A006038 -id:A006038 - cp's OEIS frontend

A136479 Odd primitive abundant numbers n such that n = x^2 + x + y^2 with y^2 < 2*x and x and y primes, subsequence of A136476.

9555, 51765, 17787315

Offset: 1

Pierre CAMI, Dec 31 2007

x values in A136477.

			9555 = 97^2+97+7^2 and 9555 odd primitive abundant number A136476(1), 97 and 7 primes so A136479(1) = 9555.

Cf. A006038, A136476, A136477, A136478.

A136477 Numbers x such that for some y < sqrt(2x), x^2 + x + y^2 is an odd primitive abundant number, A136476(n).

Original entry on oeis.org

97, 112, 122, 135, 144, 179, 202, 207, 214, 217, 227, 354, 359, 477, 507, 569, 612, 632, 639, 732, 832, 2124, 2359, 2362, 2440, 2466, 2517, 2970, 3097, 3247, 3342, 3367, 3374, 3419, 3425, 3518, 3545, 3562, 3644, 3672, 3699, 3789, 3879, 3969, 3985, 4050

Offset: 1

Pierre CAMI, Dec 31 2007

nonn

The corresponding y-values are listed in A136478. (Unlike the x-values listed here, y is not increasing with A136476(n).)

			97^2 + 97 + 7^2 = 9555 = A136476(1) is an odd primitive abundant number, so a(1) = 97.

Cf. A006038, A136476, A136478, A136479.

is(x,n=x^2+x+1,f)={forstep(y=1,sqrtint(2*x),2, sigma(n+=y*4-4,-1)>2 || next; for(i=1, #f=factor(n)[,1], sigma(n\f[i], -1)>2 && next(2)); return(1))} \\ M. F. Hasler, Feb 22 2017

Edited by M. F. Hasler, Feb 22 2017

A136478 Smallest y such that for x = A136477(n), x^2 + x + y^2 is an odd primitive abundant number, A136476(n).

Original entry on oeis.org

7, 7, 3, 15, 15, 15, 7, 3, 5, 7, 3, 15, 15, 27, 3, 15, 3, 27, 15, 27, 13, 3, 49, 17, 55, 27, 27, 15, 53, 77, 63, 77, 15, 45, 15, 69, 45, 77, 15, 57, 75, 27, 75, 63, 55, 75, 49, 85, 7, 3

Offset: 1

Pierre CAMI, Dec 31 2007

nonn

See A136477 and A136476 for the x-values and the abundant numbers x^2 + x + y^2.

			97^2+97+7^2 = 9555 = A136476(1) is an odd primitive abundant number, therefore a(1) = 7.

Cf. A006038, A136476, A136477, A136479.

{for(x=1, 5000, my(n=x^2+x+1, f); forstep(y=1, sqrtint(2*x), 2, sigma(n+=y*4-4, -1)>2 || next; for(i=1, #f=factor(n)[, 1], sigma(n\f[i], -1)>2 && next(2)); print1(y","); break))} \\ M. F. Hasler, Feb 22 2017

a(n) = sqrt(A136476(n) - A136477(n)^2 - A136477(n)). - M. F. Hasler, Feb 22 2017

Edited by M. F. Hasler, Feb 22 2017

A275449 Least odd primitive abundant number with n prime factors, counted with multiplicity.

Original entry on oeis.org

945, 7425, 81081, 78975, 1468935, 6375105, 85930875, 307879299, 1519691625, 8853249375, 17062700625, 535868474337, 2241870572475, 12759034818375, 64260996890625, 866566808687853, 2964430488515625, 23849823423763953, 100139192108634825, 772934641006640625, 2696807941801171875

Offset: 5

M. F. Hasler, Jul 27 2016

nonn

See A188342 = (945, 3465, 15015, 692835, 22309287, ...) for the least odd primitive abundant number (A006038) with n distinct prime factors.
At least up to a(11), the greatest prime factor gpf(a(n)) = Q(a(n)/gpf(a(n))), where Q(N) = floor(sigma(N)/(2N-sigma(N))). In general one has to apply the precprime() function A007917 to this integer.
The above holds also for a(12)-a(15). Lars Blomberg, Apr 09 2018

			We have:   a(5) = 945 = 3^3 * 5   * 7,
          a(6) = 7425 = 3^3 * 5^2 * 11,
         a(7) = 81081 = 3^4 *  7  * 11 * 13,
        a(8) =  78975 = 3^5 * 5^2 * 13,
       a(9) = 1468935 = 3^6 * 5   * 13 * 31,
      a(10) = 6375105 = 3^7 * 5   * 11 * 53,
     a(11) = 85930875 = 3^6 * 5^3 * 23 * 41,
    a(12) = 307879299 = 3^7 * 7^2 * 13^2 * 17,
   a(13) = 1519691625 = 3^8 * 5^3 * 17 * 109,
   a(14) = 8853249375 = 3^8 * 5^4 * 17 * 127,
  a(15) = 17062700625 = 3^9 * 5^4 * 19 * 73.

Cf. A006038, A188342, A188439.

a(n)=for(i=1,#A=A006038,bigomega(A[i])==n&&return(A[i])) \\ Provided that A006038 is defined as a set with enough elements. - M. F. Hasler, Jul 27 2016

generate(A, B, n) = A=max(A, 3^n); (f(m, p, k) = my(list=List()); if(sigma(m) > 2*m, return(list)); if(k==1, forprime(q=max(p, ceil(A/m)), B\m, my(t=m*q); if(sigma(t) > 2*t, my(F=factor(t)[,1], ok=1); for(i=1, #F, if(sigma(t\F[i], -1) > 2, ok=0; break)); if(ok, listput(list, t)))), forprime(q = p, sqrtnint(B\m, k), list=concat(list, f(m*q, q, k-1)))); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=3^n, y=2*x); while(1, my(v=generate(x, y, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Feb 10 2024

a(12)-a(15) from Lars Blomberg, Apr 09 2018

a(16)-a(25) from Daniel Suteu, Feb 10 2024

A287728 Number of odd primitive abundant numbers with n prime factors, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 0, 121, 15772, 102896101, 3475842606319962

Offset: 1

M. F. Hasler, May 30 2017

There is no odd abundant number (A005231) with less than 5 prime factors counted with multiplicity (cf. A001222).
Sequence A188439 lists the odd primitive abundant numbers (A006038) sorted by increasing number of distinct prime factors. It is known that there are 576 such terms with r = 3 distinct prime factors, but their number for any larger r = omega(x) appears to be unknown as of today.
It appears that a(n) is just slightly larger than A287590(n), the number of squarefree odd primitive abundant numbers (A249263) with n prime factors. Those with a prime factor to a higher power become less frequent because there are increasingly many terms of the form m*p_r where m has abundancy slightly less than 2, and p_r can be any prime between gpf(m) and 1/(2/A(m)-1) which becomes very large as A(m) -> 2. This also makes difficult the computation of a(n) for n >= 8: The lexicographic smallest choice of (p_1,...,p_8) has p_7 = 128194589 and then 128194601 <= p_8 <= 566684450325179, and calculation of primepi(566'684'450'325'179) takes very long.

Gianluca Amato, Primitive Weirds and Abundant Numbers, GitHub.
Gianluca Amato, Maximilian F. Hasler, Giuseppe Melfi, and Maurizio Parton, Primitive abundant and weird numbers with many prime factors, arXiv:1802.07178 [math.NT], 2018.

Cf. A005231, A006038, A188439, A249263, A287590, A001222.

SageMath
```
# See GitHub link.
```

a(7) from Gianluca Amato, Jun 26 2017

a(8) from Gianluca Amato, Feb 26 2018

A381547 Odd numbers with an odd number of abundant divisors.

Original entry on oeis.org

945, 1575, 2205, 3465, 4095, 4725, 5355, 5775, 5985, 6435, 6615, 6825, 7245, 7425, 8085, 8415, 8505, 8925, 9135, 9555, 9765, 10395, 11025, 11655, 12285, 12705, 12915, 13545, 14175, 14805, 15015, 16065, 16695, 17955, 18585, 19215, 19635, 19845, 21105, 21735, 21945

Offset: 1

Amiram Eldar, Feb 26 2025

Odd numbers k such that A080224(k) is odd.
Also, odd numbers with an odd sum of abundant divisors.
The odd primitive abundant numbers (A006038) are all terms of this sequence since A080224(A006038(n)) = 1 for all n.

			945 is a term since it is odd, and it has only one abundant divisor, 945 itself.
4725 is a term since it is odd, and it has 3 abundant divisors, 945, 1575 and 4725.
14175 is a term since it is odd, and it has 5 abundant divisors, 945, 1575, 2835, 4725 and 14175.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A080224, A187795.
Intersection of A005408 and A381546.
Subsequence of A005231.
Subsequences: A006038, A381548, A381549.

q[n_] := OddQ[DivisorSum[n, 1 &, DivisorSigma[-1, #] > 2 &]]; Select[Range[1, 22000, 2], q]

isok(k) = if(k % 2, sumdiv(k, d, (sigma(d, -1) > 2)) % 2, 0);

A287646 Irregular triangle read by rows where row n lists all odd primitive abundant numbers with n prime factors, counted with multiplicity.

Original entry on oeis.org

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 8085, 8415, 8925, 9135, 9555, 9765, 11655, 12705, 12915, 13545, 14805, 15015, 16695, 18585, 19215, 19635, 21105, 21945, 22365, 22995, 23205, 24885, 25935, 26145, 26565, 28035, 30555, 31395, 31815, 32445, 33495

Offset: 5

M. F. Hasler, May 30 2017

This triangle is the analog of A188439 for A001222 ("bigomega", total number of prime factors) instead of A001221 ("omega", distinct prime divisors). It starts with row 5, since there is no odd primitive abundant number, N in A006038, with less than A001222(N) = 5 prime factors (counted with multiplicity).
Sequence A287728 gives the row lengths: Row 5 has 121 terms (945, 1575, 2205, 3465, 4095, ..., 430815, 437745, 442365). This mostly equals the initial terms of A006038, except for those with indices {12, 39, 40, 45, 48, 54, ..., 87}. These are in turn mostly (except for the 17th and 18th term) those of the subsequent row 6 which has 15772 terms, (7425, 28215, 29835, 33345, 34155, ..., 13443355695, 13446051465, 13455037365).
Sequences A275449 and A287581 give the smallest and largest* element of each row (*assuming that the largest term in the row is squarefree). Accordingly, row 7 starts with A275449(7) = 81081, and ends with A287581(7) = 1725553747427327895.

Cf. A188439, A001222, A006038, A287728, A275449, A287581, A338133.

A287646_row( r, p=3, a=2, n=1/(a-1))={ r>1 || return(if(n>=p, primes([p,n]))); p(p-1)*a && p-1/p^(r-1)<(p-1)*a,[p^r],[]),ap=1,np=nextprime(p+1)); until( 0, if( (1+1/np)^(r-e) > (aa = a/ap += 1/p^e) && aa > 1, if(n=A287646_row(r-e,np,aa), if(e>1, my(aaa=a/(ap-1/p^e)); n=select(t->sigma(t,-1)1 || n || break; np=nextprime((e=ap=1)+p=np)); S}

A347939 Primitive terms of A347936: terms of A347936 that are not multiples of other terms of A347936.

Original entry on oeis.org

155925, 225225, 259875, 294525, 297675, 363825, 405405, 429975, 496125, 552825, 562275, 571725, 606375, 628425, 694575, 760725, 765765, 921375, 945945, 987525, 1044225, 1167075, 1195425, 1334025, 1412775, 1447875, 1486485, 1507275, 1526175, 1611225, 1640925, 1645875

Offset: 1

Amiram Eldar, Sep 20 2021

nonn

Since any positive odd multiple of a term of A347936 is also a term of A347936, the sequence A347936 consists of the positive odd multiple of this sequence.

			The first 8 terms of this sequence are the same as those of A347936. But A347936(9) = 467775 = 3 * 155925 = 3 * A347936(1) is not a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence A005101, A005231, A347936 and A347938.

Cf. A006038, A187795.

abQ[n_] := DivisorSigma[1, n] > 2*n; s[n_] := DivisorSum[n, # &, abQ[#] &]; q[n_] := s[n] > 2*n && AllTrue[Most @ Divisors[n], ! q[#] &]; Select[Range[1, 300000, 2], q]

A360356 Primitive terms of A360332: terms of A360332 with no proper divisor in A360332.

Original entry on oeis.org

56, 104, 196, 304, 364, 368, 464, 532, 644, 812, 1036, 1184, 1204, 1316, 1376, 1484, 1504, 1696, 1708, 1952, 1988, 2044, 2212, 2492, 2716, 2828, 2884, 2996, 3164, 3496, 3668, 3836, 3892, 4172, 4228, 4408, 4544, 4564, 4672, 4676, 4844, 5056, 5068, 5336, 5404, 5516

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

If m is a term then k*m is a term of A360332 for all k in A320628.

Analogous to primitive abundant numbers (A091191) with divisors that are restricted to numbers that have only nonprime-indexed prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A360332.
Cf. A320628.
Similar sequences: A006038, A091191, A249263, A302574, A360355.

f[p_, e_] := If[PrimeQ[PrimePi[p]], 1, (p^(e + 1) - 1)/(p - 1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; primQ[n_] := s[n] > 2*n && AllTrue[Divisors[n], # == n || s[#] <= 2*# &]; Select[Range[6000], primQ]

isab(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i = 1, #p, if(isprime(primepi(p[i])), 1, (p[i]^(e[i]+1)-1)/(p[i]-1))) > 2*n;}
is(n) = {if(!isab(n), return(0)); fordiv(n, d, if(d < n && isab(d), return(0))); return(1)};

A133595 Odd primitive abundant numbers with no prime factors == -1 (mod 6).

Original entry on oeis.org

10130211, 12090897, 13024557, 42154749, 53545401, 56439747, 59800923, 62228439, 74272653, 82488861, 88277553, 96960591, 98361081, 105363531, 105643629, 114326667, 115727157, 124939269, 140375781, 145199691, 149058819, 173302857, 182670579, 197048943, 202090707

Offset: 1

Pierre CAMI, Dec 27 2007; corrected Jan 02 2008, Jan 09 2008

nonn

			10130211 = 3^3*7^2*13*19*31 has no prime factors == -1 (mod 6), so 10130211 is a term.

Jinyuan Wang, Table of n, a(n) for n = 1..125

Subsequence of A006038.

Several missing terms inserted by Michel Marcus, Sep 15 2019

Several missing terms inserted by Jinyuan Wang, Mar 06 2020

cp's OEIS Frontend

A136479 Odd primitive abundant numbers n such that n = x^2 + x + y^2 with y^2 < 2*x and x and y primes, subsequence of A136476.

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A136477 Numbers x such that for some y < sqrt(2x), x^2 + x + y^2 is an odd primitive abundant number, A136476(n).

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A136478 Smallest y such that for x = A136477(n), x^2 + x + y^2 is an odd primitive abundant number, A136476(n).

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A275449 Least odd primitive abundant number with n prime factors, counted with multiplicity.

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A287728 Number of odd primitive abundant numbers with n prime factors, counted with multiplicity.

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A381547 Odd numbers with an odd number of abundant divisors.

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A287646 Irregular triangle read by rows where row n lists all odd primitive abundant numbers with n prime factors, counted with multiplicity.

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A347939 Primitive terms of A347936: terms of A347936 that are not multiples of other terms of A347936.

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A360356 Primitive terms of A360332: terms of A360332 with no proper divisor in A360332.