A379949 -id:A379949 - cp's OEIS frontend

A006038 Odd primitive abundant numbers.

945, 1575, 2205, 3465, 4095, 5355, 5775, 5985, 6435, 6825, 7245, 7425, 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, 28215

Offset: 1

N. J. A. Sloane

nonn

Dickson proves that there are only a finite number of odd primitive abundant numbers having n distinct prime factors. Sequence A188342 lists the smallest such numbers. - T. D. Noe, Mar 28 2011
Sequence A188439 sorts the numbers in this sequence by the number of distinct prime factors. Eight numbers have exactly three prime factors; 576 have exactly four prime factors. - T. D. Noe, Apr 04 2011
Any multiple of an abundant number (A005101) is again an abundant number. Primitive abundant numbers (A091191) are those not of this form, i.e., without an abundant proper divisor. We don't know any odd perfect number (A000396), so the (odd) terms here have only deficient proper divisors (A071395), and their prime factors p are less than sigma(n/p)/deficiency(n/p). See A005231 (odd abundant numbers) for an explanation why all terms have at least 3 distinct prime factors, and 5 prime factors when counted with multiplicity (A001222), whence a(1) = 3^3*5*7. All known terms are semiperfect (A005835, and thus in A006036): no odd weird number (A006037) is known, but if it exists, the smallest one is in this sequence. - M. F. Hasler, Jul 28 2016
So far, a(173) = 351351 is the only known term of A122036, i.e., which can't be written as sum of its proper divisors > 1. - M. F. Hasler, Jan 26 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
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.
R. K. Guy, Letter to N. J. A. Sloane with attachment, Jun. 1991
Jacob Liddy, An algorithm to determine all odd primitive abundant numbers with d prime divisors, Honors Research Projects (2018), 728.
Eric Weisstein's World of Mathematics, Primitive Abundant Number

Cf. A005101, A005231. Subsequence of A091191.

Cf. A000203, A027751, A379949 (subsequence of square terms).

a006038 n = a006038_list !! (n-1)
a006038_list = filter f [1, 3 ..] where
   f x = sum pdivs > x && all (<= 0) (map (\d -> a000203 d - 2 * d) pdivs)
         where pdivs = a027751_row x
-- Reinhard Zumkeller, Jan 31 2014

isA005101 := proc(n) is(numtheory[sigma](n) > 2*n ); end proc:
isA005100 := proc(n) is(numtheory[sigma](n) < 2*n ); end proc:
isA006038 := proc(n) local d; if type(n,'odd') and isA005101(n) then for d in numtheory[divisors](n) minus {1,n} do if not isA005100(d) then return false; end if; end do: return true;else false; end if; end proc:
n := 1 ; for a from 1 by 2 do if isA006038(a) then printf("%d %d\n",n,a) ; n := n+1 ; end if; end do: # R. J. Mathar, Mar 28 2011

t = {}; n = 1; While[Length[t] < 50, n = n + 2; If[DivisorSigma[1, n] > 2 n && Intersection[t, Divisors[n]] == {}, AppendTo[t, n]]]; t (* T. D. Noe, Mar 28 2011 *)

is(n)=n%2 && sumdiv(n,d,sigma(d,-1)>2)==1 \\ Charles R Greathouse IV, Jun 10 2013

is_A006038(n)=bittest(n,0) && sigma(n)>2*n && !for(i=1,#f=factor(n)[,1],sigma(n\f[i],-1)>2&&return) \\ More than 5 times faster. - M. F. Hasler, Jul 28 2016

A156942 Odd abundant numbers whose abundance is odd.

Original entry on oeis.org

11025, 99225, 245025, 275625, 342225, 540225, 893025, 1334025, 1863225, 2205225, 2480625, 3080025, 3186225, 3980025, 4601025, 4862025, 5832225, 6125625, 6890625, 7868025, 8037225, 8555625, 9272025, 9828225, 10595025, 10989225

Offset: 1

Robert G. Wilson v, Feb 18 2009

nonn

Number of terms <10^n: 0, 0, 0, 0, 2, 7, 24, 83, 250, 792, 2484, 7988, 25383, 80082, ..., . Not all are a multiple of 25, i.e.; 81162081 = 9009^2 = (9*7*11*13)^2. See A156943.
Any term must be an odd square. Square roots are in A174830.
Indeed, the sum of divisors of any number isn't odd unless it's a square or twice a square (A028982), and to get the abundance, twice the number is subtracted, so the parity remains the same. - M. F. Hasler, Jan 26 2020
Question: Is this a subsequence of A379503? (Is A379504(a(n)) > 0 for all n? See A379951). The first 15000 terms are all included there. - Amiram Eldar and Antti Karttunen, Jan 06 2025
Question 2: Is A379505(a(n)) > 1 for all n, especially if there are no quasiperfect numbers (numbers k such that sigma(k) = 2k+1)? - Antti Karttunen, Jan 06 2025
From Amiram Eldar, Jan 16 2025: (Start)
The least term that is not divisible by 5 is a(75) = 81162081.
The least term that is not divisible by 3 is a(296889) = 1382511906801025.
The least term that is coprime to 15 is 15285071557677427358507559514565648611799881. (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Robert G. Wilson v)

Cf. A005101, A005231, A028982, A033880, A156903, A174830, A379503, A379504, A379505, A379951 [= A379504(a(n))].

Subsequences: A156943, A325311 (thus also A379490), A347890, A379949 (terms that are primitive abundant).

fQ[n_] := Block[{ds = DivisorSigma[1, n] - 2 n}, ds > 0 && OddQ@ ds]; Select[ Range[1, 12006223, 2], fQ @# &]

is(n)=my(s=sigma(n)); n%2 && s>2*n && (s-2*n)%2 \\ Charles R Greathouse IV, Feb 21 2017

a(n) = A174830(n)^2. - M. F. Hasler, Jan 26 2020

Edited by Robert G. Wilson v at the suggestion of T. D. Noe, Mar 30 2010

A306796 Primitive abundant numbers (A071395) that are squares.

Original entry on oeis.org

342225, 280495504, 1029447225, 1148667664, 1435045924, 1596961444, 1757705625, 2177622225, 14787776025, 18114198921, 32871503025, 45018230625, 150897287025, 245485566225, 272993710144, 296006724225, 705373218225, 1126920249225, 1329226832241, 1358425215225

Offset: 1

Amiram Eldar, Mar 10 2019

nonn

The square roots of the terms are 585, 16748, 32085, 33892, 37882, 39962, 41925, 46665, 121605, 134589, ...

Amiram Eldar, Table of n, a(n) for n = 1..5400

Intersection of A000290 and A071395.

Cf. A306797, A379949 (most likely gives the odd terms).

abQ[f_] := Times@@((f[[;;,1]]^(f[[;;,2]]+1)-1)/(f[[;;,1]]-1)) > 2*Times@@Power@@@f;
nondefQ[f_,g_] := Times@@((f^(g+1)-1)/(f-1)) >= 2*Times@@(f^g);
sub[f_,k_] := Module[{g=f[[;;,2]]}, n=Length[g]; kk=k-1; Do[g[[i]] = Mod[kk, f[[i,2]]+1]; kk=(kk-g[[i]])/(f[[i,2]]+1), {i,1,n}]; g];
paQ[f_] := abQ[f] && Module[{nd = Times@@(f[[;;,2]]+1), ans=True}, Do[g=sub[f,k]; If[nondefQ[f[[;;,1]], g], ans=False; Break[]], {k,1,nd-1}]; ans];
papowerQ[n_, e_] := Module[{f=FactorInteger[n]}, f[[;;,2]]*=e; paQ[f]];
s={}; e=2; Do[If[papowerQ[m, e], AppendTo[s, m^e]], {m, 2, 50000}]; s

is1(k) = {my(f = factor(k)); for(i = 1, #f~, f[i, 2] *= 2); if(sigma(f, -1) <= 2, return(0)); for(i = 1, #f~, f[i, 2] -= 1; if(sigma(f, -1) >= 2, return(0)); f[i, 2] += 1); 1;}
list(lim) = for(k = 1, lim, if(is1(k), print1(k^2, ", "))); \\ Amiram Eldar, Mar 12 2025

A379950 Numbers k such that k^2 is an odd primitive abundant number (A006038).

Original entry on oeis.org

585, 32085, 41925, 46665, 121605, 134589, 181305, 212175, 388455, 495465, 544065, 839865, 1061565, 1152921, 1165515, 1243275, 1247103, 1335411, 1676829, 2151075, 2290869, 2478075, 2771835, 2838165, 3016725, 3122847, 3156795, 4571415, 4738041, 5153841, 5558985, 6125049, 7471425, 7676775, 7780101, 7822425, 8259867

Offset: 1

Antti Karttunen, Jan 07 2025

nonn

Amiram Eldar, Table of n, a(n) for n = 1..3500 (terms 1..725 from Antti Karttunen)

Square roots of A379949.
Subsequence of A174830.
Cf. A000196, A006038.

isok(k) = if(!(k % 2), 0, my(f = factor(k)); for(i = 1, #f~, f[i, 2] *= 2); if(sigma(f, -1) <= 2, return(0)); for(i = 1, #f~, f[i, 2] -= 1; if(sigma(f, -1) > 2, return(0)); f[i, 2] += 1); 1); \\ Amiram Eldar, Mar 12 2025

a(n) = A000196(A379949(n)).

A381742 Numbers k such that k^2 is abundant but d*k is nonabundant for any proper divisor d of k.

Original entry on oeis.org

14, 124, 585, 1016, 16748, 32085, 33892, 37882, 39962, 41925, 46665, 121605, 134589, 181305, 212175, 388455, 495465, 522488, 524224, 544065, 839865, 1061565, 1152921, 1165515, 1243275, 1247103, 1335411, 1676829, 1943638, 2151075, 2290869, 2478075, 2625514, 2673998

Offset: 1

Amiram Eldar, Mar 06 2025

nonn

Numbers k such that k^2 is primitive abundant number (A091191).

If p is an odd Mersenne exponent (A174265), then 2^((p-1)/2) * (2^p-1) is a term.

Amiram Eldar, Table of n, a(n) for n = 1..775

Subsequence of A381738.
A379950 is a subsequence.
Cf. A005101, A091191, A174265, A263837, A341358, A379949.

q[k_] := DivisorSigma[-1, k^2] > 2 &&  AllTrue[Divisors[k], DivisorSigma[-1, #*k] <= 2 || # == k &]; Select[Range[10^6], q]

isok(k) = fordiv(k, d, if(d < k && sigma(d*k, -1) > 2, return(0))); sigma(k^2, -1) > 2;

A379474 Number of prime factors of the form p^e || n : [p == 1 (mod 8), e == 1 (mod 4)] or [p == 5 (mod 8), e == -1 (mod 4)].

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1

Offset: 1

Antti Karttunen, Jan 07 2025

nonn

In the definition p^e || n indicates that p^e divides n, but p^(1+e) does not divide n.

In Prasad and Sunitha paper prime factors of the form {p^2e : [p == 1 (mod 8), e == 1 (mod 4)] or [p == 5 (mod 8), e == -1 (mod 4)]} are called "special factors", and it is shown there that a necessary condition for an odd primitive abundant square (A379949) to be a quasiperfect number (number x such that sigma(x) = 2x+1) is that it has an odd number of such factors.

			a(17) = 1 because 17 is of the form 8m+1 and its exponent 1 is of the form 4m+1.
a(697) = 2, as 697 = 17^1 * 41^1, a product of two primes of the form 8m+1 with exponents of the form 4m+1.
a(2125) = 2 because 2125 = 17^1 * 5^3, the first factor is a prime of the form 8m+1 with exponent of the form 4m+1, and the second factor is a prime of the form 8m+5 with exponent of the form 4m+3.
a(50881) = 3 as 50881 = 17^1 * 41^1 * 73^1, a product of three primes of the form 8m+1 with exponents of the form 4m+1.

Antti Karttunen, Table of n, a(n) for n = 1..100000
V. Siva Rama Prasad and C. Sunitha, On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.

Cf. A379949.

A379474(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, if(1==(f[i, 1]%8), 1==(f[i, 2]%4), if(5==(f[i, 1]%8), 3==(f[i, 2]%4), 0))));

A379474(n) = if(n<=1, 0, my(f=factor(n), is(p,e) = (((1==p) || (5==p)) && (2*e == (1+p)))); sum(i=1, #f~, is(f[i, 1]%8, f[i, 2]%4)));

A379490 Odd squares s such that 2s is equal to bitwise-AND of 2s and sigma(s).

Original entry on oeis.org

399736269009, 1013616036225, 1393148751631700625, 2998748839068013955625, 3547850289210724050225

Offset: 1

Antti Karttunen, Jan 13 2025

If there are any quasiperfect numbers, i.e., numbers x for which sigma(x) = 2*x+1, then they should occur also in this sequence.
Square roots of these terms are: 632247, 1006785, 1180317225, 54760833075, 59563833735.
Question: Are there any solutions to similar equations "Odd squares s such that 2*s is equal to bitwise-AND of 2*s and A001065(s)" and "Odd squares s such that 3*s is equal to bitwise-AND of 3*s and sigma(s)"? Such sequences would contain odd triperfect numbers, if they exist (cf. A005820, A347391, A347884). - Antti Karttunen, Aug 19 2025
a(6) > 4*10^21. - Giovanni Resta, Aug 19 2025

Paolo Cattaneo, Sui numeri quasiperfetti, Bollettino dell’Unione Matematica Italiana, Serie 3, Vol.6(1951), n.1, p. 59-62.
P. Hagis and G. L. Cohen, Some Results Concerning Quasiperfect Numbers, J. Austral. Math. Soc. Ser. A 33, 275-286, 1982.
V. Siva Rama Prasad and C. Sunitha, On quasiperfect numbers, Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 73-78.

Odd squares in A324647.
Intersection of A016754 and A324647.
Subsequence of A325311, which is a subsequence of A005231.
Cf. A000203, A001065, A004198, A318468.
Cf. also A336700, A336701, A337339, A337342, A348742, A379474, A379503, A379505, A379949 for other conditions that quasiperfect numbers should satisfy.

k=0; forstep(n=1,oo,2, if(!((n-1)%(2^27)),print1("("n")")); if(!isprime(n) && omega(n)>=3, f = factor(n); sq=n^2; sig=prod(i=1,#f~,((f[i,1]^(1+(2*f[i,2])))-1) / (f[i,1]-1)); if(((2*sq)==bitand(2*sq, sig)), k++; print1(sq,", "))));

a(4) and a(5) from Giovanni Resta, Aug 19 2025

A379951 a(n) = A379504(A156942(n)).

Original entry on oeis.org

816, 28471698, 714837, 8719965, 5969, 4385405, 830994029375, 241550466668344580

Offset: 1

Antti Karttunen, Jan 09 2025

Conjecture: a(n) > 0 for all n. See comments in A156942 and in A379949.

Cf. A156942, A379504.

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A006038 Odd primitive abundant numbers.

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A156942 Odd abundant numbers whose abundance is odd.

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A306796 Primitive abundant numbers (A071395) that are squares.

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A379950 Numbers k such that k^2 is an odd primitive abundant number (A006038).

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A381742 Numbers k such that k^2 is abundant but d*k is nonabundant for any proper divisor d of k.

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A379474 Number of prime factors of the form p^e || n : [p == 1 (mod 8), e == 1 (mod 4)] or [p == 5 (mod 8), e == -1 (mod 4)].

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A379490 Odd squares s such that 2*s is equal to bitwise-AND of 2*s and sigma(s).

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A379951 a(n) = A379504(A156942(n)).

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A379490 Odd squares s such that 2s is equal to bitwise-AND of 2s and sigma(s).