Sven Simon - cp's OEIS frontend

A267076 Amicable numbers with property that both members of the amicable pair have the same sopfr (as defined in A001414).

98671027044178875, 110707233329488965, 116112602631024285, 138000805229957475, 347701424633005443, 373158429641554557, 581028000695478105, 650656605928265895, 1057340336068271871, 1192941584025936129, 1317952177931347245, 1597858331524012755

Offset: 1

Sven Simon, Jan 10 2016

nonn

Both members of the amicable pair with same sopfr are listed in the sequence but are not necessarily adjacent.
The terms shown have 17, 18 digits (7 terms) and 19 digits (four terms).
Comment from N. J. A. Sloane, Jun 07 2016: (Start)
Sergei Chernykh has conducted several searches for amicable pairs in the 18-digit range (here p and q are the largest prime factors):
1) All pairs of the form (m*p^k1, n*q^k2) where k1 > 1 OR k2 > 1
2) All pairs of the form (m*p, n*q) where m < 2*10^11 AND n < 2*10^11
3) Current exhaustive search has already found all pairs of the form (m*p^k1, n*q^k2) where p < 21818622 for any k1, q, k2
If we combine the results of these searches it is easy to see that the remaining undiscovered pairs can only have the form (m*p, n*q) where their largest prime factors are p > 21818622 and q < 10000000 (2*10^18 / 2*10^11), so they can't have the same sopfr.
This means that all 18-digit members of A267076 are already known. There are no new ones. (End)
Sergei Chernykh with BOINC completed the amicable pairs list with 20 digits.
In their ongoing search for 21-digit amicable numbers  Sergei Chernykh and BOINC have so far found the following numbers: 130292188156891334007, 137813613144174393993, 208010335478545813941, 220018224493331050059, 250217395764910459875, 271313659794405815325, 276109509594435349833, 349735430520058090167, 370496519153268119073, 402333253352868456927, 781727874026691579075, 886084603302962180925. - Sven Simon Feb 26 2021

			To illustrate that 347701424633005443 and 373158429641554557 belong to the sequence:
347701424633005443 = 3*7^3*11*13*19*37*41*59*131*10607, sopfr(n) = 10942.
373158429641554557 = 3*7^2*11*13*19*311*383*839*9349, sopfr(n) = 10942.

Sven Simon, Table of n, a(n) for n = 1..22
Sergei Chernykh, Amicable pairs list

Cf. A001414, A063990, A091340, A259180.

Entry revised by Michel Marcus and N. J. A. Sloane, Jan 23 2016

A261717 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups of which the n-th largest sporadic simple group is a subquotient.

Original entry on oeis.org

19, 12, 2, 16, 4, 10, 7, 1, 5, 7, 3, 1, 3, 1, 3, 4, 5, 3, 1, 3, 4, 2, 1, 2, 2, 1

Offset: 1

Sven Simon, Charles R Greathouse IV, Franklin T. Adams-Watters, Oct 24 2015

The sum of all elements of the sequence is the same as that of A263447.

			The sporadic group Fi_23 is a subquotient of the sporadic groups Fi_23 (itself), Fi_24, the Baby Monster and the Monster, so a(21) = 4.

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites]. Page 238.

Wikipedia, Sporadic group

Cf. A001228, A263447.

A263447 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups which are subquotients of the n-th largest sporadic simple group.

Original entry on oeis.org

1, 2, 1, 1, 1, 3, 3, 1, 5, 3, 1, 1, 4, 3, 7, 6, 4, 5, 4, 1, 6, 12, 6, 9, 12, 20

Offset: 1

Sven Simon and Charles R Greathouse IV, Oct 18 2015

A group is a subquotient of itself, so a(n) >= 1.
It is well-known that a(26) = 20, the so-called "happy family". Trivially a(1) = 1 and a(2) = 2 since M_11 is a subquotient of M_12.
The sequence was generated from the diagram of subquotient relations on page 238 of the ATLAS, together with the update that J_1 is not involved in M (which replaces the single question mark in the table with a plus sign).

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites]. See page 238.

Wikipedia, Sporadic group

Cf. A001228, A261717 (another version).

Terms confirmed by N. J. A. Sloane, Oct 19 2015

A190635 Index of A190637(n) in the sequence A103431 of Gaussian primes.

Original entry on oeis.org

4, 280, 3469989, 34317382, 73414350, 178935505

Offset: 1

Sven Simon, May 15 2011

nonn

			A190637(2)=43 is the a(2)=280th entry in the sequence A103431 (which orders the Gaussian primes of the first quadrant).

Cf. A103431, A190637, A190634.

a(5)-a(6) from Sven Simon, Jun 19 2011

A190634 Indices of primes from A190637.

Original entry on oeis.org

2, 14, 969, 2831, 4050, 6167

Offset: 1

Sven Simon, May 15 2011

nonn

See A190637.

Cf. A103431 (Gaussian primes in first quadrant), A190635 (index of same prime as Gaussian prime), A190637 (primes == 3 mod 4).

a(5)-a(6) from Sven Simon, Jun 19 2011

A190637 Primes p == 3 mod 4 whose index as prime divides their index as a Gaussian prime (in the first quadrant, as defined in A103431, for example).

Original entry on oeis.org

3, 43, 7639, 25703, 38371, 61291

Offset: 1

Sven Simon, May 15 2011

The index of a prime p = 3 mod 4 as a Gaussian prime is well defined, it is summed up by 1 for the complex prime 1+i (as factor of prime 2 = -i*(1+i)^2).
The count of primes (3 mod 4) <= p, which remain unchanged as they cannot be factored further into complex primes 2 times the count of primes (1 mod 4) <= p**2 (such primes p1 are split into two distinct complex primes of the first quadrant with size sqrt(p1)).
As the result from the splitting of the primes 1 mod 4, the indices of primes 3 mod 4 as Gaussian prime grows rather rapidly against their index as normal prime.
Interesting numerical effects: the prime index of 43 is 14, with 3*14+1 = 43. 43 is the upper part of twin prime with 41 (which would be 14*3 - 1 with an index 14, if 1 was counted as prime). 4241 and 4243 are both primes.
The ratio f between both indices can be estimated as f = (p^2 / log(p^2)) / (p / log(p)) = p/2. - Sven Simon, May 26 2011

			The prime 3 has index 2, as a Gaussian prime it has index 4 (the list is 1+i, 1+2i, 2i+1, 3, ...), and 2 divides 4.

Cf. A103431 (Gaussian primes in first quadrant), A190634 (prime index), A190635 (index as Gaussian prime).

Changed name definition which was a bit wrong, the index is not a prime number

A122436 Imaginary parts of the complex numbers defined in A122435. The numbers are ordered by their norm and the size of the real part when the norms are equal.

Original entry on oeis.org

0, 0, 0, 0, 12, 34, 17, 3, 38, 14, 44, 14, 6, 39, 40, 57, 1, 48, 39, 64, 28, 69, 57, 46, 18, 60, 15, 64, 44, 70, 38, 84, 4, 48, 70, 60, 54, 0, 89, 53, 90, 52, 105, 5, 108, 104, 42, 90, 70, 18

Offset: 1

Sven Simon, Sep 04 2006

nonn

Sven Simon, Readable list of A122435/A122436

Cf. A122435.

A122435 Complex numbers a+bi which are divisible by the sum of their complex prime factors with repetition (complex sopfr). The prime factors are from the first quadrant only, as defined in A103431. This sequence gives the real part a of these numbers, the imaginary part b is in A122436.

Original entry on oeis.org

10, 15, 16, 27, 30, 17, 34, 39, 14, 38, 14, 44, 48, 39, 40, 1, 57, 45, 54, 28, 64, 21, 46, 57, 72, 45, 75, 44, 64, 38, 70, 4, 84, 72, 60, 70, 78, 100, 53, 89, 52, 90, 5, 105, 6, 42, 104, 70, 90, 114

Offset: 1

Sven Simon, Sep 04 2006

nonn

In the case of the complex sopfr it seems best to use only primes in the first quadrant because it is easy to get a well-defined function.

			a(1) = 10 is the real part of 10 = (-1)*(1+i)*(1+i)*(1+2i)*(2+i) = (1-i)(5+5i); (1+i)+(1+i)+(1+2i)+(2+i) = 5+5i;
a(5) = 30 is the real part of 30+12i = (-i)*3*(1+i)*(1+i)*(5+2i) = 3*(10+4i); 3+(1+i)+(1+i)+(5+2i) = 10+4i.

Sven Simon, Readable list of A122435/A122436
R. Spira, The Complex Sum Of Divisors, American Mathematical Monthly, 1961 Vol. 68, pp. 120-124.

Cf. A103431, A103432.

A114887 Multiperfect numbers sigma(n) = k*n, which are divisible by the sum of their prime factors without repetition.

Original entry on oeis.org

120, 672, 32760, 2178540, 1379454720, 14182439040, 518666803200, 30823866178560, 71065075104190073088, 154345556085770649600, 9186050031556349952000, 680489641226538823680000

Offset: 1

Sven Simon, Feb 19 2006

nonn

From a list of about 5000 multiperfect numbers, 38 numbers were found with the property, all having k <= 9, the largest was the only one having k=9. A091443 uses sopfr with repetition.

Conjecture: the sequence is finite.

			a(0) = 120 = 2^3*3*5, sopf(120) = 2+3+5 = 10.

Sven Simon, Table of n, a(n) for n = 1..38 [Conjectured to be complete]
Achim Flammenkamp, The Multiply Perfect Numbers Page (See here for the latest information about the search)
Eric Weisstein's World of Mathematics, Multiperfect numbers

Cf. A091443.

Intersection of A007691 and A089352. - Michel Marcus, Oct 08 2017

A114888 Multiperfect numbers, sigma(n) = k*n, which are divisible by their sums of prime factors with and without repetition.

Original entry on oeis.org

1379454720, 14182439040, 680489641226538823680000, 15229814702070563916152832000, 34111227434420791224041472000, 1423615645454579571642097845619575340668304766095889698652160000000

Offset: 1

Sven Simon, Feb 19 2006

Conjecture: the sequence is finite.

			a(1)= 1379454720 = 2^8*3*5*7*19*37*73;
sopfr(1379454720), with repetition = 160 = 2^5*5;
sopfr(1379454720), without repetition = 146 = 2*73.

Achim Flammenkamp, The Multiply Perfect Numbers Page (See here for the latest information about the search)
Eric Weisstein's World of Mathematics, Multiperfect numbers

Cf. A091443, A114887.

cp's OEIS Frontend

User: Sven Simon

A267076 Amicable numbers with property that both members of the amicable pair have the same sopfr (as defined in A001414).

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A261717 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups of which the n-th largest sporadic simple group is a subquotient.

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A263447 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups which are subquotients of the n-th largest sporadic simple group.

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A190635 Index of A190637(n) in the sequence A103431 of Gaussian primes.

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A190634 Indices of primes from A190637.

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A190637 Primes p == 3 mod 4 whose index as prime divides their index as a Gaussian prime (in the first quadrant, as defined in A103431, for example).

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A122436 Imaginary parts of the complex numbers defined in A122435. The numbers are ordered by their norm and the size of the real part when the norms are equal.

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A122435 Complex numbers a+bi which are divisible by the sum of their complex prime factors with repetition (complex sopfr). The prime factors are from the first quadrant only, as defined in A103431. This sequence gives the real part a of these numbers, the imaginary part b is in A122436.

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A114887 Multiperfect numbers sigma(n) = k*n, which are divisible by the sum of their prime factors without repetition.

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A114888 Multiperfect numbers, sigma(n) = k*n, which are divisible by their sums of prime factors with and without repetition.

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