A006038 -id:A006038 - cp's OEIS frontend

A286042 Largest prime factor of A285993(n), the largest odd abundant number (A005231) equal to the product of n consecutive primes.

13, 17, 19, 23, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353

Offset: 5

M. F. Hasler, May 01 2017

nonn

The smallest term is a(5), there is no odd abundant number (A005231) equal to the product of less than 5 consecutive primes.

The corresponding abundant numbers are A285993(n) = prime(k-n+1)*...*prime(k), with prime(k) = a(n).

			For n < 5, there is no odd abundant number equal to the product of n distinct primes.
For 5 <= n <= 8, the largest odd abundant number equal to the product of n consecutive primes is 3*...*a(n) with a(n) = prime(n+1).
For 9 <= n <= 17, the largest odd abundant number equal to the product of n consecutive primes is 5*...*a(n) with a(n) = prime(n+2).
For 18 <= n <= 30, the largest odd abundant number equal to the product of n consecutive primes is 7*...*a(n) with a(n) = prime(n+3).
For 31 <= n <= 45, the largest odd abundant number equal to the product of n consecutive primes is 11*...*a(n) with a(n) = prime(n+4).
For 46 <= n <= 66, the largest odd abundant number equal to the product of n consecutive primes is 13*...*a(n) with a(n) = prime(n+5).

Amiram Eldar, Table of n, a(n) for n = 5..10000

Cf. A285993, A005231, A006038, A007707, A007708, A007741.

a(r,f=vector(r,i,prime(i+1)),o)={ while(sigma(factorback(f),-1)>2, o=f; f=concat(f[^1],nextprime(f[r]+1)));o[#o]} \\ Intentionally throws an error when n < 5.

a(n) = A006530(A285993(n)) >= A151800(a(n-1)) = nextprime(a(n-1)), with strict inequality for n = 9, 18, 31, 46, 67, ..., in which case a(n) = nextprime(nextprime(a(n-1))). This is the case if A285993(n) is in A007741.

a(66) corrected by Amiram Eldar, Sep 24 2019

A303933 a(n) is the number of odd primitive abundant numbers with n distinct prime divisors.

Original entry on oeis.org

0, 0, 8, 576, 3913172

Offset: 1

Jacob Liddy, May 02 2018

The 6th term is possibly 59687996404445, but this is at present unverified. - Jacob Liddy, Oct 16 2018

Valdas Diciunas, On the number of odd primitive abundant numbers with five and six distinct prime factors, Vilnius Conference in Combinatorics and Number Theory, page 12, 2017.

Valdas Diciunas, On the number of odd primitive abundant numbers with five and six distinct prime factors, slides, 2017.
L. E. Dickson, Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors, American Journal of Mathematics 35 (1913), pp. 413-422.
Jacob Liddy, OPANS (Odd Primitive Abundant NumberS), 2018. The downloadable file is a Zip-archive, 640 Megabytes, uncompresses to 10 Gigabytes.

Cf. A005231 (odd abundant), A006038 (odd primitive abundant).

Row lengths of A188439.

A360355 Primitive terms of A360328: terms of A360328 with no proper divisor in A360328.

Original entry on oeis.org

7425, 8415, 46035, 76725, 101475, 182655, 355725, 669735, 1411425, 1606275, 2352375, 2891295, 3592215, 3650625, 4079295, 4861575, 5053455, 5870205, 6093225, 6636465, 6920595, 7732395, 8750835, 9120375, 9783675, 9850005, 9958905, 10155375, 11298375, 11532375, 12120075

Offset: 1

Amiram Eldar, Feb 04 2023

nonn

If m is a term then k*m is a term of A360328 for all k in A076610.

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

Subsequence of A360328.
Cf. A076610.
Similar sequences: A006038, A091191, A249263, A302574, A360356.

f[p_, e_] := If[PrimeQ[PrimePi[p]], (p^(e + 1) - 1)/(p - 1), 1]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; primQ[n_] := s[n] > 2*n && AllTrue[Divisors[n], # == n || s[#] <= 2*# &]; Select[Range[10^6], primQ]

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

A381550 Numbers whose sum of abundant divisors is odd.

Original entry on oeis.org

945, 1575, 1890, 2205, 3150, 3465, 3780, 4095, 4410, 4725, 5355, 5775, 5985, 6300, 6435, 6615, 6825, 6930, 7245, 7425, 7560, 8085, 8190, 8415, 8505, 8820, 8925, 9135, 9450, 9555, 9765, 10395, 10710, 11025, 11550, 11655, 11970, 12285, 12600, 12705, 12870, 12915

Offset: 1

Amiram Eldar, Feb 26 2025

Numbers k such that A187795(k) is odd.
Numbers whose odd part has an odd number of abundant divisors, i.e., numbers k such that A080224(A000265(k)) is odd.
If m is an odd term then 2^k * m is a term for all k >= 0. Therefore, the primitive terms of this sequence are the odd terms, that are also the odd numbers whose number of abundant divisors is odd (A381547).
Are there two consecutive integers in this sequence? There are none below 10^10.

			945 is a term since its sum of abundant divisors is 945, which is odd.
4725 is a term since its sum of abundant divisors is 945 + 1575 + 4725 = 7245, which is odd.

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

Cf. A000265, A080224, A187795.
Subsequence of A005101.
Subsequences: A006038, A381547.

q[n_] := OddQ[DivisorSum[n, # &, DivisorSigma[-1, #] > 2 &]]; Select[Range[13000], q]

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

A133688 Least odd primitive abundant number with 3^n as a divisor, but not 3^(n+1).

Original entry on oeis.org

5391411025, 5775, 1575, 945, 81081, 78975, 1468935, 6375105, 436444281, 5356826865, 21873816315, 371922783705, 2241870572475, 158639164165575, 297836412308955, 1429674513582825, 13431279259253115, 100139192108634825

Offset: 0

Pierre CAMI, Jan 04 2008

			5391411025=3^0 * 5^2 * 7 * 11 * 13 * 17 * 19 * 23 * 29 least odd abundant number with no factor 3.
5775 = 3^1 * 5^2 * 7 * 11.
1575 = 3^2 * 5^2 * 13.
945 = 3^3 * 5 * 7.
81081 = 3^4 * 7 * 11 * 13.
78975 = 3^5 * 5^2 * 13.
1468935 = 3^6 * 5 * 13 * 31.
6375105 = 3^7 * 5 * 11 * 53.
436444281 = 3^8 * 7 * 13 * 17 * 43.

Cf. A006038 (odd primitive abundant numbers).

Cf. A115414 (odd abundant numbers not divisible by 3).

isprab(v) = {my(sig = sigma(v)); if (sig < 2*v, return (0)); if (sig == 2*v, return (1)); fordiv (v, d, if ((d != v) && (sigma(d)>=2*d), return (0));); return (1);}
a(n) = {my(p = 3^n, k = 1); while (1, if (k % 3 != 0, v = p * k; if (isprab(v), return (v));); k += 2;);}
\\ Michel Marcus, Mar 07 2013

Some terms corrected and a(9)-a(13) from Michel Marcus, Mar 07 2013

a(14)-a(17) from David A. Corneth, Oct 26 2024

A133778 Odd primitive abundant numbers of the form (2P)^2+2P+p^2 with P and p primes and p^2<4*P.

Original entry on oeis.org

15015, 41055, 46035, 5581695, 507263295, 756387555, 982365075, 1817713359, 1909933155, 2689720275, 2853976275, 4838941575, 5948000415, 6782922531, 7100171715, 13666199547, 13871855151, 14694980355, 16033285395, 16901775891, 22183878915, 22416851655, 24141235275

Offset: 1

Pierre CAMI, Jan 02 2008

nonn

			15015   = 2*61*2*61     + 2*61   + 3*3;
41055   = 2*101*2*101   + 2*101  + 7*7;
46035   = 2*107*2*107   + 2*107  + 5*5;
5581695 = 2*1181*2*1181 + 2*1181 + 17*17.

Subsequence of A006038.

ispoa(n) = (n%2) && (sumdiv(n, d, sigma(d, -1)>2)==1); \\ A006038
lista(nn) = {forprime(P=2, nn, forprime(p=2, sqrtint(4*P), x = (2*P)^2+2*P+p^2; if (ispoa(x), print1(x, ", "));););} \\ Michel Marcus, Sep 15 2019

More terms from Michel Marcus, Sep 15 2019

A316116 Least odd primitive abundant number having its prime signature.

Original entry on oeis.org

945, 1575, 2205, 3465, 5775, 7425, 8085, 12705, 15015, 28215, 47025, 49875, 69825, 78975, 81081, 103455, 131625, 153153, 182325, 189189, 297297, 342225, 351351, 363375, 387345, 392445, 474045, 532875, 570375, 692835, 742203, 793611, 1102725, 1380825, 1468935, 1612875

Offset: 1

David A. Corneth, Aug 18 2018

nonn

Ordering of exponents matters; 1575 and 2205 have unordered prime signatures (2, 2, 1) and (2, 1, 2) respectively.

			1575 = 3^2 * 5^2 * 7 has prime signature (2, 2, 1) and is an odd primitive abundant number (A006038). Since 1575 is the smallest such number, it is in the sequence. - _Michael B. Porter_, Nov 24 2018

Cf. A006038, A091191.

lsig={}; lpab = {}; seq={}; Do[ d=Divisors[n]; If[Total[d] > 2 n && Intersection[ lpab, d] == {},AppendTo[lpab, n]; sig=FactorInteger[n][[;;,2]]; If[!MemberQ[ lsig,sig], AppendTo[seq, n]; AppendTo[lsig,sig]]],{n,3,1700000,2}]; seq (* Amiram Eldar, Dec 09 2018 *)

cp's OEIS Frontend

A286042 Largest prime factor of A285993(n), the largest odd abundant number (A005231) equal to the product of n consecutive primes.

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A303933 a(n) is the number of odd primitive abundant numbers with n distinct prime divisors.

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A360355 Primitive terms of A360328: terms of A360328 with no proper divisor in A360328.

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A381550 Numbers whose sum of abundant divisors is odd.

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A133688 Least odd primitive abundant number with 3^n as a divisor, but not 3^(n+1).

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A133778 Odd primitive abundant numbers of the form (2*P)^2+2*P+p^2 with P and p primes and p^2<4*P.

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A316116 Least odd primitive abundant number having its prime signature.

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A133778 Odd primitive abundant numbers of the form (2P)^2+2P+p^2 with P and p primes and p^2<4*P.