A326064 -id:A326064 - cp's OEIS frontend

A325981 Odd composites for which gcd(A325977(n), A325978(n)) is equal to abs(A325977(n)).

45, 495, 585, 765, 855, 1305, 18837, 21525, 31635, 38295, 45315, 50445, 51255, 60435, 63495, 68085, 77265, 96615, 1403115, 2446353, 3411975, 3999465, 4091745, 4233537, 4287255, 4631319, 10813425, 10967085, 11490345, 15578199, 16143309, 16329645, 16633071, 17179515, 17311203, 17355915, 21159075, 21933975, 22579725

Offset: 1

Antti Karttunen, Jun 02 2019

nonn

Provided that A325977 and A325978 are never zero on same n, these are odd composite numbers n such that A325977(n) is not zero and divides A325978(n).
Based on the first 147 terms it seems that this sequence is a subsequence of A072357, that is each term has exactly one prime factor with exponent 2, with one or more other prime factors that are all unitary (i.e., each term satisfies A001222(x) - A001221(x) = 1). On the other hand, it has been proved that no odd perfect number, if such numbers exist at all, can have such a factorization (see A326137 and a link to P. P. Nielsen's paper there).
Nineteen initial terms factorize as:
= 3^2 * 5^1,
= 3^2 * 5^1 * 11^1,
= 3^2 * 5^1 * 13^1,
= 3^2 * 5^1 * 17^1,
= 3^2 * 5^1 * 19^1,
= 3^2 * 5^1 * 29^1,
= 3^2 * 7^1 * 13^1 * 23^1,
= 3^1 * 5^2 * 7^1 * 41^1,
= 3^2 * 5^1 * 19^1 * 37^1,
= 3^2 * 5^1 * 23^1 * 37^1,
= 3^2 * 5^1 * 19^1 * 53^1,
= 3^2 * 5^1 * 19^1 * 59^1,
= 3^2 * 5^1 * 17^1 * 67^1,
= 3^2 * 5^1 * 17^1 * 79^1,
= 3^2 * 5^1 * 17^1 * 83^1,
= 3^2 * 5^1 * 17^1 * 89^1,
= 3^2 * 5^1 * 17^1 * 101^1,
= 3^2 * 5^1 * 19^1 * 113^1,
  1403115 = 3^1 * 5^1 * 7^2 * 23^1 * 83^1,
and the 62nd term as a(62) = 2919199437 = 3^2 * 7^1 * 11^1 * 43^1 * 163^1 * 601^1.
If we select a subsequence of terms for which the quotient A325978(n)/A325977(n) is positive, then we are left with the following numbers: 495, 585, 31635, 38295, 45315, 51255, 60435, 63495, 1403115, 3999465, etc. which is a subsequence of A326070.

Antti Karttunen (terms 1-62) & Giovanni Resta, Table of n, a(n) for n = 1..147
Index entries for sequences where any odd perfect numbers must occur

Cf. A072357, A228058, A325973, A325974, A325975, A325977, A325978, A325979, A326064, A326070, A326071, A326137.

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);
A048250(n) = factorback(apply(p -> p+1,factor(n)[,1]));
A325313(n) = (A048250(n) - n);
A325977(n) = ((A034460(n)+A325313(n))/2);
A162296(n) = sumdiv(n, d, d*(1-issquarefree(d)));
A325314(n) = (n - A162296(n));
A048146(n) = (sigma(n)-A034448(n));
A325814(n) = (n-A048146(n));
A325978(n) = ((A325314(n)+A325814(n))/2);
A325975(n) = gcd(A325977(n), A325978(n));
isA325981(n) = ((n%2)&&!isprime(n)&&(A325975(n)==abs(A325977(n))));
\\ Or alternatively as:
isA325981(n) = if(!(n%2)||isprime(n),0,my(x = A325977(n), y = A325978(n)); (!x&&!y)||(x&&!(y%x)));

A326062 a(1) = gcd((sigma(n)-A032742(n))-n, n-A032742(n)), where A032742 gives the largest proper divisor of n.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 6, 1, 1, 1, 10, 2, 12, 1, 2, 1, 16, 3, 18, 2, 2, 1, 22, 12, 1, 1, 2, 14, 28, 3, 30, 1, 2, 1, 2, 1, 36, 1, 2, 10, 40, 3, 42, 2, 6, 1, 46, 4, 1, 1, 2, 2, 52, 3, 2, 4, 2, 1, 58, 6, 60, 1, 2, 1, 2, 3, 66, 2, 2, 1, 70, 3, 72, 1, 2, 2, 2, 3, 78, 2, 1, 1, 82, 14, 2, 1, 2, 4, 88, 9, 2, 2, 2, 1, 2, 12, 96, 1, 6, 1, 100, 3, 102, 2, 2

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

See comments in A326063 and A326064.

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

Cf. A000203, A032742, A060681, A318505, A326061, A326063, A326064.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060.

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
A326062(n) = gcd((sigma(n)-A032742(n))-n, n-A032742(n));

a(1) = 1; for n > 1, a(n) = gcd(A060681(n), A318505(n)).

a(n) = gcd((A000203(n)-A032742(n))-n, n-A032742(n)).

A326074 Numbers n for which A326073(n) is equal to abs(1+A326146(n)).

Original entry on oeis.org

3, 6, 28, 221, 391, 496, 1189, 1421, 1961, 2419, 5429, 7811, 8128, 11659, 15049, 18871, 36581, 44461, 48689, 57721, 80851, 86519, 98431, 107869, 117739, 146171, 169511, 181829, 207761, 235421, 240199, 280151, 312131, 387349, 437669, 497951, 525991, 637981, 685801, 735349, 752249, 804101, 885119, 950821, 1009009

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

Numbers n such that 1+(A001065(n)-A020639(n)) is not zero and divides 1+n-A020639(n).
Note that whenever n is even, then the above condition reduces to "(even) numbers n such that A048050(n) is not zero and divides n-1", which is a condition satisfied only by the even terms of A000396.
a(375) = 360866239 = 449 * 509 * 1579 is the first term with more than two distinct prime factors, the second is a(392) = 413733139 = 199 * 239 * 8699, and the third is a(485) = 718660177 = 41 * 853 * 20549.
Question: Are any of these terms present also in A326064 and A326148? None of the first 564 terms are. If such intersections are empty, then there are no odd perfect numbers.
If one selects only semiprimes from this sequence, one is left with 6, 221, 391, 1189, 1961, 2419, 5429, 7811, 11659, 15049, 18871, 36581, ... (555 terms out of the first 564 terms). Their smaller prime factors are: 2, 13, 17, 29, 37, 41, 61, 73, 89, 101, 113, 157, 173, 181, 197, 233, 241, 257, 269, 281, 313, ... while their larger prime factors are: 3, 17, 23, 41, 53, 59, 89, 107, 131, 149, 167, 233, 257, 269, 293, 347, 359, 383, 401, 419, 467, 503, 521, ..., and both sequences of primes seem to be monotonic.

Cf. A000203, A001065, A020639, A048050, A061228, A325960, A325961, A326064, A326073, A326146, A326148.

Cf. A000396 (a subsequence, the even terms of this sequence if there are no odd perfect numbers).

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326073(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n);
A326146(n) = (sigma(n)-A020639(n)-n);
isA326074(n) = (A326073(n)==abs(1+A326146(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));

A326148 Odd numbers > 1, not powers of primes, for which A326147(n) is equal to abs(A326146(n)).

Original entry on oeis.org

15, 91, 207, 703, 847, 1023, 1891, 2701, 2725, 5551, 12403, 15043, 18721, 19359, 38503, 49141, 79003, 88831, 104653, 146611, 148951, 188191, 218791, 226801, 269011, 286903, 346957, 385003, 497503, 597871, 665281, 721801, 736291, 765703, 873181, 954271, 1056331, 1207359, 1314631, 1345873, 1373653, 1537381, 1755001

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

Odd numbers > 1, not powers of primes, for which A326146(n) [= (sigma(n)-A020639(n)-n)] is not zero and divides n-A020639(n).
Question: Are any of these terms present also in A326064 and A326074? None of the first 519 terms are. If such intersections are empty, then there are no odd perfect numbers.
Of the first 519 terms, 485 are semiprimes.

Cf. A020639, A046666, A326064, A326074, A326146, A326147.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326146(n) = (sigma(n)-A020639(n)-n);
A326147(n) = gcd(n-A020639(n), sigma(n)-A020639(n)-n);
isA326148(n) = if((n>1)&&(n%2)&&!isprimepower(n), my(s=factor(n)[1, 1], t=n-s, u=sigma(n)-s-n); (u && !(t%u)), 0);

A326063 Composite numbers n such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Original entry on oeis.org

4, 6, 9, 25, 28, 49, 117, 121, 169, 289, 361, 496, 529, 775, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 8128, 9409, 10201, 10309, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729, 51529, 52441

Offset: 1

Antti Karttunen, Jun 06 2019

nonn

Composite numbers n such that A318505(n) [sum of divisors of n excluding n itself and the second largest of them, A032742(n)] divides A060681(n) [the largest difference between consecutive divisors of n, = n - A032742(n)].
Numbers k such that A326062(k) = A318505(k).
Question: Is it possible that this sequence could contain a term with more than one non-unitary prime factor? If not, then there are no odd perfect numbers. (See e.g., A326137).

			For n = 9 = 3*3, its divisors are [1, 3, 9], thus A318505(9) = 1 and A060681(9) = 9-3 = 6, and 1 divides 6, so 9 is included, like all squares of primes.
For n = 117 = 3^2 * 13,its divisors are [1, 3, 9, 13, 39, 117], thus A318505(117) = 1+3+9+13 = 26 and A060681(117) = (117-39) = 78, which is a multiple of 26, thus 117 is included in the sequence.

Cf. A000203, A001065, A032742, A060681, A318505, A325981, A326062, A326137.

Subsequences: A000396, A001248, A326064 (odd terms that are not squares of primes).

A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
isA326063(n) = (gcd((sigma(n)-A032742(n))-n, n-A032742(n)) == (sigma(n)-A032742(n))-n);

A060681(n) = (n-A032742(n));
A318505(n) = if(1==n,0,(sigma(n)-A032742(n))-n);
isA326063(n) = { my(t=A318505(n)); (t && !(A060681(n)%t)); };

cp's OEIS Frontend

A325981 Odd composites for which gcd(A325977(n), A325978(n)) is equal to abs(A325977(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A326062 a(1) = gcd((sigma(n)-A032742(n))-n, n-A032742(n)), where A032742 gives the largest proper divisor of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A326074 Numbers n for which A326073(n) is equal to abs(1+A326146(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A326148 Odd numbers > 1, not powers of primes, for which A326147(n) is equal to abs(A326146(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A326063 Composite numbers n such that (A001065(n) - A032742(n)) divides (n - A032742(n)), where A032742 gives the largest proper divisor, and A001065 is the sum of proper divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI