A325320 -id:A325320 - cp's OEIS frontend

A325375 a(n) = gcd(A325319(n), A325320(n)).

3, 1, 9, 1, 3, 1, 1, 3, 3, 1, 9, 1, 1, 7, 1, 5, 9, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 3, 9, 1, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 9, 1, 3, 5, 3, 3, 9, 1, 1, 1, 3, 1, 3, 1, 5, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 3, 3, 1, 25, 1, 1, 9, 1, 1, 9, 1, 3, 1, 27, 1, 1, 1, 1, 3, 9, 1, 49, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

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

Cf. A228058, A325319, A325320, A325376.

up_to = 25000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n];
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A325319(n) = -A325313(A228058(n));
A325320(n) = -A325314(A228058(n));
A325375(n) = gcd(A325319(n),A325320(n));

a(n) = gcd(A325319(n), A325320(n)).

A228058 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1. (Euler's criteria for odd perfect numbers).

Original entry on oeis.org

45, 117, 153, 245, 261, 325, 333, 369, 405, 425, 477, 549, 605, 637, 657, 725, 801, 833, 845, 873, 909, 925, 981, 1017, 1025, 1053, 1233, 1325, 1341, 1377, 1413, 1421, 1445, 1525, 1557, 1573, 1629, 1737, 1773, 1805, 1813, 1825, 2009, 2057, 2061, 2097, 2169

Offset: 1

T. D. Noe, Aug 13 2013

It has been proved that if an odd perfect number exists, it belongs to this sequence. The first term of the form p^5 * n^2 is 28125 = 5^5 * 3^2, occurring in position 520.
Sequence A228059 lists the subsequence of these numbers that are closer to being perfect than smaller numbers. - T. D. Noe, Aug 15 2013
Sequence A326137 lists terms with at least five distinct prime factors. See further comments there. - Antti Karttunen, Jun 13 2019

T. D. Noe, Table of n, a(n) for n = 1..10000
Charles Greathouse and Eric W. Weisstein, MathWorld: Odd perfect number
Oliver Knill, The oldest open problem in mathematics, Handout for NEU Math Circle, December 2, 2007
P. P. Nielsen, Odd Perfect Numbers Have At Least Nine Distinct Prime Factors, arXiv:math/0602485 [math.NT], 2006.
Wikipedia, Perfect number: Odd perfect numbers
Index entries for sequences where any odd perfect numbers must occur

Subsequence of A191218, and also of A228056 and A228057 (simpler versions of this sequence).
For various subsequences with additional conditions, see A228059, A325376, A325380, A325822, A326137 (with omega(n)>=5), A324898 (conjectured, subsequence if it does not contain any prime powers), A354362, A386425 (conjectured), A386427 (nondeficient terms), A386428 (powerful terms), A386429 U A351574.
Cf. A027748, A124010, A005408, A324647, A325319, A325320, A325375, A325377, A325378, A325379, A325819, A325823, A325824.

import Data.List (partition)
a228058 n = a228058_list !! (n-1)
a228058_list = filter f [1, 3 ..] where
   f x = length us == 1 && not (null vs) &&
         fst (head us) `mod` 4 == 1 && snd (head us) `mod` 4 == 1
         where (us,vs) = partition (odd . snd) $
                         zip (a027748_row x) (a124010_row x)
-- Reinhard Zumkeller, Aug 14 2013

nn = 100; n = 1; t = {}; While[Length[t] < nn, n = n + 2; {p, e} = Transpose[FactorInteger[n]]; od = Select[e, OddQ]; If[Length[e] > 1 && Length[od] == 1 && Mod[od[[1]], 4] == 1 && Mod[p[[Position[e, od[[1]]][[1,1]]]], 4] == 1, AppendTo[t, n]]]; t (* T. D. Noe, Aug 15 2013 *)

up_to = 1000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, Apr 22 2019

From Antti Karttunen, Apr 22 2019 & Jun 03 2019: (Start)
A325313(a(n)) = -A325319(n).
A325314(a(n)) = -A325320(n).
A001065(a(n)) = A325377(n).
A033879(a(n)) = A325379(n).
A034460(a(n)) = A325823(n).
A325814(a(n)) = A325824(n).
A324213(a(n)) = A325819(n).
(End)

Note in parentheses added to the definition by Antti Karttunen, Jun 03 2019

A325314 a(n) = n - A162296(n), where A162296(n) is the sum of divisors of n that have a square factor.

Original entry on oeis.org

1, 2, 3, 0, 5, 6, 7, -4, 0, 10, 11, -4, 13, 14, 15, -12, 17, -9, 19, -4, 21, 22, 23, -24, 0, 26, -9, -4, 29, 30, 31, -28, 33, 34, 35, -43, 37, 38, 39, -32, 41, 42, 43, -4, -9, 46, 47, -64, 0, -25, 51, -4, 53, -54, 55, -40, 57, 58, 59, -36, 61, 62, -9, -60, 65, 66, 67, -4, 69, 70, 71, -111, 73, 74, -25, -4, 77, 78, 79, -88, -36, 82, 83

Offset: 1

Antti Karttunen, Apr 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A013661, A033879, A162296, A228058, A325313, A325315, A325320.

Array[# - DivisorSigma[1, #] + Times @@ (1 + FactorInteger[#][[;; , 1]]) - Boole[# == 1] &, 83] (* Michael De Vlieger, Jun 06 2019, after Amiram Eldar at A162296 *)

A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));

a(n) = n - A162296(n).
a(n) = A033879(n) + A325313(n).
a(A228058(n)) = -A325320(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 1 - zeta(2)/2 = 0.1775329665... . - Amiram Eldar, Feb 22 2024

A325313 a(n) = A048250(n) - n, where A048250(n) is the sum of squarefree divisors of n.

Original entry on oeis.org

0, 1, 1, -1, 1, 6, 1, -5, -5, 8, 1, 0, 1, 10, 9, -13, 1, -6, 1, -2, 11, 14, 1, -12, -19, 16, -23, -4, 1, 42, 1, -29, 15, 20, 13, -24, 1, 22, 17, -22, 1, 54, 1, -8, -21, 26, 1, -36, -41, -32, 21, -10, 1, -42, 17, -32, 23, 32, 1, 12, 1, 34, -31, -61, 19, 78, 1, -14, 27, 74, 1, -60, 1, 40, -51, -16, 19, 90, 1, -62, -77, 44, 1, 12, 23, 46

Offset: 1

Antti Karttunen, Apr 21 2019

sign

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A033879, A048250, A126795, A228058, A325314, A325315, A325319, A325320.

Array[DivisorSum[#, # &, SquareFreeQ] - # &, 86] (* Michael De Vlieger, Apr 21 2019 *)

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);

a(n) = A048250(n) - n.
a(n) = A325314(n) - A033879(n).
a(A228058(n)) = -A325319(n).

A325379 a(n) = A033879(A228058(n)).

Original entry on oeis.org

12, 52, 72, 148, 132, 216, 172, 192, 84, 292, 252, 292, 412, 476, 352, 520, 432, 640, 592, 472, 492, 672, 532, 552, 748, 412, 672, 976, 732, 576, 772, 1132, 1048, 1128, 852, 1284, 892, 952, 972, 1324, 1460, 1356, 1624, 1720, 1132, 1152, 1192, -36, 1660, 1272, 1068, 1332, 1812, 1372, 1888, 1392, 2116, 1452, 1972, 2040, 1552, 2116

Offset: 1

Antti Karttunen, Apr 22 2019

sign

The negative terms -36, -1692, -2388, -34944, -16596, -38628, -512, ..., occur at n = 48, 378, 1744, 2255, 2745, 2870, 3555, ..., where A228058(n) is 2205, 19845, 108045, 143325, 178605, 187425, 236925, ..., one of the odd abundant numbers, A005231.

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

Cf. A005231, A033879, A228058, A228059, A325310, A325319, A325320, A325378.

A033879(n) = (n+n-sigma(n));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
k=0; n=0; while(k<100,n++; if(isA228058(n), k++; print1(A033879(n), ", ")));

a(n) = A033879(A228058(n)).
a(n) = A325319(n) - A325320(n).
A001511(abs(a(n))) = A325310(A228058(n)), assuming there are no odd perfect numbers, in which case A001511(abs(a(n))) >= 3 for all n. That is, all terms are multiples of 4.

A325319 a(n) = -A325313(A228058(n)).

Original entry on oeis.org

21, 61, 81, 197, 141, 241, 181, 201, 381, 317, 261, 301, 533, 525, 361, 545, 441, 689, 761, 481, 501, 697, 541, 561, 773, 997, 681, 1001, 741, 1305, 781, 1181, 1337, 1153, 861, 1405, 901, 961, 981, 1685, 1509, 1381, 1673, 1841, 1141, 1161, 1201, 2013, 1685, 1281, 2229, 1341, 1837, 1381, 1913, 1401, 2165, 1461, 2501, 2065, 1561, 2141

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

All terms are of the form 4k+1, A016813.

If a(n) is never equal to A325320(n), then there are no odd perfect numbers.

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

Cf. A016813, A048250, A228058, A325313, A325320, A325378, A325379.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
k=0; n=0; while(k<100,n++; if(isA228058(n), k++; print1(-A325313(n), ", ")));

a(n) = -A325313(A228058(n)) = A228058(n) - A048250(A228058(n)).

a(n) = A325320(n) + A325379(n) = A325378(n) - A325320(n).

A325378 a(n) = A162296(A228058(n)) - A048250(A228058(n)).

Original entry on oeis.org

30, 70, 90, 246, 150, 266, 190, 210, 678, 342, 270, 310, 654, 574, 370, 570, 450, 738, 930, 490, 510, 722, 550, 570, 798, 1582, 690, 1026, 750, 2034, 790, 1230, 1626, 1178, 870, 1526, 910, 970, 990, 2046, 1558, 1406, 1722, 1962, 1150, 1170, 1210, 4062, 1710, 1290, 3390, 1350, 1862, 1390, 1938, 1410, 2214, 1470, 3030

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

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

Cf. A048250, A162296, A228058, A325319, A325320, A325379.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
k=0; n=0; while(k<100,n++; if(isA228058(n), k++; print1(A162296(n) - A048250(n),", ")));

a(n) = A162296(A228058(n)) - A048250(A228058(n)).

a(n) = A325319(n) + A325320(n).

A325824 a(n) = A325814(A228058(n)).

Original entry on oeis.org

27, 75, 99, 203, 171, 255, 219, 243, 171, 335, 315, 363, 539, 539, 435, 575, 531, 707, 767, 579, 603, 735, 651, 675, 815, 507, 819, 1055, 891, 675, 939, 1211, 1343, 1215, 1035, 1419, 1083, 1155, 1179, 1691, 1547, 1455, 1715, 1859, 1371, 1395, 1443, 759, 1775, 1539, 1179, 1611, 1935, 1659, 2015, 1683, 2219, 1755, 2507, 2175, 1875, 2255

Offset: 1

Antti Karttunen, May 23 2019

sign

First negative term occurs as a(16307) = -210973, with A228058(16307) = 1289925. The next negative terms occurs as a(20807) = -242901, with A228058(20807) = 1686825.

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

Cf. A228058, A325379, A325814, A325822, A325823.

Cf. also A325320.

up_to = 10000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, May 23 2019
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325824(n) = A325814(A228058(n));

a(n) = A325814(A228058(n)).

a(n) = A325379(n) + A325823(n).

A325376 Terms k of A228058 such that gcd(k - A048250(k), A162296(k) - k) = A162296(k) - k.

Original entry on oeis.org

153, 477, 801, 1773, 2097, 2421, 3725, 4041, 4689, 4753, 5013, 5337, 6309, 6957, 7281, 7929, 8577, 8725, 9549, 9873, 11225, 11493, 13437, 14357, 14409, 14733, 15381, 17001, 17973, 18621, 19269, 19917, 21213, 21537, 23481, 24777, 25101, 25749, 26073, 26225, 26721, 27369, 28989, 29161, 29313, 29961, 31225, 32229, 32553, 33849

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

Also, terms of this sequence are A228058(k) for all such k that A325375(k) = A325320(k).
In range 1 .. 2^27 there are no such terms k of A228058 that gcd(k-A048250(k), A162296(k)-k) = k - A048250(k).
If any odd perfect number exists, then it must occur in this sequence, but should also satisfy the other condition given above.

Antti Karttunen, Table of n, a(n) for n = 1..12532
Index entries for sequences where any odd perfect numbers must occur

Cf. A048250, A162296, A228058, A325313, A325314, A325320, A325375, A325822.

A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
k=0; n=0; while(k<100,n++; if(isA228058(n) && (gcd(n-A048250(n),A162296(n)-n) == A162296(n)-n),k++; print1(n,", ")));

A325823 Sum of unitary proper divisors of A228058(n): a(n) = A034460(A228058(n)).

Original entry on oeis.org

15, 23, 27, 55, 39, 39, 47, 51, 87, 43, 63, 71, 127, 63, 83, 55, 99, 67, 175, 107, 111, 63, 119, 123, 67, 95, 147, 79, 159, 99, 167, 79, 295, 87, 183, 135, 191, 203, 207, 367, 87, 99, 91, 139, 239, 243, 251, 795, 115, 267, 111, 279, 123, 287, 127, 291, 103, 303, 535, 135, 323, 139, 327, 187, 715, 111, 119, 347, 359, 363, 123, 383

Offset: 1

Antti Karttunen, May 23 2019

nonn

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

Cf. A034448, A034460, A228058, A325379, A325824.

Cf. also A325319, A325320.

up_to = 10000;
isA228058(n) = if(!(n%2)||(omega(n)<2),0,my(f=factor(n),y=0); for(i=1,#f~,if(1==(f[i,2]%4), if((1==y)||(1!=(f[i,1]%4)),return(0),y=1), if(f[i,2]%2, return(0)))); (y));
A228058list(up_to) = { my(v=vector(up_to), k=0, n=0); while(kA228058(n), k++; v[k] = n)); (v); };
v228058 = A228058list(up_to);
A228058(n) = v228058[n]; \\ Antti Karttunen, May 23 2019
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); }; \\ After code in A034448
A034460(n) = (A034448(n) - n);
A325823(n) = A034460(A228058(n));

a(n) = A034460(A228058(n)).

a(n) = A325824(n) - A325379(n).

cp's OEIS Frontend

A325375 a(n) = gcd(A325319(n), A325320(n)).

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A228058 Odd numbers of the form p^(1+4k) * r^2, where p is prime of the form 1+4m, r > 1, and gcd(p,r) = 1. (Euler's criteria for odd perfect numbers).

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A325314 a(n) = n - A162296(n), where A162296(n) is the sum of divisors of n that have a square factor.

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A325313 a(n) = A048250(n) - n, where A048250(n) is the sum of squarefree divisors of n.

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A325379 a(n) = A033879(A228058(n)).

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A325319 a(n) = -A325313(A228058(n)).

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A325378 a(n) = A162296(A228058(n)) - A048250(A228058(n)).

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A325824 a(n) = A325814(A228058(n)).

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