A326074 -id:A326074 - cp's OEIS frontend

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

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);

A326146 a(n) = sigma(n) - A020639(n) - n, where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.

Original entry on oeis.org

-1, -1, -2, 1, -4, 4, -6, 5, 1, 6, -10, 14, -12, 8, 6, 13, -16, 19, -18, 20, 8, 12, -22, 34, 1, 14, 10, 26, -28, 40, -30, 29, 12, 18, 8, 53, -36, 20, 14, 48, -40, 52, -42, 38, 30, 24, -46, 74, 1, 41, 18, 44, -52, 64, 12, 62, 20, 30, -58, 106, -60, 32, 38, 61, 14, 76, -66, 56, 24, 72, -70, 121, -72, 38, 46, 62, 12, 88

Offset: 1

Antti Karttunen, Jun 10 2019

sign

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

Cf. A000203, A001065, A020639, A061228, A105086, A326074, A326147, A326148.

A020639(n) = if(1==n, n, factor(n)[1, 1]);
A326146(n) = (sigma(n)-A020639(n)-n);

a(n) = A000203(n) - A020639(n) - n = A000203(n) - A061228(n).
a(n) = A001065(n) - A020639(n).
For n > 1, a(n) = A105086(n) - n.

A326073 a(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n (and 1 for 1), and sigma is the sum of divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 7, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen, Jun 10 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A020639, A326074.

Cf. also A009194, A325385, A325813, A325975, A326046, A326047, A326048, A326056, A326057, A326060, A326062, A326129, A326130, A326140, A326144, A326147.

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

a(n) = gcd(1+n-A020639(n), 1+A000203(n)-A020639(n)-n).

cp's OEIS Frontend

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

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

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A326146 a(n) = sigma(n) - A020639(n) - n, where A020639 gives the smallest prime factor of n, and sigma is the sum of divisors of n.

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A326073 a(n) = gcd(1+n-A020639(n), 1+sigma(n)-A020639(n)-n), where A020639 gives the smallest prime factor of n (and 1 for 1), and sigma is the sum of divisors of n.

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