A097706 -id:A097706 - cp's OEIS frontend

A072436 Remove prime factors of form 4*k+3.

1, 2, 1, 4, 5, 2, 1, 8, 1, 10, 1, 4, 13, 2, 5, 16, 17, 2, 1, 20, 1, 2, 1, 8, 25, 26, 1, 4, 29, 10, 1, 32, 1, 34, 5, 4, 37, 2, 13, 40, 41, 2, 1, 4, 5, 2, 1, 16, 1, 50, 17, 52, 53, 2, 5, 8, 1, 58, 1, 20, 61, 2, 1, 64, 65, 2, 1, 68, 1, 10, 1, 8, 73, 74, 25, 4, 1, 26, 1, 80, 1, 82, 1, 4, 85, 2, 29

Offset: 1

Reinhard Zumkeller, Jun 17 2002

a(n) <= n; a(a(n)) = a(n); for all factors p^m of a(n): p=2 or p=4*k+1.

			a(90) = a(2*3*3*5) = a(2*(4*0+3)^2*(4*1+1)^1) = 2*1^2*5 = 10.

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

Cf. A072438, A072437, A002144, A002145, A065338.

Equals n / A097706(n).

a:= n-> mul(`if`(irem(i[1], 4)=3, 1, i[1]^i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 09 2014

a[n_] := n/Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}];
Array[a, 100] (* Jean-François Alcover, May 29 2019 *)

a(n) = my(f=factor(n)); for (k=1, #f~, if ((f[k,1] % 4) == 3, f[k,1]=1)); factorback(f); \\ Michel Marcus, May 08 2017

from sympy import factorint
from operator import mul
def a(n):
    f = factorint(n)
    return 1 if n == 1 else reduce(mul, [1 if i%4==3 else i**f[i] for i in f])# Indranil Ghosh, May 08 2017

Multiplicative with a(p) = (if p==3 (mod 4) then 1 else p).

A343431 Part of n composed of prime factors of the form 6k-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 11, 1, 1, 1, 5, 1, 17, 1, 1, 5, 1, 11, 23, 1, 25, 1, 1, 1, 29, 5, 1, 1, 11, 17, 5, 1, 1, 1, 1, 5, 41, 1, 1, 11, 5, 23, 47, 1, 1, 25, 17, 1, 53, 1, 55, 1, 1, 29, 59, 5, 1, 1, 1, 1, 5, 11, 1, 17, 23, 5, 71, 1, 1, 1, 25, 1, 11, 1, 1, 5, 1, 41, 83, 1, 85, 1, 29, 11, 89, 5

Offset: 1

Peter Munn, Apr 15 2021

Completely multiplicative with a(p) = p if p is of the form 6k-1 and a(p) = 1 otherwise.

Largest term of A259548 that divides n.

Equivalent sequence for distinct prime factors: A170825.
Equivalent sequences for prime factors of other forms: A000265 (2k+1), A343430 (3k-1), A170818 (4k+1), A097706 (4k-1), A248909 (6k+1), A065330 (6k+/-1), A065331 (<= 3), A355582 (<= 5).
Range of terms: A259548.

f[p_, e_] := If[Mod[p, 6] == 5, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* after Amiram Eldar at A248909 *)

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((f[i, 1] + 1) % 6, f[i, 1] = 1); ); factorback(f); } \\ after Michel Marcus at A248909

from math import prod
from sympy import factorint
def A343431(n): return prod(p**e for p, e in factorint(n).items() if not (p+1)%6) # Chai Wah Wu, Dec 26 2022

a(n) = n / A065331(n) / A248909(n) = A065330(n) / A248909(n).

A170817 a(n) = product of distinct primes of form 4k+1 that divide n.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 13, 1, 5, 1, 17, 1, 1, 5, 1, 1, 1, 1, 5, 13, 1, 1, 29, 5, 1, 1, 1, 17, 5, 1, 37, 1, 13, 5, 41, 1, 1, 1, 5, 1, 1, 1, 1, 5, 17, 13, 53, 1, 5, 1, 1, 29, 1, 5, 61, 1, 1, 1, 65, 1, 1, 17, 1, 5, 1, 1, 73, 37, 5, 1, 1, 13, 1, 5, 1, 41, 1, 1, 85, 1, 29, 1

Offset: 1

N. J. A. Sloane, Dec 22 2009

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A170818, A170819, A097706, A083025, A170824, A170825.

a:= n-> mul (i, i=map (x-> x[1], select (x-> isprime (x[1]) and irem (x[1], 4)=1, ifactors(n)[2]))): seq (a(n), n=1..120);

Table[Times@@Select[Transpose[FactorInteger[n]][[1]],Mod[#,4]==1&], {n,90}] (* Harvey P. Dale, Dec 07 2012 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] % 4 == 1, f[i,1], 1));} \\ Amiram Eldar, Jun 09 2025

Corrected and extended with Maple program by Alois P. Heinz, Dec 23 2009

A260730 Numbers n for which A065339(n) > A260728(n).

Original entry on oeis.org

21, 33, 42, 57, 66, 69, 77, 84, 93, 105, 114, 129, 132, 133, 138, 141, 154, 161, 165, 168, 177, 186, 189, 201, 209, 210, 213, 217, 228, 231, 237, 249, 253, 258, 264, 266, 273, 276, 282, 285, 297, 301, 308, 309, 321, 322, 329, 330, 336, 341, 345, 354, 357, 372, 378, 381, 385, 393, 399, 402, 413, 417, 418, 420, 426, 429, 434, 437, 441, 453, 456, 462, 465, 469, 473, 474, 483, 489, 497, 498, 501, 506, 513, 516, 517, 525, 528, 532, 537, 546, 552

Offset: 1

Antti Karttunen, Aug 12 2015

nonn

Numbers n such that when the exponents in the prime factorization of A097706(n) are added in base-2 they produce at least one carry-bit. In other words, in that set of exponents {e1, e2, ..., en} there is at least one pair e_i, e_j that their binary representations have at least one 1-bit in the same position. (Here i and j are distinct as e_i and e_j are exponents of different primes, although e_i could be equal to e_j. See the examples.)

This differs from A119973 for the first time at n=30 where a(30)=231, term which is not present in A119973. Note that n=231 is the first position where the difference A065339(n) - A260728(n) > 1 as 231 = 3*7*11, a product of three distinct 4k+3 primes, thus A065339(231) = 3, while A260728(231) = 1.

			21 = 3^1 * 7^1 is present, because in its prime factors of the form 4k+3 (which are 3 and 7) the exponents 1 and 1 have at least one 1-bit in the same position, thus producing a carry-bit when summed in base-2.
63 = 3^2 * 7^1 is NOT present, because in its prime factors of the form 4k+3 the exponents 2 and 1 ("10" and "1" in binary) do NOT produce a carry-bit when summed in base-2, as those binary representations do not have any 1's in a common position.
189 = 3^3 * 7^1 is present, because in its prime factors of the form 4k+3 the exponents 3 and 1 ("11" and "1" in binary) have at least one 1-bit in the same position, thus producing a carry-bit when summed in base-2.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A065339, A260728, A097706, A119973.

A363340 a(n) is the smallest positive integer such that a(n) * n is the sum of two squares.

Original entry on oeis.org

1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 11, 3, 1, 7, 3, 1, 1, 1, 19, 1, 21, 11, 23, 3, 1, 1, 3, 7, 1, 3, 31, 1, 33, 1, 7, 1, 1, 19, 3, 1, 1, 21, 43, 11, 1, 23, 47, 3, 1, 1, 3, 1, 1, 3, 11, 7, 57, 1, 59, 3, 1, 31, 7, 1, 1, 33, 67, 1, 69, 7, 71, 1, 1, 1, 3, 19, 77, 3, 79

Offset: 1

Peter Schorn, May 28 2023

Using Fermat's two-squares theorem it is easy to see that a(n) is the product of all prime factors of n that are congruent to 3 modulo 4 and have an odd exponent.
This implies that a(n) is also the smallest positive integer such that n / a(n) is the sum of two squares.
Equivalently, a(n) is the product of all primes of the form 4k+3 that divide the squarefree part of n. If we use the squarefree kernel instead, we get A170819. - Peter Munn, Aug 06 2023

			a(1) = a(2) = 1 since 1 and 2 are sums of two squares.
a(3) = 3 since 3 and 6 are not sums of two squares but 3*3 is.
a(6) = 3 since 6 and 12 are not sums of two squares but 3*6 = 3^2 + 3^2.

Cf. A001481 (positions of 1's), A167181 (range of values).
Fixed points: A167181.
Cf. A000265, A002145, A007913, A059897, A097706, A170819.

a(n) = my(r=1); foreach(mattranspose(factor(n)), f, if(f[1]%4==3&&f[2]%2==1, r*=f[1])); r

Multiplicative with a(p^e) = p if p^e == 3 (mod 4), otherwise 1. - Peter Munn, Jul 03 2023
From Peter Munn, Aug 06 2023: (Start)
a(n) = A007913(A097706(n)) = A097706(A007913(n)).
a(n) == A000265(n) (mod 4).
a(A059897(n, k)) = A059897(a(n), a(k)).
(End)

A102574 a(n) is the sum of the distinct norms of the divisors of n over the Gaussian integers.

Original entry on oeis.org

1, 7, 10, 31, 31, 70, 50, 127, 91, 217, 122, 310, 183, 350, 310, 511, 307, 637, 362, 961, 500, 854, 530, 1270, 781, 1281, 820, 1550, 871, 2170, 962, 2047, 1220, 2149, 1550, 2821, 1407, 2534, 1830, 3937, 1723, 3500, 1850, 3782, 2821, 3710, 2210, 5110, 2451

Offset: 1

Yasutoshi Kohmoto, Feb 25 2005

Also sum of divisors of n^2 which are the sum of two squares (A001481). For example the divisors of 3^2 are 1, 3, 9 of which only 1 and 9 are in A001481 and a(3) = 1 + 9 = 10. - Jianing Song, Aug 03 2018

			Let ||i|| denote the norm of i.
a(2) = 1 + ||1+i|| + 2^2 = 1 + 2 + 4 = 7.
a(5) = 1 + ||1+2i|| + 5^2 = 1 + 5 + 25 = 31. Note that ||1+2i|| = ||2+i|| so their norm (5) is only counted once.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)
Wikipedia, Gaussian integer.

Cf. A000203 (sigma), A001157 (sigma_2), A001481, A097706, A103230, A243380.

b[n_] := Product[{p, e} = pe; If[Mod[p, 4] == 3, p^e, 1], {pe, FactorInteger[n]}];
a[n_] := With[{r = b[n]}, DivisorSigma[2, r] DivisorSigma[1, (n/r)^2]];
a /@ Range[50] (* Jean-François Alcover, Sep 20 2019, from PARI *)

\\ here b(n) is A097706.
b(n)={my(f=factor(n)); my(r=prod(i=1, #f~, my([p,e]=f[i,]); if(p%4==3, p^e, 1))); r}
a(n)={my(r=b(n)); sigma(r,2)*sigma((n/r)^2)} \\ Andrew Howroyd, Aug 03 2018

from math import prod
from sympy import factorint
def A102574(n): return prod((q := int(p & 3 == 3))*(p**(2*(e+1))-1)//(p**2-1) + (1-q)*(p**(2*e+1)-1)//(p-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 28 2022

a(n) = sigma_2(A097706(n)) * sigma((n/A097706(n))^2). - Andrew Howroyd, Aug 03 2018
Multiplicative with a(p^e) = sigma(p^(2e)) = (p^(2e+1) - 1)/(p - 1) if p = 2 or p == 1 (mod 4); sigma_2(p^e) = (p^(2e+2) - 1)/(p^2 - 1) if p == 3 (mod 4). - Jianing Song, Aug 03 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = (5/12) * zeta(3) * A243380 = 0.52812367275583317729... . - Amiram Eldar, Feb 13 2024

Corrected and extended by David Wasserman, Apr 08 2008
Keyword:mult added by Andrew Howroyd, Aug 03 2018
Name clarified by Jianing Song, Aug 03 2018

A097707 Part of n! composed of prime factors of form 4k+3.

Original entry on oeis.org

1, 1, 3, 3, 3, 9, 63, 63, 567, 567, 6237, 18711, 18711, 130977, 392931, 392931, 392931, 3536379, 67191201, 67191201, 1411015221, 15521167431, 356986850913, 1070960552739, 1070960552739, 1070960552739, 28915934923953

Offset: 1

Ralf Stephan, Aug 30 2004

nonn

log(a(n)) ~ 1/2 log n!.

P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 75. [?Broken link]
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 75.

Equals A097706(A000142(n)).

a(n)=local(f); f=factor(n!); prod(k=1, matsize(f)[1], if(f[k, 1]%4<>3, 1, f[k, 1]^f[k, 2]))

Typo in PARI code fixed by Colin Barker, Mar 12 2015

cp's OEIS Frontend

A072436 Remove prime factors of form 4*k+3.

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A343431 Part of n composed of prime factors of the form 6k-1.

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A170817 a(n) = product of distinct primes of form 4k+1 that divide n.

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A260730 Numbers n for which A065339(n) > A260728(n).

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A363340 a(n) is the smallest positive integer such that a(n) * n is the sum of two squares.

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A102574 a(n) is the sum of the distinct norms of the divisors of n over the Gaussian integers.

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A097707 Part of n! composed of prime factors of form 4k+3.

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