A047802 -id:A047802 - cp's OEIS frontend

A358418 Least number k coprime to 2, 3, and 5 such that sigma(k)/k >= n.

1, 20169691981106018776756331

Offset: 1

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(2) = 7^2*11^2*13*...*67 ~ 2.01697*10^25. a(3) = 7^3*11^3*13^2*17^2*19^2*23^2*29^2*31*...*569 ~ 2.54562*10^239 and a(4) = 7^5*11^3*13^3*17^3*19^3*23^2*...*97^2*101*...*4561 ~ 1.11116*10^1986 are too large to display.

			a(2) = A047802(3) = 20169691981106018776756331 is the smallest abundant number coprime to 2, 3, and 5.
Even if there is a number k coprime to 2, 3, and 5 with sigma(k)/k = 3, we have that k is a square since sigma(k) is odd. If omega(k) = m, then 3 = sigma(k)/k < Product_{i=4..m+3} (prime(i)/(prime(i)-1)) => m >= 97, and we have k >= prime(4)^2*...*prime(100)^2 ~ 2.46692*10^436 > A358413(3) ~ 2.54562*10^239. So a(3) = A358413(3).
Even if there is a number k coprime to 2, 3, and 5 with sigma(k)/k = 4, there can be at most 2 odd exponents in the prime factorization of k (see Theorem 2.1 of the Broughan and Zhou link). If omega(k) = m, then 4 = sigma(k)/k < Product_{i=4..m+3} (prime(i)/(prime(i)-1)) => m >= 606, and we have k >= prime(4)^2*...*prime(607)^2*prime(608)*prime(609) ~ 6.54355*10^3814 > A358414(3) ~ 1.11116*10^1986. So a(4) = A358414(3).

Jianing Song, Table of n, a(n) for n = 1..3
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, author’s version, Research Commons.
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, Journal of Number Theory 128 (2008) 1566-1575.
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), A358414 (k=4).

Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), this sequence (p=7), A358419 (p=11).

A358419 Least number k coprime to 2, 3, 5, and 7 such that sigma(k)/k >= n.

Original entry on oeis.org

1, 49061132957714428902152118459264865645885092682687973

Offset: 1

Jianing Song, Nov 14 2022

Data copied from the Hi.gher. Space link where Mercurial, the Spectre calculated the terms. We have a(2) = 11^2*13^2*17*...137 ~ 4.90611*10^52. a(3) = 11^3*13^3*17^2*...*47^2*53*...*1597 ~ 3.99515*10^688 and a(4) = 11^4*13^4*17^3*19^3*23^3*29^3*31^3*37^2*...*181^2*191*...*18493 ~ 2.99931*10^8063 are too large to display.

			a(2) = A047802(4) = 49061132957714428902152118459264865645885092682687973 is the smallest abundant number coprime to 2, 3, 5, and 7.
Even if there is a number k coprime to 2, 3, 5, and 7 with sigma(k)/k = 3, we have that k is a square since sigma(k) is odd. If omega(k) = m, then 3 = sigma(k)/k < Product_{i=5..m+4} (prime(i)/(prime(i)-1)) => m >= 240, and we have k >= prime(5)^2*...*prime(244)^2 ~ 1.60834*10^1297 > A358413(4) ~ 3.99515*10^688. So a(3) = A358413(4).
Even if there is a number k coprime to 2, 3, 5, and 7 with sigma(k)/k = 4, there can be at most 2 odd exponents in the prime factorization of k (see Theorem 2.1 of the Broughan and Zhou link). If omega(k) = m, then 4 = sigma(k)/k < Product_{i=5..m+4} (prime(i)/(prime(i)-1)) => m >= 2096, and we have k >= prime(5)^2*...*prime(2098)^2*prime(2099)*prime(2100) ~ 6.21439*10^15801 > A358414(4) ~ 2.99931*10^8063. So a(4) = A358414(4).

Jianing Song, Table of n, a(n) for n = 1..3
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, author’s version, Research Commons.
Kevin A. Broughan and Qizhi Zhou, Odd multiperfect numbers of abundancy 4, Journal of Number Theory 128 (2008) 1566-1575.
Mercurial, the Spectre, Abundant numbers coprime to n, Hi.gher. Space.

Smallest k-abundant number which is not divisible by any of the first n primes: A047802 (k=2), A358413 (k=3), A358414 (k=4).

Least p-rough number k such that sigma(k)/k >= n: A023199 (p=2), A119240 (p=3), A358412 (p=5), A358418 (p=7), this sequence (p=11).

A338133 Primitive nondeficient numbers sorted by largest prime factor then by increasing size. Irregular triangle T(n, k), n >= 2, k >= 1, read by rows, row n listing those with largest prime factor = prime(n).

Original entry on oeis.org

6, 20, 28, 70, 945, 1575, 2205, 88, 550, 3465, 5775, 7425, 8085, 12705, 104, 572, 650, 1430, 2002, 4095, 6435, 6825, 9555, 15015, 78975, 81081, 131625, 189189, 297297, 342225, 351351, 570375, 63126063, 99198099, 117234117, 272, 748, 1870, 2210, 5355, 8415, 8925, 11492

Offset: 2

David A. Corneth and Peter Munn, Oct 11 2020

For definitions and further references/links, see A006039, the main entry for primitive nondeficient numbers.
Rows are finite: row n is a subset of the divisors of any of the products formed by multiplying 2^(A035100(n)-1) by a member of the first n finite sets described in the Dickson reference.
Column 1 includes the even perfect numbers.
The largest number in rows 2..n (therefore the largest that is prime(n)-smooth) is A338427(n). - Peter Munn, Sep 07 2021

			Row 1 is empty as there exists no primitive nondeficient number of the form prime(1)^k = 2^k.
Row 2 is (6) as 6 is the only primitive nondeficient number of the form prime(1)^k * prime(2)^m = 2^k * 3^m that is a multiple of prime(2) = 3.
Irregular triangle T(n, k) begins:
  n   prime(n)  row n
  2      3      6;
  3      5      20;
  4      7      28, 70, 945, 1575, 2205;
  5     11      88, 550, 3465, 5775, 7425, 8085, 12705;
  ...
See also the factorization of initial terms below:
      6 = 2 * 3,
     20 = 2^2 * 5,
     28 = 2^2 * 7,
     70 = 2 * 5 * 7,
    945 = 3^3 * 5 * 7,
   1575 = 3^2 * 5^2 * 7,
   2205 = 3^2 * 5 * 7^2,
     88 = 2^3 * 11,
    550 = 2 * 5^2 * 11,
   3465 = 3^2 * 5 * 7 * 11,
   5775 = 3 * 5^2 * 7 * 11,
   7425 = 3^3 * 5^2 * 11,
   8085 = 3 * 5 * 7^2 * 11,
  12705 = 3 * 5 * 7 * 11^2,
    104 = 2^3 * 13,
    572 = 2^2 * 11 * 13,
    650 = 2 * 5^2 * 13,
   1430 = 2 * 5 * 11 * 13,
   2002 = 2 * 7 * 11 * 13,
   4095 = 3^2 * 5 * 7 * 13,
  ...

L. E. Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math., 35 (1913), 413-426.
Index entries for sequences where any odd perfect numbers must occur

A000040, A006530 are used to define this sequence.
Permutation of A006039.
A047802\{12}, A308710 are subsequences.
Cf. A000043, A000079, A000668, A035100, A059305, A061652, A338427.

rownupto(n, u) = { my(res = List(), pr = primes(n), e = vector(n, i, logint(u, pr[i]))); vu = vector(n, i, [0, e[i]]); vu[n][1] = 1; forvec(x = vu, c = prod(i = 1, n, pr[i]^x[i]); if(c <= u && isprimitive(c), listput(res, c) ) ); Set(res) }
isprimitive(n) = { my(f = factor(n), c); if(sigma(f) < 2*n, return(0)); for(i = 1, #f~, c = n / f[i,1]; if(sigma(c) >= c * 2, return(0) ) ); 1 }
for(i = 2, 7, print(rownupto(i, 10^9)))

A006530(T(n, k)) = A000040(n).
T(n, 1) = A308710(n-1) [provided there is no least deficient number that is not a power of 2, as described in A000079].
For m >= 1, T(A059305(m), 1) = A000668(m) * 2^(A000043(m)-1) = A000668(m) * A061652(m).

A110585 Smallest number k of consecutive primes > p_n such that p_n^2 * p_(n+1) * p_(n+2) * ... * p_(n+k) is an abundant number.

Original entry on oeis.org

1, 3, 7, 16, 29, 44, 65, 89, 120, 155, 192, 236, 282, 332, 390, 453, 520, 589, 666, 746, 832, 927, 1026, 1131, 1239, 1350, 1467, 1592, 1725, 1867, 2017, 2161, 2313, 2469, 2634, 2800, 2975, 3155, 3339, 3532, 3729, 3931, 4143, 4356, 4579, 4809, 5051, 5291

Offset: 1

Igor Schein, Sep 13 2005

nonn

The sequence arose while solving puzzle 329 from Carlos Rivera's Prime Puzzles & Problems Connection site.

			a(2)=3 because the second prime being 3, then 3^2 * 5 * 7 * 11 = 3465 and sigma(3465) - 2*3465 = 558, a positive number (i.e., 3465 is abundant), but 3^2 * 5 * 7 = 315 and sigma(315) - 2*315 = -6, a nonpositive number (i.e., 315 is not abundant).

Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3.

Cf. A005101.

Cf. A005231, A047802.

abQ[n_] := DivisorSigma[1, n] > 2n; f[0] = 0; f[n_] := f[n] = Block[{k = f[n - 1]}, p = Fold[Times, Prime[n], Prime[ Range[n, n + k]]]; While[ !abQ[p], k++; p = p*Prime[n + k]]; k]; Table[ f[n], {n, 48}] (* Robert G. Wilson v *)

forprime(p=2,100,k=0;while(k++,if(sigma(n=p^2*prod(j=1,k,prime(j+primepi(p))))-n>n,print(k);break)))

Edited and extended by Robert G. Wilson v, Sep 15 2005

A341508 a(n) = 0 if n is nonabundant, otherwise a(n) is the number of abundant divisors of the last abundant number in the iteration x -> A003961(x) (starting from x=n) before a nonabundant number is reached.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Antti Karttunen, Feb 20 2021

nonn

Question: Is a(A336389(n)) = 1 for all n >= 2? Note that all the terms of A047802 are obviously primitively abundant (in A091191).

			Starting from n = 120 = 2^3 * 3 * 5, the number of its abundant divisors is A080224(120) = 7. Then we apply a prime shift (A003961) to obtain the next number, 3^3 * 5 * 7 = 945, which has A080224(945) = 1 abundant divisors (as 945 is a term of A091191). The next prime shift gives 5^3 * 7* 11 = 9625, which has zero abundant divisors (as it is nonabundant, in A263837), so A080224(9625) = 0, and a(120) = 1, the last nonzero value encountered.

Cf. A003961, A047802, A091191, A336389, A336835, A337468.
Cf. A263837 (positions of zeros), A005101 (and of nonzeros).
Differs from A080224 for the first time at n=120, with a(120) = 1, while A080224(120) = 7.

A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
A080224(n) = sumdiv(n, d, sigma(d)>2*d);
A341508(n) = { my(t, u=0); while((t=A080224(n))>0, u=t; n = A003961(n)); (u); };

A112644 Odd and squarefree abundant numbers not divisible by 5.

Original entry on oeis.org

22309287, 28129101, 30069039, 34051017, 35888853, 36399363, 38057019, 39768729, 40681641, 41708667, 43444401, 45588543, 45894849, 48141093, 48555507, 50489439, 51294243, 51408357, 53804751, 54777723, 55186131, 56429373, 57228171, 58555497, 59168109

Offset: 1

Labos Elemer, Sep 20 2005

nonn

The least term that is not divisible by 3 is 73#/5# = Product_{k=4..21} prime(k) = 1357656019974967471687377449. - Amiram Eldar, Aug 15 2024

			99906807 = 3*7*11*13*17*19*103 is a term since it is an odd squarefree number that is not divisible by 5, and sigma(99906807) = 201277440 > 2*99906807.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A087248, A046391, A112643, A112642, A112640.

Cf. A005231, A047802.

ta={{0}};Do[g=n;s=DivisorSigma[1, n]-2*n; If[Greater[s, 0]&&Equal[Abs[MoebiusMu[n]], 1]&& !Equal[Mod[n, 2], 0]&&!Equal[Mod[n, 5], 0], Print[n, PrimeFactorList[n], s];ta=Append[ta, n]], {n, 10000000, 100000000}];{ta=Delete[ta, 1], g}

issfab(k) = my(f = factor(k)); issquarefree(f) && sigma(f, -1) > 2;
is(k) = gcd(k, 10) == 1 && issfab(k); \\ Amiram Eldar, Aug 15 2024

A387165 Nondeficient numbers k for which A324644(k)/A324198(k) = 2.

Original entry on oeis.org

38745, 77805, 78435, 118755, 141075, 157815, 210735, 237195, 241605, 294975, 300105, 323505, 364455, 371925, 390195, 409185, 455715, 475335, 499905, 567945, 607635, 660825, 701415, 733005, 766395, 806085, 809325, 872235, 885465, 891135, 937755, 964845, 978705, 1101555, 1150065, 1201095, 1229445, 1265355, 1293705

Offset: 1

Antti Karttunen, Aug 28 2025

First three nonmultiples of 5 occur at a(138), a(276), a(356) = 4446981, 8909901, 11234223. (Cf. A005231, A064001).

Intersection of A023196 and A364286.
Cf. A000203, A276086.
Cf. also A371082, A386422, A387163 and A047802, A064001, A115414.

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
is_A387165(n) = if(sigma(n)<2*n, 0, my(u=A276086(n)); (gcd(sigma(n),u)==2*gcd(n,u)));

{k | sigma(k) >= 2*k, A324644(k) = 2*A324198(k)}.

A114371 Smallest abundant number relatively prime to n.

Original entry on oeis.org

12, 945, 20, 945, 12, 5391411025, 12, 945, 20, 81081, 12, 5391411025, 12, 6435, 56, 945, 12, 5391411025, 12, 81081, 20, 945, 12, 5391411025, 12, 945, 20, 6435, 12, 20169691981106018776756331, 12, 945, 20, 945, 12, 5391411025, 12, 945, 20, 81081

Offset: 1

Franklin T. Adams-Watters, Feb 09 2006

nonn

If we require only non-deficient, all the 12's become 6's and the 56's become 28's, plus some other changes for large n (such as n = 3234846615 = 3*5*7*11*13*17*19*23*29, 16256 becomes 8128).

Cf. A005101, A005231, A047802, A000396, A114809.

Numbers confirmed by Hugo van der Sanden, Feb 10 2006, who reports that a(210) = 49061132957714428902152118459264865645885092682687973

A114809 Smallest abundant number with some prime powers fixed by n.

Original entry on oeis.org

12, 945, 20, 18, 12, 5391411025, 12, 12, 12, 81081, 12, 70, 12, 6435, 56, 24, 12, 5775, 12, 18, 20, 945, 12, 20, 20, 945, 18, 18, 12, 20169691981106018776756331, 12, 48, 20, 945, 12, 30, 12, 945, 20, 12, 12, 366005822969340125, 12, 18, 12, 945

Offset: 1

Hugo van der Sanden, Feb 09 2006

nonn

If n = prod{ p_i^x_i }, a(n) = prod{ p_i^y_i } then we require y_i = x_i - 1 whenever x_i > 0.
Same as A114371 for squarefree n.
If k is abundant then k = a(k * prod{ p | k }).

			12 = 2^2.3^1, so a(12)=70 is the least abundant 2^1.3^0.k with (k,2.3)=1.

Cf. A005101, A005231, A047802, A114371.

A133849 Least odd primitive abundant numbers with no factor 3 and with 5^n but not 5^(n+1) as a factor.

Original entry on oeis.org

20169691981106018776756331, 33426748355, 5391411025, 26957055125, 134785275625, 673926378125, 3369631890625, 16848159453125, 84240797265625, 421203986328125, 2106019931640625, 10530099658203125, 52650498291015625, 263252491455078125, 1316262457275390625, 6581312286376953125

Offset: 0

Pierre CAMI, Jan 06 2008

nonn

A subsequence of A115414, odd abundant numbers (A005231) not divisible by 3. The smallest of these equals a(2). All subsequent terms are a(n) = 5*a(n-1). - M. F. Hasler, Jul 28 2016

			a(0) = 20169691981106018776756331 = 5^0*7^2*11^2*13*17*19*23*29*31*37*41*43*47*53*59*61*67 = A047802(3), the least odd abundant number with no factor 3 or 5.
a(1) = 33426748355 = 5^1*7*11*13*17*19*23*29*31.
a(2) = 5391411025 = 5^2*7*11*13*17*19*23*29 = A115414(1) = A047802(2), the least odd abundant number with no factor 3.

Cf. A115414, A005231, A005101.

A133849(n)=215656441*if(n>1,5^n,[3016998806898461,5][n+1]*31) \\ M. F. Hasler, Jul 28 2016

For all n >= 2, a(n) = 5^n*7*11*13*17*19*23*29. This can be seen from sigma[-1](5^n) = (5-1/5^n)/4 and sigma[-1](29#/5#) = 1.615... > 2/sigma[-1](5^n) for all n >= 2 (but not for n = 1), while sigma[-1](23#/5#) = 1.56... < 2*4/5 (and idem for any other factor omitted) is never large enough. - M. F. Hasler, Jul 28 2016

Edited, a(3) corrected, and more terms added by M. F. Hasler, Jul 28 2016

cp's OEIS Frontend

A358418 Least number k coprime to 2, 3, and 5 such that sigma(k)/k >= n.

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A358419 Least number k coprime to 2, 3, 5, and 7 such that sigma(k)/k >= n.

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A338133 Primitive nondeficient numbers sorted by largest prime factor then by increasing size. Irregular triangle T(n, k), n >= 2, k >= 1, read by rows, row n listing those with largest prime factor = prime(n).

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A110585 Smallest number k of consecutive primes > p_n such that p_n^2 * p_(n+1) * p_(n+2) * ... * p_(n+k) is an abundant number.

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A341508 a(n) = 0 if n is nonabundant, otherwise a(n) is the number of abundant divisors of the last abundant number in the iteration x -> A003961(x) (starting from x=n) before a nonabundant number is reached.

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A112644 Odd and squarefree abundant numbers not divisible by 5.

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A387165 Nondeficient numbers k for which A324644(k)/A324198(k) = 2.

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A114371 Smallest abundant number relatively prime to n.

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A114809 Smallest abundant number with some prime powers fixed by n.

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