A228058 -id:A228058 - cp's OEIS frontend

A325320 Sum of proper divisors of A228058(n) that are not squarefree; a(n) = -A325314(A228058(n)).

9, 9, 9, 49, 9, 25, 9, 9, 297, 25, 9, 9, 121, 49, 9, 25, 9, 49, 169, 9, 9, 25, 9, 9, 25, 585, 9, 25, 9, 729, 9, 49, 289, 25, 9, 121, 9, 9, 9, 361, 49, 25, 49, 121, 9, 9, 9, 2049, 25, 9, 1161, 9, 25, 9, 25, 9, 49, 9, 529, 25, 9, 25, 9, 169, 2381, 49, 1449, 9, 9, 9, 1593, 9, 25, 9, 121, 9, 49, 9, 2889, 9, 25, 289, 9, 2997, 9

Offset: 1

Antti Karttunen, Apr 22 2019

nonn

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

If a(n) is never equal to A325319(n), then there are no odd perfect numbers.

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

Cf. A016813, A162296, A228058, A325313, A325319, A325378, A325379.

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

a(n) = -A325314(A228058(n)) = A162296(A228058(n)) - A228058(n).
a(n) = A325319(n) - A325379(n) = A325378(n) - A325319(n).
a(n) < A001065(A228058(n)) for all 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).

A326137 Numbers with at least five distinct prime factors that satisfy Euler's criterion (A228058) for odd perfect numbers.

Original entry on oeis.org

17342325, 22678425, 31674825, 38686725, 41420925, 45090045, 49358925, 51740325, 54033525, 54695025, 67660425, 68939325, 70703325, 75818925, 76392225, 77106645, 78217425, 81375525, 92400525, 96316605, 97383825, 98750925, 99147825, 102284325, 107694405, 113656725, 115420725, 117890325, 118728225, 120536325, 127766925

Offset: 1

Antti Karttunen, Jun 12 2019

nonn

P. P. Nielsen's 2006 paper shows that any odd perfect number must have at least nine distinct prime factors, thus if such numbers exist at all, they must occur in this sequence.

I conjecture that it is eventually possible to find an easy proof that this sequence has no common terms with A325981, and/or several other sequences (A326064, A326074, A326141, A326148, etc.) listed under index entry "sequences where odd perfect numbers must occur", thus settling the question about the existence of such numbers.

Antti Karttunen, Table of n, a(n) for n = 1..1032; all terms < 2^31
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 A228058.

Cf. A001221, A325981, A326064, A326074, A326141, A326148.

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));
isA326137(n) = ((omega(n)>=5)&&isA228058(n));

A349755 Numbers k for which the 3-adic valuations of k and sigma(k) are equal, and that also satisfy Euler's criterion for odd perfect numbers (see A228058).

Original entry on oeis.org

153, 325, 801, 925, 1525, 1573, 1773, 1825, 2097, 2205, 2425, 2725, 3757, 3825, 3925, 4041, 4477, 4525, 4689, 4825, 5013, 5725, 6025, 6877, 6925, 6957, 7381, 7605, 7825, 7929, 8125, 8425, 8577, 8725, 8833, 9325, 9549, 9873, 9925, 10225, 10525, 10693, 10825, 10933, 11425, 11493, 11737, 12789, 13189, 13437, 13525

Offset: 1

Antti Karttunen, Dec 02 2021

Obviously, all odd perfect numbers x, if such numbers exist at all, have to satisfy not only the famous condition given by Euler (see A228058), but also valuation(sigma(x), p) = valuation(x, p) for all odd primes p = 3, 5, 7, 11, etc. See also comments in A349752.

a(109), a(283), a(440) = 31213, 88837, 146461, are the first terms not occurring in A387162. - Antti Karttunen, Aug 27 2025

Intersection of A228058 and A349749.
Cf. A387162 (subsequence).
Cf. A000203, A007949, A349752.

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));
isA349755(n) = (isA228058(n)&&valuation(sigma(n), 3)==valuation(n, 3));

A354362 Intersection of A228058 and A260021.

Original entry on oeis.org

45, 49005, 597861, 715473, 1538757, 1891593, 1893213, 2714877, 3067713, 3890997, 4126221, 4479057, 5302341, 5465313, 5793525, 5890437, 6008013, 6478461, 6596073, 6882525, 7184133, 7419357, 8595477, 9771597, 10712493, 11300553, 11771001, 11888613, 12123837, 12947121, 13535181, 14240853, 15887421, 16240257, 17181153

Offset: 1

Antti Karttunen, May 24 2022

nonn

Cf. A228058, A260021.
Subsequence of A353679.
Cf. also A354106.

isA354362(n) = (isA228058(n) && isA260021(n)); \\ Uses the programs from A228058 and A260021.

A386429 Odd composites k such that A342926(k) is even and A342926(2*k) is a multiple of 3 and which satisfy Euler's condition for odd perfect numbers (A228058).

Original entry on oeis.org

45, 153, 261, 325, 369, 405, 477, 801, 909, 925, 1017, 1233, 1341, 1377, 1525, 1557, 1573, 1773, 1825, 2097, 2205, 2313, 2349, 2421, 2425, 2529, 2637, 2725, 2853, 3177, 3321, 3501, 3609, 3645, 3757, 3825, 3925, 4041, 4149, 4293, 4477, 4525, 4581, 4689, 4825, 5013, 5121, 5337, 5445, 5553, 5725, 5733, 5769, 5877, 6025

Offset: 1

Antti Karttunen, Aug 18 2025

Sequence contains also some terms of A386428: 28125, 253125, 1378125, 2278125, 3341637, 3403125, 4753125, etc.

Intersection of A228058 and A347874.
Conjectured to be also the intersection of A228058 and A349751.
Setwise difference A228058 \ A351574.
Cf. also A349755, A387162.

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A342926(n) = (A003415(sigma(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));
isA347874(n) = ((n%2)&&!isprime(n)&&!(A342926(n)%2)&&!(A342926(2*n)%3));
isA386429(n) = (isA228058(n) && isA347874(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,", ")));

cp's OEIS Frontend

A325320 Sum of proper divisors of A228058(n) that are not squarefree; a(n) = -A325314(A228058(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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A326137 Numbers with at least five distinct prime factors that satisfy Euler's criterion (A228058) for odd perfect numbers.

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A349755 Numbers k for which the 3-adic valuations of k and sigma(k) are equal, and that also satisfy Euler's criterion for odd perfect numbers (see A228058).

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A354362 Intersection of A228058 and A260021.

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A386429 Odd composites k such that A342926(k) is even and A342926(2*k) is a multiple of 3 and which satisfy Euler's condition for odd perfect numbers (A228058).

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