A071395 -id:A071395 - cp's OEIS frontend

A330872 Numbers k such that k and k+1 are both primitive abundant numbers (A071395).

82004, 158235, 516704, 2921535, 5801984, 10846016, 12374144, 12603824, 18738224, 24252074, 32409530, 33696975, 35356544, 36149295, 41078114, 42541190, 43485584, 65090864, 88304475, 90725775, 181480695, 183872535, 213261795, 233762528, 242301344, 254502495, 254630144

Offset: 1

Amiram Eldar, Apr 29 2020

nonn

Not to be confused with A283418 in which the primitive abundant numbers can have perfect numbers as divisors (as defined in A091191).

			82004 is a term since both 82004 and 82005 are abundant, and all of their proper divisors are deficient numbers.

Amiram Eldar, Table of n, a(n) for n = 1..100

Subsequence of A005101, A071395, A096399 and A283418.

primAbQ[n_] := DivisorSigma[1, n] > 2 n && AllTrue[Most @ Rest @ Divisors[n], DivisorSigma[1, #] < 2*# &]; q1 = False; seq = {}; Do[q2 = primAbQ[n]; If[q1 && q2, AppendTo[seq, n - 1]]; q1 = q2, {n, 2, 6*10^6}]; seq

A295369 Number of squarefree primitive abundant numbers (A071395) with n prime factors.

Original entry on oeis.org

0, 0, 1, 18, 610, 216054, 12566567699

Offset: 1

Gianluca Amato, Feb 12 2018

Here primitive abundant number means an abundant number all of whose proper divisors are deficient numbers (A071395). The alternative definition (an abundant number having no abundant proper divisor, see A091191) would yield an infinite count for a(3): since 2*3 = 6 is perfect, all numbers of the kind 2*3*p with p > 3 would be primitive abundant.
See A287590 for the number of squarefree ODD primitive abundant numbers with n prime factors.
The actual numbers are listed in A298973. - M. F. Hasler, Feb 16 2018

			For n=3, the only squarefree primitive abundant number (SFPAN) is 2*5*7 = 70, which is also a primitive weird number, see A002975.
For n=4, the 18 SFPAN range from 2*5*11*13 = 1430 to 2*5*11*53 = 5830.
For n=5, the 610 SFPAN range from 3*5*7*11*13 = 15015 to 2*5*11*59*647 = 4199030.

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. A071395 (primitive abundant numbers), A287590 (counts of odd SFPAN), A298973, A249242 (using A091191).

A295369(n, p=1, m=1, sigmam=1) = {
  my(centerm = sigmam/(2*m-sigmam), s=0);
  if (n==1,
    if (centerm > p, primepi(ceil(centerm)-1) - primepi(p), 0),
    p = max(floor(centerm),p); while (0A295369(n-1, p=nextprime(p+1), m*p, sigmam*(p+1)), s+=c); s
  )
}

def A295369(n, p=1, m=1, sigmam=1):
  centerm = sigmam/(2*m-sigmam)
  if n==1:
    return prime_pi(ceil(centerm)-1) - prime_pi(p) if centerm > p else 0
  else:
    p = max(floor(centerm), p)
    s = 0
    while True:
       p = next_prime(p)
       c = A295369(n-1, p, m*p, sigmam*(p+1))
       if c <= 0: return s
       s+=c

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

A306797 Primitive abundant numbers (A071395) that are cubes.

Original entry on oeis.org

6886512413632368153, 8815747507513708671, 334845050584968548307656, 1254177078562232856388071, 27869863573964698956703125, 108182814324640834480192546875, 384852900473651366592567235048, 520616176957487045802123463832, 567962434462802770687173681448, 1389387861291307410644039382069

Offset: 1

Amiram Eldar, Mar 10 2019

nonn

The cube roots of the terms are 1902537, 2065791, 69440786, 107841591, 303187725, ...

Amiram Eldar, Table of n, a(n) for n = 1..24

Intersection of A000578 and A071395.

Cf. A306796.

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=3; Do[If[papowerQ[m, e], AppendTo[s, m^e]], {m, 2, 7*10^7}]; s

is1(k) = {my(f = factor(k)); for(i = 1, #f~, f[i, 2] *= 3); 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^3, ", "))); \\ Amiram Eldar, Mar 12 2025

a(6)-a(10) from Amiram Eldar, Mar 12 2025

A334419 Primitive abundant numbers (A071395) with a record gap to the next primitive abundant number.

Original entry on oeis.org

20, 104, 945, 2210, 2584, 8415, 10184, 12104, 15368, 86272, 133484, 135470, 140668, 643336, 700256, 1149952, 2410816, 2434888, 5924032, 6100605, 7623872, 8531144, 8760424, 9405045, 10471755, 14803216, 16283085, 21506432, 26919250, 34441946, 35622016, 36064964

Offset: 1

Amiram Eldar, Apr 29 2020

nonn

The record gap values are 50, 168, 239, 260, 406, 510, ... (see the link for more values).

			The first 5 terms of A071395 are 20, 70, 88, 104 and 272. The differences between these terms are 50, 18, 16, and 168. The record gaps are 50 and 168, which occur after the terms 20 and 104.

Amiram Eldar, Table of n, a(n) for n = 1..59
Amiram Eldar, Table of n, a(n), gap(n) for n = 1..59

Cf. A071395, A330872, A334418.

Similar sequences: A306747, A306748, A306953.

primAbQ[n_] := DivisorSigma[1, n] > 2 n && AllTrue[Most @ Rest @ Divisors[n], DivisorSigma[1, #] < 2*# &]; seq = {}; m = 20; dm = 0; Do[If[primAbQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 21, 10^6}]; seq

A363175 Primitive abundant numbers (A071395) that are powerful numbers (A001694).

Original entry on oeis.org

342225, 570375, 3172468, 4636684, 63126063, 99198099, 117234117, 171991125, 280495504, 319600125, 327921075, 404529741, 581549787, 635689593, 762155163, 1029447225, 1148667664, 1356949503, 1435045924, 1501500375, 1558495125, 1596961444, 1757705625, 1778362047

Offset: 1

Amiram Eldar, May 19 2023

nonn

The least cubefull (A036966) term is a(154) = A363177(1) = 26376098024367 = 3^6 * 7^4 * 13^3 * 19^3.

Amiram Eldar, Table of n, a(n) for n = 1..2145 (terms below 10^18)
Wikipedia, Powerful number.
Wikipedia, Primitive abundant number.

Intersection of A001694 and A071395.
Subsequence of A363169 and A363176.
Subsequences: A306796, A306797, A363177.
Cf. A036966.

f1[p_, e_] := (p^(e + 1) - 1)/(p^(e + 1) - p^e); f2[p_, e_] := (p^(e + 1) - p)/(p^(e + 1) - 1);
primAbQ[n_] := (r = Times @@ f1 @@@ (f = FactorInteger[n])) > 2 && r * Max @@ f2 @@@ f < 2;
seq[max_] := Module[{pow = Union[Flatten[Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]]}, Select[Rest[pow], primAbQ]]; seq[10^10]

isPrimAb(n) = {my(f = factor(n), r, p, e); r = sigma(f, -1); r > 2 && vecmax(vector(#f~, i, p = f[i, 1]; e = f[i, 2]; (p^(e + 1) - p)/(p^(e + 1) - 1))) * r < 2; }
lista(lim) = {my(pow = List(), t); for(j=1, sqrtnint(lim\1, 3), for(i=1, sqrtint(lim\j^3), listput(pow, i^2*j^3))); select(x->isPrimAb(x), Set(pow)); }

A306987 Primitive abundant numbers (A071395) that are pseudoperfect (A005835).

Original entry on oeis.org

20, 88, 104, 272, 304, 368, 464, 550, 572, 650, 748, 945, 1184, 1312, 1376, 1430, 1504, 1575, 1696, 1870, 1888, 1952, 2002, 2090, 2205, 2210, 2470, 2530, 2584, 2990, 3128, 3190, 3230, 3410, 3465, 3496, 3770, 3944, 4070, 4095, 4216, 4288, 4408, 4510, 4544, 4672

Offset: 1

Amiram Eldar, Mar 18 2019

nonn

By definition these numbers are also primitive pseudoperfect (A006036).

Benkoski and Erdős proved that this sequence is infinite, since it includes all the numbers of the form 2^k * p with p a prime such that 2^k < p < 2^(k+1).

Amiram Eldar, Table of n, a(n) for n = 1..10000
S. J. Benkoski and P. Erdős, On weird and pseudoperfect numbers, Math. Comp. 28 (1974), 617-623. Corrigendum: Math. Comp. 29 (1975), 673-674.

Cf. A005835, A006036, A071395.

paQ[n_]:=DivisorSigma[1, n] > 2n && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ n] == 1; psQ[n_]:=Module[{d= Most[Divisors[n] ]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0]; Select[Range[5000], paQ[#]&&psQ[#]&] (* after Michael De Vlieger at A071395 and T. D. Noe at A005835 *)

A333967 Subsequence of A071395. The extra constraint is m is not a term if m*q/p is abundant where prime p|m and q is the least prime larger than p.

Original entry on oeis.org

70, 2002, 3230, 4030, 5830, 8415, 8925, 20482, 32445, 45885, 51765, 83265, 107198, 131054, 133042, 178486, 206770, 253270, 253946, 258970, 270470, 310930, 330310, 334305, 362710, 442365, 474045, 497835, 513890, 544310, 567765, 589095, 592670, 602175, 617265, 631670, 654675

Offset: 1

David A. Corneth, Jul 05 2020

nonn

			70 is in the sequence as it's abundant. Its prime factorization is 2 * 5 * 7. Each of 3 * 5 * 7, 2 * 7 * 7 and 2 * 5 * 11 are deficient and no divisor of 70 is in this sequence.

David A. Corneth, Table of n, a(n) for n = 1..1317

Cf. A071395, A071927, A335557.

primabQ[n_] := DivisorSigma[1, n] > 2n && AllTrue[Most @ Divisors[n], DivisorSigma[1, #] < 2# &]; seqQ[n_] := Module[{f = FactorInteger[n]}, p = f[[;; , 1]]; e = f[[;; , 2]]; q = NextPrime[p]; AllTrue[n*(q/p), DivisorSigma[1, #] <= 2# &]]; Select[Range[10^5], primabQ[#] && seqQ[#] &] (* Amiram Eldar, Jul 05 2020 *)

A337538 a(n) is the least k such that A003961(k*A071395(n)) is abundant.

Original entry on oeis.org

6, 6, 15, 15, 15, 15, 15, 15, 3, 6, 6, 6, 9, 2, 15, 15, 15, 3, 15, 2, 15, 3, 15, 15, 6, 3, 2, 3, 3, 3, 9, 3, 9, 3, 3, 3, 2, 15, 6, 15, 6, 3, 2, 15, 15, 15, 3, 15, 15, 15, 3, 15, 15, 3, 15, 15, 2, 15, 15, 15, 15, 2, 3, 15, 2, 15, 15, 15, 2, 15, 15, 15, 15, 2, 15, 15, 15, 15, 15, 15, 2, 2, 15, 15, 15, 15, 2

Offset: 1

Antti Karttunen and Peter Munn, Sep 07 2020

nonn

A071395(n) is the n-th primitive abundant number. A003961(k) replaces each prime factor of k with the next larger prime.

See also the table in the example section of A337469.

Antti Karttunen, Table of n, a(n) for n = 1..13037

Cf. A000203, A003961, A003973, A071395, A337386, A337469, A337479.

Map[Block[{k = 1}, While[DivisorSigma[1, #] <= 2 # &[Times @@ Map[#1^#2 & @@ # &, FactorInteger[k #] /. {p_, e_} /; e > 0 :> {Prime[PrimePi@ p + 1], e}]], k++]; k] &, Select[Range[10^4], DivisorSigma[1, #] > 2 # && Times @@ Boole@ Map[DivisorSigma[1, #] < 2 # &, Most@ Divisors@ #] == 1 &]] (* Michael De Vlieger, Oct 05 2020 *)

isA071395(n) = if(sigma(n) <= 2*n, 0, fordiv(n, d, if((d != n)&&(sigma(d) >= 2*d), return(0))); (1)); \\ After code in A071395
A003961(n) = { my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };
isA337386(n) = { my(x=A003961(n)); (sigma(x)>=2*x); };
for(n=1,2^13,if(isA071395(n), i=0; for(k=1,oo,if(isA337386(k*n),i++; print1(k,", "); break))));

a(n) = A337469(n) / A071395(n).

A363177 Primitive abundant numbers (A071395) that are cubefull numbers (A036966).

Original entry on oeis.org

26376098024367, 33912126031329, 1910383099764867, 2792098376579421, 5229860083034911875, 6886512413632368153, 8815747507513708671, 28966027524687899919, 42200802302982406288, 89594138836162749375, 224439112362213402759, 288564573037131517833, 512767531125033485625

Offset: 1

Amiram Eldar, May 19 2023

nonn

It seems that this sequence is also the intersection of A036966 and A091191 (checked up to 10^27).

Are there terms that are 4-full numbers (A036967)? There are none below 10^27.

Amiram Eldar, Table of n, a(n) for n = 1..42 (terms below 10^27)
Wikipedia, Primitive abundant number.

Intersection of A036966 and A071395.
Subsequence of A363169 and A363175.
A306797 is a subsequence.
Cf. A036967, A091191.

cp's OEIS Frontend

A330872 Numbers k such that k and k+1 are both primitive abundant numbers (A071395).

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A295369 Number of squarefree primitive abundant numbers (A071395) with n prime factors.

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A306796 Primitive abundant numbers (A071395) that are squares.

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A306797 Primitive abundant numbers (A071395) that are cubes.

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A334419 Primitive abundant numbers (A071395) with a record gap to the next primitive abundant number.

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A363175 Primitive abundant numbers (A071395) that are powerful numbers (A001694).

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A306987 Primitive abundant numbers (A071395) that are pseudoperfect (A005835).

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A333967 Subsequence of A071395. The extra constraint is m is not a term if m*q/p is abundant where prime p|m and q is the least prime larger than p.

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