A191217 -id:A191217

A191218

Odd numbers n such that sigma(n) is congruent to 2 modulo 4.

+10
26

5, 13, 17, 29, 37, 41, 45, 53, 61, 73, 89, 97, 101, 109, 113, 117, 137, 149, 153, 157, 173, 181, 193, 197, 229, 233, 241, 245, 257, 261, 269, 277, 281, 293, 313, 317, 325, 333, 337, 349, 353, 369, 373, 389, 397, 401, 405, 409, 421, 425, 433, 449, 457, 461, 477

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Exactly the numbers of the form p^{4k+1}*m^2 with p a prime congruent to 1 modulo 4 and m a positive integer coprime with p. The odd perfect numbers are all of this form.

See A228058 for the terms where m > 1. - Antti Karttunen, Apr 22 2019

LINKS

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

Index entries for sequences related to sigma(n)

EXAMPLE

For n=3 one has a(3)=17 since sigma(17) = 18 = 4*4 +2 is congruent to 2 modulo 4

MAPLE

with(numtheory): genodd := proc(b) local n, s, d; for n from 1 to b by 2 do s := sigma(n);

if modp(s, 4)=2 then print(n); fi; od; end;

MATHEMATICA

Select[Range[1, 501, 2], Mod[DivisorSigma[1, #], 4]==2&] (* Harvey P. Dale, Nov 12 2017 *)

PROG

(PARI) forstep(n=1, 10^3, 2, if(2==(sigma(n)%4), print1(n, ", "))) \\ Joerg Arndt, May 27 2011

(PARI) list(lim)=my(v=List()); forstep(e=1, logint(lim\=1, 5), 4, forprimestep(p=5, sqrtnint(lim, e), 4, my(pe=p^e); forstep(m=1, sqrtint(lim\pe), 2, if(m%p, listput(v, pe*m^2))))); Set(v) \\ Charles R Greathouse IV, Feb 16 2022

CROSSREFS

Subsequence of A191217.

Cf. A228058, A324898 (subsequences).

Cf. A000203, A191219, A324647, A324718, A324719.

KEYWORD

nonn,easy

AUTHOR

Luis H. Gallardo, May 26 2011

STATUS

approved

A105824

a(n) = sigma(n) mod 4.

+10
11

1, 3, 0, 3, 2, 0, 0, 3, 1, 2, 0, 0, 2, 0, 0, 3, 2, 3, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 3, 0, 2, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 3, 2, 2, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 3, 0, 1, 2, 0, 0, 2, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences related to sums of divisors

FORMULA

a(n) = A010873(A000203(n)). - Antti Karttunen, Nov 07 2017

MAPLE

A105824:= n-> (numtheory[sigma](n) mod 4):

seq (A105824(n), n=1..105); # Jani Melik, Jan 26 2011

MATHEMATICA

Table[Mod[DivisorSigma[1, n], 4], {n, 100}] (* Wesley Ivan Hurt, Nov 07 2017 *)

PROG

(PARI) a(n)=sigma(n)%4

CROSSREFS

Cf. A000203, A010873, A248150, A072461, A072462, A191217.

Sequences sigma(n) mod k: A053866 (k=2), A074941 (k=3), A105824 (k=4), A105825 (k=5), A084301 (k=6), A105826 (k=7), A105827 (k=8).

KEYWORD

easy,nonn

AUTHOR

Shyam Sunder Gupta, May 05 2005

STATUS

approved

A230779

Numbers which are uniquely decomposable into a sum of two squares, the unique decomposition being with two distinct nonzero squares.

+10
8

5, 10, 13, 17, 20, 26, 29, 34, 37, 40, 41, 45, 52, 53, 58, 61, 68, 73, 74, 80, 82, 89, 90, 97, 101, 104, 106, 109, 113, 116, 117, 122, 136, 137, 146, 148, 149, 153, 157, 160, 164, 173, 178, 180, 181, 193, 194, 197, 202, 208, 212, 218, 226, 229, 232, 233, 234, 241, 244, 245, 257, 261, 269, 272

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Numbers with exactly one prime factor of form 4*k+1, that must have multiplicity one, and no prime factor of the form 4*k+3 with odd multiplicity. There is thus no square in the sequence.

These are the primitive elements of A004431, the integers which are the sum of two nonzero distinct squares.

Numbers such that A004018(a(n)) = 8.

The square of these numbers is also uniquely decomposable into a sum of two squares, thus this sequence is a subsequence of A084645.

Also a subsequence of A191217: the two sequences are equal up to a(76) = 320, then A191217(77) = 325, the value which is missing from this sequence, as a(77) = 328 = A191217(78). (3125 is also missing from this sequence, although present in A191217, and it is the 31st such number). - Corrected by Antti Karttunen, May 14 2022.

Numbers n such that n^3 is the sum of two nonzero squares in exactly two ways. - Altug Alkan, Jul 01 2016

Sequence A125022 (numbers with a unique partition as the sum of 2 squares x^2 + y^2), but without any terms of A028982 (squares and twice squares) that might occur there. - Antti Karttunen, May 14 2022

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000 (extending the previous b-file from Jean-Christophe Hervé, which contained terms up to the 1647th term 10009, but accidentally missed terms 8992 and 9376)

Eric Weisstein's World of Mathematics, Square Number

Wikipedia (fr), Théorème des deux carrés

G. Xiao, Two squares

Index entries for sequences related to sums of squares

FORMULA

Terms are obtained by the products A125853(k)*A002144(p) for k, p > 0, ordered by increasing values.

{k | A004018(k) = 8}.

EXAMPLE

a(1) = 5 = 4+1, a(2) = 10 = 9+1, a(3) = 13 = 9+4. However 2 = 1+1, 4 = 4+0, 8 = 4+4 are excluded because the unique decomposition of these numbers in two squares is not with two distinct nonzero squares; 25, 50, 100 are also excluded because there are two decompositions of these numbers in two squares (including one with equal or zero squares).

PROG

(PARI) isok(n) = {f = factor(n); nb1 = 0; for (i=1, #f~, p = f[i, 1]; ep = f[i, 2]; if (p % 4 == 1, nb1 ++; if (ep != 1, return (0))); if (p % 4 == 3, if (ep % 2, return (0))); ); return (nb1 == 1); } \\ Michel Marcus, Nov 17 2013

CROSSREFS

Cf. A001481, A004431, A002144, A028982, A353813 (characteristic function).

Subsequence of A004431, of A084645, of A125022, and of A191217.

Cf. A004018, A022554 (A004018 = 0), A125853 (A004018 = 4).

KEYWORD

nonn

AUTHOR

Jean-Christophe Hervé, Nov 16 2013

STATUS

approved

A353812

a(n) = 1 if sigma(n) is of the form 4m+2, otherwise 0.

+10
7

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

LINKS

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

Index entries for characteristic functions

Index entries for sequences related to sigma(n)

FORMULA

a(n) = A347870(n) - A353811(n).

a(n) >= A353813(n).

MATHEMATICA

a[n_] := If[Mod[DivisorSigma[1, n], 4] == 2, 1, 0]; Array[a, 100] (* Amiram Eldar, May 13 2022 *)

PROG

(PARI) A353812(n) = (2==(sigma(n)%4));

CROSSREFS

Characteristic function of A191217.

Cf. A000203, A347870, A353811.

Differs from A353813 for the first time at n=325, where a(325) = 1, while A353813(325) = 0.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 13 2022

STATUS

approved

A332226

Numbers k such that sigma(k) is congruent to 2 modulo 8.

+10
6

10, 17, 20, 26, 40, 41, 52, 58, 73, 74, 80, 89, 90, 97, 104, 106, 113, 116, 122, 137, 148, 153, 160, 180, 193, 202, 208, 212, 218, 232, 233, 234, 241, 244, 257, 281, 296, 298, 313, 314, 320, 325, 337, 346, 353, 360, 362, 369, 394, 401, 404, 409, 416, 424, 433, 436, 449, 457, 458, 464, 468, 488, 490, 521, 522, 538, 554

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..25000

Index entries for sequences related to sigma(n)

PROG

(PARI) isA332226(n) = (2==(sigma(n)%8));

CROSSREFS

Cf. A000203.

Subsequence of A191217.

Subsequences: A332227 (odd terms), A332228.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 13 2020

STATUS

approved

A354106

Numbers k for which A354102(k) = A354102(sigma(k)).

+10
6

1, 24, 2475, 2520, 2728, 5347, 6683, 8184, 8307, 8568, 9108, 9306, 10106, 11484, 12974, 16041, 17892, 20049, 23265, 25265, 26199, 30318, 32256, 32435, 38922, 39618, 40918, 44010, 44576, 44872, 50976, 55224, 55720, 56516, 58817, 63720, 63952, 64890, 65689, 66528, 67356, 69860, 72072, 73409, 74448, 75795, 79101, 83160

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

No common terms with A006872 in range a(2) .. a(1001).

Among the 1001 initial terms, only 3119744 and 13890816 occur also in A191217.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..1001

Index entries for sequences where odd perfect numbers must occur, if they exist at all

Index entries for sequences related to sigma(n)

PROG

(PARI)

\\ Needs also the program from A267099:

A354102(n) = eulerphi(A267099(n));

isA354106(k) = (A354102(k)==A354102(sigma(k)));

CROSSREFS

Cf. A000010, A000203, A354102.

Cf. also A006872, A191217, A354189, A354362.

KEYWORD

nonn

AUTHOR

Antti Karttunen, May 18 2022

STATUS

approved

A324814

Numbers n > 1 for which sigma(A156552(n)) == 2 (mod 4).

+10
4

6, 14, 15, 18, 33, 35, 38, 42, 54, 58, 65, 69, 70, 75, 77, 90, 93, 102, 106, 110, 119, 130, 143, 145, 150, 162, 165, 174, 177, 185, 190, 209, 217, 221, 230, 231, 234, 242, 245, 273, 285, 287, 290, 294, 299, 305, 323, 330, 338, 350, 357, 375, 378, 390, 407, 410, 414, 434, 437, 450, 455, 465, 469, 473, 483, 493, 494, 507

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Sequence A005940(1+A191217(n)), n >= 1, sorted into ascending order.

LINKS

Table of n, a(n) for n=1..68.

Index entries for sequences related to binary expansion of n

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

PROG

(PARI)

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552 by David A. Corneth

A324824(n) = ((n>1)&&(2==(sigma(A156552(n))%4)));

A324824(n) = (2==(A323243(n)%4)); \\ Alternatively.

for(n=1, oo, if(A324824(n), print1(n, ", ")));

CROSSREFS

Cf. A000203, A005940, A191217, A324812, A324813, A324824 (characteristic function).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 16 2019

STATUS

approved

A324824

a(n) = 1 if n>1 and sigma(A156552(n)) is congruent to 2 modulo 4, otherwise a(n) = 0.

+10
4

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000 (based on Hans Havermann's factorization of A156552)

Index entries for sequences related to binary expansion of n

Index entries for characteristic functions

Index entries for sequences computed from indices in prime factorization

Index entries for sequences related to sigma(n)

PROG

(PARI)

A156552(n) = {my(f = factor(n), p2 = 1, res = 0); for(i = 1, #f~, p = 1 << (primepi(f[i, 1]) - 1); res += (p * p2 * (2^(f[i, 2]) - 1)); p2 <<= f[i, 2]); res}; \\ From A156552 by David A. Corneth

A324824(n) = ((n>1)&&(2==(sigma(A156552(n))%4)));

(PARI) A324824(n) = (2==(A323243(n)%4)); \\ This needs code also from A323243.

CROSSREFS

Characteristic function of A324814.

Cf. A156552, A191217, A324822, A324823.

KEYWORD

nonn

AUTHOR

Antti Karttunen, Mar 16 2019

STATUS

approved

A292763

Numbers m such that sigma(m)+phi(m) == 2 mod 4.

+10
3

1, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 20, 22, 23, 26, 27, 29, 31, 34, 37, 38, 40, 41, 43, 45, 46, 47, 52, 53, 54, 58, 59, 61, 62, 67, 68, 71, 73, 74, 79, 80, 82, 83, 86, 89, 90, 94, 97, 101, 103, 104, 106, 107, 109, 113, 116, 117, 118, 122, 127, 131, 134, 136, 137, 139, 142, 146, 148, 149, 151, 153, 157

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Union of {1}, A191217 and A292762.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, Part I, Part 2, Slides. (Mentions this sequence)

MATHEMATICA

Do[If[Mod[DivisorSigma[1, n] + EulerPhi[n], 4]==2, Print[n]], {n, 1, 10^3}] (* Vincenzo Librandi, Oct 02 2017 *)

PROG

(PARI) isok(m) = ((sigma(m)+eulerphi(m)) % 4) == 2; \\ Michel Marcus, Oct 02 2017

CROSSREFS

Cf. A000010, A000203.

Cf. A191217, A292762.

A065091 (odd primes) is a subsequence. - Michel Marcus, Oct 02 2017

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Sep 26 2017

STATUS

approved

A351550

Numbers of the form 4u+2 in A351551.

+10
2

2, 10, 34, 106, 1666, 8746, 26242, 134946, 2125762, 3997714, 8298346, 156057642, 323814834, 672166026, 3901441050

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

This is the intersection of A016825 and A351551. Possibly after the initial 2 also the intersection of A191217 and A351551, or more precisely, of 2*A191218 and A351551. However, so far no common terms with A351538 encountered.

LINKS

Table of n, a(n) for n=1..15.

CROSSREFS

Cf. A016825, A191217, A191218, A351538, A351551.

Cf. also A349745.

KEYWORD

nonn,more

AUTHOR

Antti Karttunen, Feb 17 2022

STATUS

approved