A054520 -id:A054520 - cp's OEIS frontend

A057949 Numbers with more than one factorization into S-primes. See A054520 and A057948 for definition.

441, 693, 1089, 1197, 1449, 1617, 1881, 1953, 2205, 2277, 2541, 2709, 2793, 2961, 3069, 3249, 3381, 3465, 3717, 3933, 3969, 4221, 4257, 4389, 4473, 4557, 4653, 4761, 4977, 5229, 5301, 5313, 5445, 5733, 5841, 5929, 5985, 6237, 6321, 6417, 6489, 6633

Offset: 1

Jud McCranie, Oct 14 2000

nonn

Numbers with k >= 4 prime factors (with multiplicity) that are congruent to 3 mod 4, no k-1 of which are equal. - Charlie Neder, Nov 03 2018

			2205 is in S = {1,5,9, ... 4i+1, ...}, 2205 = 5*9*49 = 5*21^2; 5, 9, 21 and 49 are S-primes (A057948).

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A054520, A057948, A057950.

Cf. A343826 (only 1 way), A343827 (exactly 2 ways), A343828 (exactly 3 ways).

ok(n)={if(n%4==1, my(f=factor(n)); my(s=[f[i,2] | i<-[1..#f~], f[i,1]%4==3]); vecsum(s)>=4 && vecmax(s)Andrew Howroyd, Nov 25 2018

def A057949_list(bound) :
    numterms = (bound-1)//4 + 1
    M = [1] * numterms
    for k in range(1, numterms) :
        if M[k] == 1 :
            kpower = k
            while kpower < numterms :
                step = 4*kpower+1
                for j in range(kpower, numterms, step) :
                    M[j] *= 4*k+1
                kpower = 4*kpower*k + kpower + k
    # Now M[k] contains the product of the terms p^e where p is an S-prime
    # and e is maximal such that p^e divides 4*k+1
    return [4*k+1 for k in range(numterms) if M[k] > 4*k+1]
# Eric M. Schmidt, Dec 11 2016

Offset corrected by Eric M. Schmidt, Dec 11 2016

A057948 S-primes: let S = {1,5,9, ... 4i+1, ...}; then an S-prime is in S but is not divisible by any members of S except itself and 1.

Original entry on oeis.org

5, 9, 13, 17, 21, 29, 33, 37, 41, 49, 53, 57, 61, 69, 73, 77, 89, 93, 97, 101, 109, 113, 121, 129, 133, 137, 141, 149, 157, 161, 173, 177, 181, 193, 197, 201, 209, 213, 217, 229, 233, 237, 241, 249, 253, 257, 269, 277, 281, 293, 301, 309, 313, 317, 321, 329

Offset: 1

Jud McCranie, Oct 14 2000

nonn

Factorization in S is not unique. See related sequences.
Kostrikin calls these numbers quasi-primes. - Arkadiusz Wesolowski, Aug 19 2017
a(n) is a prime of the form 4*n + 1 or a product of 2 primes of the form 4*n + 3. - David A. Corneth, Nov 10 2018

			21 is of the form 4i+1, but it is not divisible by any smaller S-primes, so 21 is in the sequence.

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 101, problem 1.
A. I. Kostrikin, Introduction to Algebra, universitext, Springer, 1982.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Hilbert Number [From _Eric W. Weisstein_, Sep 15 2008]

Union of A002144 and A107978. - Charlie Neder, Nov 03 2018

Cf. A054520, A057949, A057950.

N:= 1000: # to get all terms <= N
S:= {seq(4*i+1,i=1..floor((N-1)/4))}:
for n from 1 while n <= nops(S) do
  r:= S[n];
  S:= S minus {seq(i*r,i=2..floor(N/r))};
od:
S; # Robert Israel, Dec 14 2014

nn = 100; Complement[Table[4 k + 1, {k, 1, nn}], Union[Flatten[ Table[Table[(4 k + 1) (4 j + 1), {k, 1, j}], {j, 1, nn}]]]] (* Geoffrey Critzer, Dec 14 2014 *)

is(n) = if(n % 2 == 0, return(0)); if(n%4 == 1 && isprime(n), return(1)); f = factor(n); if(vecsum(f[, 2]) != 2, return(0)); for(i = 1, #f[, 1], if(f[i, 1] % 4 == 1, return(0))); n>1 \\ David A. Corneth, Nov 10 2018

a(n) ~ C n log n / log log n, where C > 2. - Thomas Ordowski, Sep 09 2012

Offset corrected by Charlie Neder, Nov 03 2018

A057950 Numbers primitive with respect to having more than one factorization into S-primes. See related sequences for definition.

Original entry on oeis.org

441, 693, 1089, 1197, 1449, 1617, 1881, 1953, 2277, 2541, 2709, 2793, 2961, 3069, 3249, 3381, 3717, 3933, 4221, 4257, 4389, 4473, 4557, 4653, 4761, 4977, 5229, 5301, 5313, 5841, 5929, 6321, 6417, 6489, 6633, 6741, 6897, 6909, 7029, 7161, 7353, 7581

Offset: 1

Jud McCranie, Oct 14 2000

nonn

A subset of A057949, removing terms that are a multiple of a smaller term.

Cubefree numbers with exactly 4 prime factors, all congruent to 3 mod 4. - Charlie Neder, Nov 26 2018

			441 is in S = {1, 5, 9, ... 4i+1, ...}, 441 = 9*49 = 21^2, 9, 21 and 49 as S-primes (A057948). 441 is primitive because it is not divisible by any smaller numbers with more than 1 factorization into S-primes. Multiples of 441 within S are not primitive.

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A054520, A057948, A057949.

Cf. A004709 (cubefree numbers).

Definition edited and offset corrected by Eric M. Schmidt, Dec 11 2016

A343826 Numbers which are the product of two S-primes (A057948) in exactly one way.

Original entry on oeis.org

25, 45, 65, 81, 85, 105, 117, 145, 153, 165, 169, 185, 189, 205, 221, 245, 261, 265, 273, 285, 289, 297, 305, 333, 345, 357, 365, 369, 377, 385, 429, 445, 465, 477, 481, 485, 493, 505, 513, 533, 545, 549, 561, 565, 605, 609, 621, 629, 637, 645, 657, 665, 685

Offset: 1

Zachary DeStefano, Apr 30 2021

nonn

There exist numbers which are the product of two S-primes in exactly 1, 2, and 3 ways; however, it is unknown if any numbers exist which are the product of two S-primes in exactly 4 ways.

			153 = 9*17 which are both S-primes, and admits no other S-prime factorizations.

Zachary DeStefano, Table of n, a(n) for n = 1..3786

Cf. A054520, A057948, A057949, A057950.

Exactly two ways: A343827. Exactly three ways: A343828.

\\ uses is(n) from A057948
isok(n) = sumdiv(n, d, (d<=n/d) && is(d) && is(n/d)) == 1; \\ Michel Marcus, May 01 2021

a(n) == 1 (mod 4). - Hugo Pfoertner, May 01 2021

A343827 Numbers which are the product of two S-primes (A057948) in exactly two ways.

Original entry on oeis.org

441, 693, 1089, 1197, 1449, 1617, 1881, 1953, 2277, 2541, 2709, 2793, 2961, 3069, 3249, 3381, 3717, 3933, 4221, 4257, 4473, 4557, 4653, 4761, 4977, 5229, 5301, 5841, 5929, 6321, 6417, 6489, 6633, 6741, 6897, 6909, 7029, 7353, 7581, 7821, 8001, 8037, 8217, 8253

Offset: 1

Zachary DeStefano, Apr 30 2021

nonn

First differs from A057950 at a(21)=4473, whereas A057950(21)=4389, which can be represented as the product of two S-primes in exactly 3 ways.

There exist numbers which are the product of two S-primes in exactly 1, 2, and 3 ways; however, it is unknown if any numbers exist which are the product of two S-primes in exactly 4 ways.

			1449=9*161=21*69 which are all S-primes (A057948), and admits no other S-prime factorizations.

Zachary DeStefano, Table of n, a(n) for n = 1..2484

Cf. A054520, A057948, A057949, A057950.

Exactly one way: A343826. Exactly three ways: A343828.

\\ uses is(n) from A057948
isok(n) = sumdiv(n, d, (d<=n/d) && is(d) && is(n/d)) == 2; \\ Michel Marcus, May 01 2021

a(n) == 1 (mod 4). - Hugo Pfoertner, May 01 2021

A343828 Numbers which are the product of two S-primes (A057948) in exactly three ways.

Original entry on oeis.org

4389, 5313, 7161, 9177, 9933, 10857, 12369, 13629, 14421, 14973, 15477, 16401, 17157, 18249, 18753, 19173, 19437, 20769, 22701, 23529, 23541, 23793, 24717, 26733, 26961, 27993, 28329, 28497, 29337, 29469, 30261, 30597, 31521, 32109, 32361, 32637, 33117, 33649

Offset: 1

Zachary DeStefano, Apr 30 2021

nonn

There exist numbers which are the product of two S-primes in exactly 1, 2, and 3 ways.

An S-prime is either a prime of the form 4k+1 or a semiprime of the form (4k+3)*(4m+3). That means the maximum number of prime factors that a number factorizable into two S-primes can have is four (all 4k + 3), and those can be combined into S-primes in at most three distinct ways. - Gleb Ivanov, Dec 07 2021

			9177 = 21*437 = 57*161 = 69*133 which are all S-primes (A057948), and admits no other S-Prime factorizations.
4389 = (3*7)*(11*19) = (3*11)*(7*19) = (3*19)*(7*11); 3,7,11,19 are the smallest primes of the form 4k + 3.

Zachary DeStefano, Table of n, a(n) for n = 1..1014

Cf. A054520, A057948, A057949, A057950.

Exactly one way: A343826. Exactly two ways: A343827.

\\ uses is(n) from A057948
isok(n) = sumdiv(n, d, (d<=n/d) && is(d) && is(n/d)) == 3; \\ Michel Marcus, May 01 2021

a(n) == 1 (mod 4). - Hugo Pfoertner, May 01 2021

A380509 Numbers of the form i+j+4ij for i,j >= 1 together with numbers of the form -i-j+4ij for i,j >= 2.

Original entry on oeis.org

6, 11, 12, 16, 19, 20, 21, 26, 29, 30, 31, 33, 36, 38, 40, 41, 42, 46, 47, 51, 52, 54, 55, 56, 61, 63, 65, 66, 68, 71, 72, 74, 75, 76, 81, 82, 83, 85, 86, 89, 90, 91, 92, 94, 96, 101, 103, 106, 107, 109, 110, 111, 116, 117, 118, 119, 120, 121, 123, 124, 126, 128, 129, 131, 132, 133

Offset: 1

Davide Rotondo, Jan 26 2025

nonn

			1+1+4*1*1 = 6; -2-2+4*2*2 = 12.

Cf. A054520, A380140, A380572 (complement), A380549, A380550.

L := 150: S := {}:
for i to L do for j to L do
if 4*i*j + i + j <= L then S := `union`(S, {4*i*j+i+j}, {4*i*j+3*i+3*j+2}) fi;
od; od;
S; # Peter Bala, Jan 30 2025

More terms from Stefano Spezia, Jan 26 2025

Missing terms 111, 121 and 126 added by Peter Bala, Jan 30 2025

A380140 Numbers of the form 4jk - j - k for j, k >= 2.

Original entry on oeis.org

12, 19, 26, 30, 33, 40, 41, 47, 52, 54, 56, 61, 63, 68, 71, 74, 75, 82, 85, 86, 89, 90, 96, 101, 103, 107, 109, 110, 116, 117, 118, 124, 128, 129, 131, 132, 138, 140, 145, 146, 147, 151, 152, 155, 159, 161, 162, 166, 173, 176, 178, 180, 182, 184, 185, 187, 191, 194, 195, 201

Offset: 1

Hugo Pfoertner, Jan 26 2025

nonn

Cf. A054520, A380509.

A321337 a(n) is the least number with n factorizations into S-primes (numbers 4k+1 with no proper divisors > 1 of form 4m+1).

Original entry on oeis.org

1, 441, 4389, 39501, 53361, 92169, 829521, 1935549, 302841, 2725569, 3041577, 27374193, 853577109, 7682193981, 3129357, 6359661, 19263321, 234201429, 230639102001, 437200389130923862144165773, 1341335457, 12072019113, 23318757, 1201975929, 28164213, 253477917, 4918230009, 101711843982441

Offset: 1

Charlie Neder, Nov 05 2018

nonn

It suffices to check only numbers with all prime factors congruent to 3 mod 4 and continuous, nonincreasing prime exponents (i.e., members of the analog of A025487 with respect to primes congruent to 3 mod 4).

			a(4) = 39501 can be factored into S-primes (A057948) in 4 distinct ways: 9 * 21 * 209, 9 * 33 * 133, 9 * 57 * 77, or 21 * 33 * 57, and it is the smallest number with this property.

Charlie Neder, Table of n, a(n) for n = 1..277
Charlie Neder, Table of n, a(n) for n = 1..600, missing 278 and 482

Cf. A054520, A057948 (S-primes), A057949 (numbers with multiple factorizations into S-primes).

a(13) corrected by David A. Corneth, Nov 10 2018

More terms from WG Zeist, Jan 09 2019

A348156 S_3-primes: let S_3 = {1,4,7,...,3i+1,...}; then an S_3-prime is in S_3 but is not divisible by any elements of S_3 except for itself and 1.

Original entry on oeis.org

4, 7, 10, 13, 19, 22, 25, 31, 34, 37, 43, 46, 55, 58, 61, 67, 73, 79, 82, 85, 94, 97, 103, 106, 109, 115, 118, 121, 127, 139, 142, 145, 151, 157, 163, 166, 178, 181, 187, 193, 199, 202, 205, 211, 214, 223, 226, 229, 235, 241, 253, 262, 265, 271, 274, 277, 283, 289, 295, 298

Offset: 1

Gleb Ivanov, Oct 03 2021

nonn

Factorization in S_3 is not unique; for example, 220 = 4 * 55 = 10 * 22.

Cf. A016777, A057948, A054520.

nn = 100; Complement[Table[3 k + 1, {k, 1, nn}], Union[Flatten[ Table[Table[(3 k + 1) (3 j + 1), {k, 1, j}], {j, 1, nn}]]]]

isok(m) = ((m % 3)==1) && (#select(x->((x%3)==1), divisors(m)) == 2); \\ Michel Marcus, Oct 06 2021

nn = 300
s = [True]*((nn)//3 + 1)
for i in range(4, nn, 3):
    if s[(i-1)//3]:
        for t in range(4, (nn)//i, 3):
            s[(i*t-1)//3] = False
print([3*i + 1 for i in range(1, (nn + 3)//3) if s[i]])

cp's OEIS Frontend

A057949 Numbers with more than one factorization into S-primes. See A054520 and A057948 for definition.

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A057948 S-primes: let S = {1,5,9, ... 4i+1, ...}; then an S-prime is in S but is not divisible by any members of S except itself and 1.

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A057950 Numbers primitive with respect to having more than one factorization into S-primes. See related sequences for definition.

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A343826 Numbers which are the product of two S-primes (A057948) in exactly one way.

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A343827 Numbers which are the product of two S-primes (A057948) in exactly two ways.

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A343828 Numbers which are the product of two S-primes (A057948) in exactly three ways.

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A380509 Numbers of the form i+j+4ij for i,j >= 1 together with numbers of the form -i-j+4ij for i,j >= 2.

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A380140 Numbers of the form 4*j*k - j - k for j, k >= 2.

A380140 Numbers of the form 4jk - j - k for j, k >= 2.