A001749 -id:A001749 - cp's OEIS frontend

A009188 Short leg of more than one Pythagorean triangle.

9, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 87, 88, 90, 91, 92, 93, 95, 96, 98, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116

Offset: 1

David W. Wilson

nonn

Values of n for which composite n X n magic squares are possible. - J. Lowell, May 20 2010
If n is in the sequence, k*n is in the sequence for all k > 1. So odd semiprimes (A046315) and numbers of the form 4*p where p is an odd prime are core subsequences which give the initial terms of arithmetic progressions in this sequence. - Altug Alkan, Nov 29 2015
Numbers appearing more than once in A009004. - Sean A. Irvine, Apr 20 2018

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

Cf. A001749, A009004, A020884, A046315, A264828.

filter:= proc(n) not isprime(n) and (n::odd or not isprime(n/2)) end proc:
select(filter, [$9 .. 10000]); # Robert Israel, Nov 30 2015

filterQ[n_] := !PrimeQ[n] && (OddQ[n] || !PrimeQ[n/2]);
Select[Range[9, 120], filterQ] (* Jean-François Alcover, Feb 28 2019, from Maple *)

forcomposite(n=9, 1e3, if(n % 2 == 1 || !isprime(n/2), print1(n, ", "))) \\ Altug Alkan, Dec 01 2015

from sympy import primepi
def A009188(n):
    def f(x): return int(n+2+primepi(x)+primepi(x>>1))
    m, k = n+2, f(n+2)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 17 2024

a(n) = A264828(n+2). - Chai Wah Wu, Oct 17 2024

A057859 Number of residue classes modulo n which contain a prime.

Original entry on oeis.org

1, 2, 3, 3, 5, 4, 7, 5, 7, 6, 11, 6, 13, 8, 10, 9, 17, 8, 19, 10, 14, 12, 23, 10, 21, 14, 19, 14, 29, 11, 31, 17, 22, 18, 26, 14, 37, 20, 26, 18, 41, 15, 43, 22, 26, 24, 47, 18, 43, 22, 34, 26, 53, 20, 42, 26, 38, 30, 59, 19, 61, 32, 38, 33, 50, 23, 67, 34, 46

Offset: 1

Henry Bottomley, Sep 08 2000

a(n) = n iff n is prime; a(2*n)<=n+1; a(4*p)=2*p for primes p>2: a(A001749(n))=A057860(A001749(n)). - Reinhard Zumkeller, Jan 11 2004

			a(30) = 11 since 30k+m can be prime if m = 2, 3 or 5 (once each with k = 0) or m = 1, 7, 11, 13, 17, 19, 23 or 29 (each for an infinite number of values of k).

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A057857, A179179.

with(numtheory):
a:= n-> phi(n)+nops(factorset(n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 14 2016

Table[EulerPhi[n] + PrimeNu[n], {n, 1, 100}] (* G. C. Greubel, May 13 2017 *)

for(n=1,100, print1(eulerphi(n) + omega(n), ", ")) \\ G. C. Greubel, May 13 2017

a(n) = A000010(n) + A001221(n) = n - A057860(n).

A075520 4*prime(n) + (prime(n) mod 4).

Original entry on oeis.org

10, 15, 21, 31, 47, 53, 69, 79, 95, 117, 127, 149, 165, 175, 191, 213, 239, 245, 271, 287, 293, 319, 335, 357, 389, 405, 415, 431, 437, 453, 511, 527, 549, 559, 597, 607, 629, 655, 671, 693, 719, 725, 767, 773, 789, 799, 847, 895, 911, 917, 933, 959, 965

Offset: 1

Reinhard Zumkeller, Sep 19 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000040, A039702.

a075520 n = a075520_list !! (n-1)
a075520_list = zipWith (+) a001749_list a039702_list
-- Reinhard Zumkeller, Feb 20 2012

4#+Mod[#,4]&/@Prime[Range[60]] (* Harvey P. Dale, Mar 10 2016 *)

a(n) = A001749(n) + A039702(n).

A177425 Integers with multiple and strictly distinct powers.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 200, 207, 208, 212, 224, 232, 236, 242, 244, 245

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 07 2010

12=2^2*3, 18=2*3^2,.. 600=2^3*3*5^2,..

Lei Zhou, Table of n, a(n) for n = 1..10000

Apart from its first term, A001749 is a subsequence.

Cf. A001358.

f[n_]:=Sort[Last/@FactorInteger[n]];lst={};Do[If[Length[f[n]]>1&&f[n]==Union@f[n],AppendTo[lst,n]],{n,0,6!}];lst

is(n)=my(f=vecsort(factor(n)[,2]~)); n>1 && Set(f)==f && #f>1 \\ Charles R Greathouse IV, Mar 20 2014

a(n) << n log n. - Charles R Greathouse IV, Mar 25 2014

A272470 7 times the primes.

Original entry on oeis.org

14, 21, 35, 49, 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329, 371, 413, 427, 469, 497, 511, 553, 581, 623, 679, 707, 721, 749, 763, 791, 889, 917, 959, 973, 1043, 1057, 1099, 1141, 1169, 1211, 1253, 1267, 1337, 1351, 1379, 1393, 1477, 1561, 1589, 1603, 1631, 1673, 1687, 1757, 1799, 1841, 1883, 1897

Offset: 1

Omar E. Pol, Apr 30 2016

Column 4 of A272214.

k times the primes (k=1..6): A000040, A100484, A001748, A001749, A001750, A138636.

7 Prime@ Range@ 58 (* Michael De Vlieger, May 01 2016 *)

a(n) = 7*prime(n); \\ Michel Marcus, May 01 2016

from sympy import prime
for n in range(1,1000):print(7*prime(n),end=", ") # Soumil Mandal, May 08 2016

a(n) = 7*prime(n) = 7*A000040(n).

A326697 a(n) is the sum of divisors d of n such that sigma(d) divides n.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 12, 1, 1, 1, 5, 1, 8, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 40, 1, 1, 1, 17, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 28

Offset: 1

Jaroslav Krizek, Jul 19 2019

nonn

a(A097603(n)) > 1.
See A173441 and A326698 for number and product of such divisors.
From Bernard Schott, Aug 13 2019: (Start)
a(n) = 1 if n is in A000961,
a(n) = 1 if n is in A006881 \ {6},
a(n) = 1 if n is in A001749 \ {12, 28}. (End)

			For n = 12, divisors d of 12: 1, 2, 3, 4, 6, 12;
corresponding sigma(d): 1, 3, 4, 7, 12, 28;
sigma(d) divides n for 4 divisors d: 1, 2, 3, 6;
a(12) = 1 + 2 + 3 + 6 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000203, A000961, A173441, A326698.

[&+[d: d in Divisors(n) | IsIntegral(n / SumOfDivisors(d))]: n in [1..100]];

a[n_] := DivisorSum[n, # &, Divisible[n, DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, Jul 21 2019 *)

a(n) = sumdiv(n, d, d*(!(n % sigma(d)))); \\ Michel Marcus, Jul 19 2019

A382255 Heinz number of the partition corresponding to run lengths in the bits of n.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 6, 5, 10, 12, 16, 12, 9, 12, 10, 7, 14, 20, 24, 18, 24, 32, 24, 20, 15, 18, 24, 18, 15, 20, 14, 11, 22, 28, 40, 30, 36, 48, 36, 30, 40, 48, 64, 48, 36, 48, 40, 28, 21, 30, 36, 27, 36, 48, 36, 30, 25, 30, 40, 30, 21, 28, 22, 13, 26, 44, 56, 42

Offset: 0

M. F. Hasler and Ali Sada, Mar 19 2025

The run lengths (number of consecutive bits that are equal) in the binary numbers in [2^(L-1), 2^L-1], i.e., of bit length L, yield all possible compositions of L, i.e., the partitions with any possible order of the parts.
Associated to any composition (p1, ..., pK) is their Heinz number prime(p1)*...*prime(pK) which depends only on the partition, i.e., not on the order of the parts.
The sequence can also be read as a table with row lengths 1, 1, 2, 4, 8, 16, 32, ... (= A011782), where row L = 0, 1, 2, 3, ... lists the 2^(L-1) compositions of L through their Heinz numbers (which will appear more than once if they contain at least two distinct parts).

			   n | binary | partition | a(n) = Heinz number
  ---+--------+-----------+--------------------
   0 |   (0)  | empty sum | 1 = empty product
   1 |     1  |     1     | 2 = prime(1)
   2 |    10  |    1+1    | 4 = prime(1) * prime(1)
   3 |    11  |     2     | 3 = prime(2)
   4 |   100  |    1+2    | 6 = prime(1) * prime(2)
   5 |   101  |   1+1+1   | 8 = 2^3 = prime(1) * prime(1) * prime(1)
   6 |   110  |    2+1    | 6 = prime(2) * prime(1)
   7 |   111  |     3     | 5 = prime(3)
   8 |  1000  |    1+3    | 10 = 2*5 = prime(1) * prime(3)
   9 |  1001  |   1+2+1   | 12 = 2^2*3 = prime(1) * prime(2) * prime(1)
  ...|   ...  |    ...    | ...
For example, n = 4 = 100[2] (in binary) has run lengths (1, 2), namely: one bit 1 followed by two bits 0. This gives a(4) = prime(1)*prime(2) = 6.
Next, n = 5 = 101[2] (in binary) has run lengths (1, 1, 1): one bit 1, followed by one bit 0, followed by one bit 1. This gives a(4) = prime(1)^3 = 8.
Then, n = 6 = 110[2] (in binary) has run lengths (2, 1): first two bits 1, then one bit 0. This is the same as for 4, just in reverse order, so it yields the same Heinz number a(6) = prime(2)*prime(1) = 6.
Then, n = 7 = 111[2] (in binary) has run lengths (3), namely: three bits 1. This gives a(5) = prime(3) = 5.
Sequence written as irregular triangle:
   1;
   2;
   4,  3;
   6,  8,  6,  5;
  10, 12, 16, 12,  9, 12, 10,  7;
  14, 20, 24, 18, 24, 32, 24, 20, 15, 18, 24, 18, 15, 20, 14, 11;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..32768

Cf. A112798 and A296150 (partitions sorted by Heinz number).
Cf. A185974, A334433, A334435, A334438, A334434, A129129, A334436 (partitions given as Heinz numbers, in Abramowitz-Stegun, Maple, Mathematica order).
For "constructive" lists of partitions see A036036 (Abramowitz and Stegun order), A036036 (reversed), A080576 (Maple order), A080577 (Mathematica order).
Row sums of triangle give A030017(n+1).
Cf. A007088 (the binary numbers).
Cf. A011782, A001747, A001749, A008578.
Cf. A101211 (the run lengths as rows of a table).

a:= proc(n) option remember; `if`(n<2, 1+n, (p->
      a(iquo(n, 2^p))*ithprime(p))(padic[ordp](n+(n mod 2), 2)))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 20 2025

Heinz(p)=vecprod([ prime(k) | k <- p ])
RL(v) = if(#v, v=Vec(select(t->t,concat([1,v[^1]-v[^-1],1]),1)); v[^1]-v[^-1])
apply( {A382255(n) = Heinz(RL(binary(n)))}, [0..99] )

a(2^n) = A001747(n+1).
a(2^n-1) = A008578(n+1).
a(2^n+1) = A001749(n-1) for n>=2.

A181794 Numbers n such that the number of even divisors of n is an even divisor of n.

Original entry on oeis.org

4, 6, 10, 12, 14, 16, 20, 22, 24, 26, 28, 34, 36, 38, 44, 46, 48, 52, 58, 62, 68, 74, 76, 80, 82, 86, 90, 92, 94, 106, 112, 116, 118, 120, 122, 124, 126, 134, 142, 144, 146, 148, 150, 158, 160, 164, 166, 168, 172, 176, 178, 180, 188, 192, 194, 198, 202, 206, 208, 212, 214, 216, 218, 226, 234, 236, 240, 244, 252, 254, 256, 262, 264, 268, 272, 274, 278

Offset: 1

Matthew Vandermast, Nov 14 2010

nonn

All terms are even, since odd numbers, even if they have an even count of divisors, don't have any even divisors.

Includes all numbers of the form A000040(m)*A001146(n).

			a(4)=12 has four even divisors (2, 4, 6, and 12), and 4 is one of those even divisors.
The number 21 is not in this sequence: it has four divisors (1, 3, 7, and 21), and 4 is not one of those divisors.

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

A100484 and A001749 are subsequences. A001146 and A100042 are also subsequences except for their initial terms.

A309131 Triangle read by rows: T(n, k) = (n - k)*prime(1 + k), with 0 <= k < n.

Original entry on oeis.org

2, 4, 3, 6, 6, 5, 8, 9, 10, 7, 10, 12, 15, 14, 11, 12, 15, 20, 21, 22, 13, 14, 18, 25, 28, 33, 26, 17, 16, 21, 30, 35, 44, 39, 34, 19, 18, 24, 35, 42, 55, 52, 51, 38, 23, 20, 27, 40, 49, 66, 65, 68, 57, 46, 29, 22, 30, 45, 56, 77, 78, 85, 76, 69, 58, 31

Offset: 1

Stefano Spezia, Jul 14 2019

T(n, k) is the k-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive prime numbers prime(1), ..., prime(n).
The h-th subdiagonal of the triangle T gives the primes multiplied by (h + 1).
The k-th column of the triangle T gives the multiples of prime(1 + k).
Also array A(n, k) = n*prime(1 + k) read by ascending antidiagonals, with 0 <= k < n. - Michel Marcus, Jul 15 2019

			The triangle T(n, k) begins:
---+-----------------------------------------------------
n\k|    0     1     2     3     4     5     6     7     8
---+-----------------------------------------------------
1  |    2
2  |    4     3
3  |    6     6     5
4  |    8     9    10     7
5  |   10    12    15    14    11
6  |   12    15    20    21    22    13
7  |   14    18    25    28    33    26    17
8  |   16    21    30    35    44    39    34    19
9  |   18    24    35    42    55    52    51    38    23
...
For n = 3 the matrix M(3) is
          2,         3,         5
    M_{2,1},         2,         3
    M_{3,1},   M_{3,2},         2
and therefore T(3, 0) = 2 + 2 + 2 = 6, T(3, 1) = 3 + 3 = 6, and T(3, 2) = 5.

Wikipedia, Toeplitz matrix

Cf. A025581, A318173, A325516.

Cf. A000040: diagonal; A001747: 1st subdiagonal; A001748: 2nd subdiagonal; A001749: 3rd subdiagonal; A001750: 4th subdiagonal; A005843: 0th column; A008585: 1st column; A008587: 2nd column; A008589: 3rd column; A008593: 4th column; A008595: 5th column; A008599: 6th column; A008601: 7th column; A014148: row sums; A138636: 5th subdiagonal; A272470: 6th subdiagonal.

[[(n-k)*NthPrime(1+k): k in [0..n-1]]: n in [1..11]]; // triangle output

a:=(n, k)->(n-k)*ithprime(1+k): seq(seq(a(n, k), k=0..n-1), n=1..11);

Flatten[Table[(n-k)*Prime[1+k],{n,1,11},{k,0,n-1}]]

T(n, k) = (n - k)*prime(1 + k);
tabl(nn) = for(n=1, nn, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ triangle output

[[(n-k)*Primes().unrank(k) for k in (0..n-1)] for n in (1..11)] # triangle output

T(n, k) = A025581(n, k)*A000040(1 + k).

A351958 a(1) = 1, followed by numbers k for which the primorial inflation of k is equal to x * p#, where p# is the primorial (A034386) of some prime p, and 1 <= x < p.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 56, 57, 58, 59, 61, 62, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 86, 87, 88, 89, 92, 93, 94, 97, 101, 102, 103, 104, 106, 107, 109, 111, 113, 114, 116, 118, 122, 123, 124

Offset: 1

Antti Karttunen, Apr 04 2022

nonn

Numbers k such that A108951(k) is in A060735.
Numbers k for which A324886(k) is a power of prime (in A000961).
Numbers k such that A108951(k) / A002110(A061395(k)) < A000040(1+A061395(k)), the next prime larger than the greatest prime dividing k.

Positions of 1's in A329040.
Cf. A000961, A002110, A061395, A034386, A060735, A108951, A324886, A351956 (characteristic function).
Subsequences: A000040, A008578, A100484, A001748 \ {9}, A001749 \ {8}.
Cf. also A344591.

isA351958(n) = A351956(n); \\ Uses the program given in A351956.

cp's OEIS Frontend

A009188 Short leg of more than one Pythagorean triangle.

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A057859 Number of residue classes modulo n which contain a prime.

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A075520 4*prime(n) + (prime(n) mod 4).

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A177425 Integers with multiple and strictly distinct powers.

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A272470 7 times the primes.

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A326697 a(n) is the sum of divisors d of n such that sigma(d) divides n.

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Magma

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A382255 Heinz number of the partition corresponding to run lengths in the bits of n.

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Maple