n),integer)
then print(n); fi; fi; od; end: P(10^5);

A308169 Numbers k such that A023896(k) mod A000203(k) is prime.

Original entry on oeis.org

3, 7, 10, 11, 16, 19, 22, 23, 25, 27, 31, 34, 43, 46, 49, 58, 59, 71, 79, 82, 83, 94, 100, 103, 106, 118, 121, 131, 139, 142, 163, 166, 178, 191, 199, 202, 208, 211, 214, 223, 226, 251, 262, 271, 274, 298, 311, 331, 334, 346, 358, 359, 379, 382, 383, 394, 419, 443, 454, 463, 466, 478, 479, 484

Offset: 1

J. M. Bergot and Robert Israel, May 15 2019

nonn

Numbers k such that (k*A000010(k)/2) mod A000203(k) is prime.
The primes in the sequence are A092109.
The even semiprimes in the sequence are A112774.

			a(3)=10 is in the sequence because A023896(10) mod A000203(10) = 20 mod 6 = 2, and 2 is prime.

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

Cf. A000010, A000203, A023896, A092109, A112774.

select(n -> isprime((n*numtheory:-phi(n)/2) mod numtheory:-sigma(n)), [$2..1000]);

isok(n) = isprime(n*eulerphi(n)/2 % sigma(n)); \\ Michel Marcus, May 15 2019

A344872 Semiprimes of the form 3m+2.

Original entry on oeis.org

14, 26, 35, 38, 62, 65, 74, 77, 86, 95, 119, 122, 134, 143, 146, 155, 158, 161, 185, 194, 203, 206, 209, 215, 218, 221, 254, 278, 287, 299, 302, 305, 314, 323, 326, 329, 335, 341, 362, 365, 371, 377, 386, 395, 398, 407, 413, 422, 437, 446, 458, 473, 482, 485, 497

Offset: 1

Peter Munn, May 31 2021

There are no square terms, as squares are congruent to 0 or 1 modulo 3.

Products of a prime of the form 3m+1 and a prime of the form 3m+2 (the former necessarily being of the form 6m+1).

			14 = 2 * 7 has 2 prime factors (counting repetitions) so is a semiprime, and 14 = 3*4 + 2, so has the form 3m+2. So 14 is in the sequence.

Intersection of A001358 and A016789.
Disjoint union of A108172 and A112772.
Complement within A001358 of A001748, A112771 and A112774.
Subsequence of A344703.

Select[Range[2,500,3],PrimeOmega@#==2&] (* Giorgos Kalogeropoulos, Jun 02 2021 *)

isok(m) = bigomega(m) == 2 && m % 3 == 2;

A167857 Numbers whose divisors are represented by an integer polynomial.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 11, 13, 17, 19, 22, 23, 25, 29, 31, 34, 37, 41, 43, 46, 47, 49, 53, 55, 58, 59, 61, 67, 71, 73, 79, 82, 83, 85, 89, 91, 94, 97, 101, 103, 106, 107, 109, 113, 115, 118, 121, 127, 131, 133, 137, 139, 142, 145, 149, 151, 157, 163, 166, 167, 169, 171

Offset: 1

T. D. Noe, Nov 13 2009

nonn

That is, these numbers n have the property that there is a polynomial f(x) with integer coefficients whose values at x=0..tau(n)-1 are the divisors of n, where tau(n) is the number of divisors of n.

Every prime has this property, as do 1 and 9, the squares of primes of the form 6k+1, and semiprimes p*q with p and q both primes of the form 3k-1 or 3k+1. Terms of the form p^2*q also appear. We can find terms of the form p^m for any m. For example, 2311^13 is the smallest 13th power that appears. For any m, it seems that p^m appears for p a prime of the form k*m#+1, where m# is the product of the primes up to m. Are there terms with three distinct prime divisors?

			The divisors of 55 are (1, 5, 11, 55). The polynomial 1+15x-17x^2+6x^3 takes these values at x=0..3.

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

Cf. A108164, A108166, A112774 (forms of semiprimes)
Cf. A002476 (primes of the form 6k+1)
Cf. A132230 (primes of the form 30k+1)
Cf. A073103 (primes of the form 210k+1)
Cf. A073917 (least prime of the form k*prime(n)#+1)

Select[Range[1000], And @@ IntegerQ /@ CoefficientList[Expand[InterpolatingPolynomial[Divisors[ # ], x+1]], x] &]

is(n)=my(d=divisors(n));denominator(content(polinterpolate([0..#d-1],d))) == 1 \\ Charles R Greathouse IV, Jan 29 2016

cp's OEIS Frontend

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (i*n*k - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A307002 Numbers > 1 not of the form (3n*k - A319929(n,k))/2 where n and k > 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Extensions

A112772 Semiprimes of the form 6n+2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A319996 Let g = A006530(n), the largest prime factor of n. This filter sequence combines (g mod 6), n/g (A052126), and a single bit A319988(n) telling whether the largest prime factor is unitary.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A232540 Numbers n such that (n(n+1)/2) modulo sigma(n) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A173178 Numbers k such that 2*k+3 is a prime of the form 3*A024893(m) + 2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A250398 Composite numbers whose sum of divisors divides the sum of their unrelated numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A308169 Numbers k such that A023896(k) mod A000203(k) is prime.

Views

Author

Keywords

Comments

Examples

A327263 Array T(n,k) in which the i-th row consists of numbers > 1 not in array U(i;n,k) = (ink - (i-2)*A319929(n,k))/2 where i >= 1, n >= 1 and k >= 1, read by antidiagonals.

A173178 Numbers k such that 2k+3 is a prime of the form 3A024893(m) + 2.