A002975 -id:A002975 - cp's OEIS frontend

A298973 Squarefree primitive abundant numbers (first definition: having only deficient proper divisors).

70, 1430, 1870, 2002, 2090, 2210, 2470, 2530, 2990, 3190, 3230, 3410, 3770, 4030, 4070, 4510, 4730, 5170, 5830, 15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 49742, 50505, 51765, 54285, 55965, 58695, 58786, 60214, 61215, 64155, 67298

Offset: 1

M. F. Hasler, Feb 16 2018

nonn

Squarefree numbers (A005117) in A071395. The number of terms with n prime factors are counted in A295369. The subsequence of odd terms is A249263.
Two variants of the present sequence are possible: the terms listed by size, or as a table whose n-th row gives all those with n prime factors (so that A295369 would be the row lengths). They would differ only from a(322) = 692835 on, which is the first term with 6 prime factors, while a(755) = 4199030 is the last term with 5 prime factors.
A subsequence of the variant A249242, squarefree primitive abundant numbers using the 2nd definition, A091191, i.e., having no abundant proper divisors.
These numbers are also primitive unitary abundant numbers: unitary abundant numbers (A034683) that are also primitive abundant numbers (A071395). A unitary abundant number k is primitive if and only if usigma(k) - 2*k < 2*k/p^e, where p^e is the largest prime power dividing k and usigma is the sum of unitary divisors function (A034448). For numbers k in this sequence limsup_{k->oo} usigma(k)/k = 2. (Prasad and Reddy, 1990). - Amiram Eldar, Jul 18 2020

			The only squarefree primitive abundant number (SFPAN) with only 3 prime factors is a(1) = 2*5*7 = 70. Indeed, this number is abundant (sigma(70) - 70 = 1 + 2 + 5 + 7 + 10 + 14 + 35 = 74) but all of 2*5, 2*7 and 5*7 are deficient. This is also the smallest (thus primitive) weird number, see A002975.
The A295369(4) = 18 SFPAN with 4 prime factors range from a(2) = 2*5*11*13 = 1430 to a(19) = 2*5*11*53 = 5830.
The A295369(5) = 610 SFPAN with 5 prime factors range from a(20) = 3*5*7*11*13 = 15015 to a(755) = 2*5*11*59*647 = 4199030, but the first term with 6 prime factors occurs already at a(322) =  3*5*11*13*17*19 = 692835.

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter III, p. 115.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from M. F. Hasler)
V. Siva Rama Prasad and D. Ram Reddy, On primitive unitary abundant numbers, Indian J. Pure Appl. Math., Vol. 21, No. 1 (1990), pp. 40-44.
D. P. Shukla and Shikha Yadav, Composition of arithmetical functions with generalization of perfect and related numbers, Commentationes Mathematicae, Vol. 52, No. 2 (2012), pp. 153-170.

Cf. A005117 (squarefree numbers), A071395 (primitive abundant numbers, first definition), A091191 (idem, second definition), A249242 (squarefree numbers in A091191).

Cf. A034448, A034683.

spaQ[n_] := SquareFreeQ[n] && DivisorSigma[1, n] > 2*n && AllTrue[Most @ Divisors[n], DivisorSigma[1, #] < 2*# &]; Select[Range[70000], spaQ] (* Amiram Eldar, Jul 18 2020 *)

is_A298973(n)=issquarefree(n)&&is_A071395(n)

A320875 Least d > 0 such that both Q = M + 2d and R = M + (M^2-1)/(Q-M) are prime, where M = 2^n - 1 = A000225(n), or 0 if there is no such d.

Original entry on oeis.org

0, 1, 2, 1, 3, 8, 2, 0, 6, 4, 66, 16, 20, 0, 6, 1, 2720, 0, 32, 0, 164, 8, 0, 524288, 153, 3573184, 2097152, 7354396, 19436, 4517888, 672, 0, 174080, 0, 262146, 1984, 48, 0, 4194296, 79, 30720, 128, 1825866, 4188889, 194396, 27227248, 0, 16384, 723, 0, 265227072, 22771712, 13982720, 134217728, 59885796, 587144, 19436, 0, 17179869152, 8388608

Offset: 1

M. F. Hasler, Nov 11 2018

nonn

It is easy to see that R can't be an integer unless M < Q < M^2 + M.
Nonzero terms yield primitive weird numbers (PWN) 2^(n-1)*Q*R, cf. A258882.
This idea was used by S. Kravitz in 1976 and 35 years later by students of CWU to find the largest known PWN, cf. links and A242025, A242993, A242998, A242999, A243003. The 226 digits mentioned in the news article correspond not to a PWN but to the prime R for a(381) = 5456. The corresponding prime Q = M(381) + 2*5456 is the 54th prime after M(381), and only the third one for which R is an integer. The 127 digit PWN they found earlier corresponds to a non-minimal solution d = 34008 for n = 109. (It is a matter of seconds to find many much larger solutions, see examples.) This news led to renewed interest in this topic and a series of recent research papers, see references in A258882 and A002975.
Sequences A242025, A242993, A242998, A242999, A243003 consider PWN of the form 2^(k-1)*Q*R(k,Q) where the prime Q is fixed to be a Mersenne prime A000668, and k is varied to find a prime R.
Zero terms do not mean that there aren't PWN of the form 2^(n-1)*p*q with M+1 = 2^n < p < 2M < q < M(M+1). For example, a(8) = 0, but there are A258333(8) = 53 weird numbers with such (p,q). However, the two primes never satisfy the relation (p-M)(q-M) = M^2-1 which is considered here for (Q,R). - M. F. Hasler, Nov 20 2018

			a(109) = 8436 yields a 62-digit prime R and a 127 digit PWN 2^108*Q*R.
a(381) = 5456 yields a 226-digit prime R and a 455 digit PWN 2^380*Q*R. (This and the preceding one are mentioned in the News articles, cf LINKS.)
a(391) = 16386 leads to a 231-digit prime R and a 466-digit PWN 2^390*Q*R.
a(409) = 12360 leads to a 242-digit prime R and a 488-digit PWN 2^408*Q*R.
a(421) = 1661 leads to a 250-digit prime R and a 504-digit PWN 2^420*Q*R.
a(430) = 10304 leads to a 255-digit prime R and a 514-digit PWN 2^429*Q*R.
a(441) = 36080 leads to a 261-digit prime R and a 526-digit PWN 2^440*Q*R.
a(505) = 20726 leads to a 300-digit prime R and a 604-digit PWN 2^504*Q*R.

Daily Record, CWU students find longest 'weird' number, and also Yakima Herald, CWU math students calculate what no mathematician has before (backup on web.archive.org, page no longer available), both from Dec. 5, 2013
S. Kravitz, A search for large weird numbers. J. Recreational Math. 9(1976), 82-85 (1977). Zbl 0365.10003

Cf. A258882, subsequence of A002975.

Cf. A242025, A242993, A242998, A242999, A243003 (all related to the case Q = 2^p-1 in A000668, p in A000043).

a(n)={my(M=2^n-1,S=M^2-1); fordiv(S+!S,D, ispseudoprime(M+D)&& ispseudoprime(M+S/D)&& return(D/2))} \\ Much faster than the variant below, but requires increasingly more stack space (allocatemem()) for larger n.

A320875(n,L=0)={my(M=2^n-1,S=M^2-1); forprime(Q=M+1,if(L,L,M<

A265418 a(1)=2; for n>1, a(n) is the least prime q greater than p = a(n-1) such that p/q reaches a new minimum.

Original entry on oeis.org

2, 3, 5, 11, 29, 79, 223, 631, 1787, 5077, 14431, 41023, 116639, 331651, 943031, 2681467, 7624649, 21680413, 61647497, 175292519, 498438203, 1417291781, 4030020143, 11459222851, 32583903763, 92651203181, 263450491193, 749112358279, 2130075077051, 6056794796849, 17222286484817

Offset: 1

Robert G. Wilson v, Dec 08 2015

nonn

Inspired by the fact that 294911/235927 = 1.2500095368..., two primes together with 2^16 form a primitive weird number (A002975(9729)).
p/q ->
0.3516835469078526298668938767771073728
...
Each pair of initial primes, p & q, will yield a different ratio.

			2/3 is 0.666... is a new low or minimum;
3/5 is 0.600... is a new minimum;
5/11 is 0.454... is a new minimum;
11/29 is 0.379... is a new minimum;
29/79 is 0.367... is a new minimum;
... 6056794796849/17222286484817 is 0.351... is a new minimum; etc.

Robert G. Wilson v, Table of n, a(n) for n = 1..1620

f[lst_List] := Block[{p = lst[[-2]], q = lst[[-1]]}, Append[lst, NextPrime[q^2/p]]]; Nest[f, {2, 3}, 29]

A265726 Primitive weird numbers whose abundance is a record.

Original entry on oeis.org

70, 836, 7192, 9272, 73616, 243892, 338572, 1188256, 1901728, 3963968, 28279232, 36228736, 91322752, 141659096, 263144192, 351295232, 664373504, 2113834496, 5522263024, 6933503488, 19179527168, 22755515392, 31574500724, 98620009472, 135895635968

Offset: 1

Douglas E. Iannucci and Robert G. Wilson v, Dec 14 2015

nonn

Although the abundance A(n) = sigma(n) - 2n is increasing, the (relative) abundancy sigma(n)/n is decreasing, except at indices {3, 6, 8, 15, 16, 19, 24 ...}. No term has larger abundancy than 2 + 2/35, that of a(1). - M. F. Hasler, Nov 14 2018

			a(1) = 70 since it is the first term in A002975; its abundance is 4.
a(2) is 836 since its abundance, 8, exceeds that of a(1); 4.
a(3) is 7192 = A002975(5) since its abundance, 16, exceeds that of a(2) and that of A002975(1..4).

Douglas E. Iannucci, On primitive weird numbers of the form 2^k*p*q, arXiv:1504.02761 [math.NT], 2015.
Giuseppe Melfi, On the conditional infiniteness of primitive weird numbers, Journal of Number Theory, Vol. 147, Feb 2015, pgs 508-514.
Wikipedia, Weird number

Cf. A002975, A258250, A258333, A258374, A258375, A258401, A258882, A258883, A258884, A258885, A265727, A265728.

(* copy the terms from A002975, assign them equal to 'lst' and then *) f[n_] := DivisorSigma[1, n] - 2n; k = 1; lsu = {}; mx = 0; While[k < 647, ds = f@ lst[[k]]; If[ds > mx, mx = ds; AppendTo[lsu, lst[[k]]]]; k++]; lsu

A295001 a(n) = nextprime(1/(2/sigma[-1](P(n)) - 1)) where P(n) = Product_{0 <= k < n} a(k), sigma[-1](x) = sigma(x)/x, a(0) = 4.

Original entry on oeis.org

4, 11, 23, 257, 13007, 44512049, 46880563785749, 125637016478802067649031191, 652182699863469019760217209096329987925268834143233, 1800254420479597976179975458181139131985404009703136640765845238082635790500153934999846722641241849

Offset: 0

M. F. Hasler, Nov 23 2017

Here, nextprime(x) = min { p > x; p prime }, prevprime(x) = max { p < x; p prime }.
The next term, a(10) ~ 3.1*10^196, is too large to be displayed above.
From a(3) on, a(n+1) has roughly twice the number of digits of a(n).
For n >= 1, a(n) is the least prime such that Product_{k=0..n} a(k) is deficient. This implies that (Product_{k=0..n-1} a(k))*prevprime(a(n)) is perfect for n = 1, and a primitive weird number (A002975) for some but not all larger n.

			Let Q(x) = 1/(2/sigma[-1](x) - 1), P(n) = Product(a(k), k=0..n-1), and start with a(0) = 4 = P(1). Then:
Q(P(1)) = 7, a(1) = 11. (4*7 is perfect, P(2) = 4*11 is deficient.)
Q(P(2)) = 21, a(2) = 23. (4*11*19 is weird, P(3) = 4*11*23 is deficient.)
Q(P(3)) = 252, a(3) = 257. (P(3)*251 is weird, P(4) = 4*11*23*257 is deficient.)
Q(P(4)) = 13003.2, a(4) = 13007. (P(4)*13003 is weird, P(5) = 4*11*23*257*13007 is deficient.)
Q(P(5)) = 44512006.7..., a(5) = 44512049. (P(5)*44511949 is weird ; P(6) = 4*11*257*44512049 is deficient.)
P(6)*prevprime(a(6)) is semiperfect, i.e., no more weird.

M. F. Hasler, Table of n, a(n) for n = 0..13

Cf. A002975 (primitive weird numbers), A000203 (sigma).
The nextprime and prevprime functions are here used for possibly non-integral arguments, but rounding these down or up allows the use of the nextprime and prevprime functions for integer arguments, A151800 and A151799.
See A262228 for the variant starting with a(0) = 1.

A295001=List(m=4);for(n=1,13,listput(A295001,p=nextprime(1\(2/sigma(m,-1)-1)+1));p>default(primelimit)&&addprimes(p);m*=p)

A357050 Number of ways A005101(n)+1 can be written as sum of a subset of the proper divisors of A005101(n), the n-th abundant number.

Original entry on oeis.org

2, 1, 1, 4, 4, 7, 2, 2, 10, 2, 2, 32, 2, 1, 26, 1, 6, 24, 1, 19, 20, 2, 1, 1, 20, 4, 1, 270, 11, 14, 1, 14, 116, 12, 9, 12, 3, 195, 1, 2, 719, 1, 42, 1, 8, 9, 8, 2, 148, 142, 6, 1, 8, 6, 6, 2154, 1, 534, 1, 6, 125, 108, 1, 6, 117, 1, 447, 4

Offset: 1

M. F. Hasler, Dec 13 2022

nonn

Obviously, for non abundant numbers (including perfect numbers) N, there is no way to write N+1 as the sum of a subset of N's proper divisors. Therefore we consider only abundant N = A005101(n) here.

The first zero appears for the seventh weird and primitive weird number A006037(7) = A002975(7) = 9272 = A005101(2310) (which surprisingly is w = A100696(1), the first weird number such that the sum of its divisors less than its abundance A033880(w) is larger than that).

Cf. A005101, A006037, A002975, A005835 (abundant, weird, primitive weird and pseudoperfect numbers).

Cf. A033880 (abundance), A100696.

{A357050(n)= sum(b=1, -1+2^#d=divisors(n)[^-1], vecsum(vecextract(d,b))==n+1)} \\ not very efficient, better use code as in is_A005835().

A299401 Number of primitive weird numbers (PWN) of the form 2^npq*r, where p,q,r are odd primes.

Original entry on oeis.org

2, 7, 12, 18, 41, 130

Offset: 1

M. F. Hasler, Feb 18 2018

The analog of A258333 for three odd factors.

Note that this sequence counts PWN with nonsquarefree odd part, which are excluded from A258883, see also A273815.

			In the sequel, p,q,r denote arbitrary odd primes.
The a(1) = 2 PWN of the form 2*p*q*r are A258883(1..2): 4030 = 2*5*13*31 and 5830 = 2*5*11*53.
The a(2) = 7 PWN of the form 2^2*p*q*r are 45356, 91388, 243892, 254012, 338572, 343876 and 388076, with (p,q,r) = (17, 23, 29), (11, 31, 67), (11, 23, 241), (11, 23, 251), (13, 17, 383), (13, 17, 389) and (13, 17, 439).
The a(3) = 12 PWN of the form 2^3*p*q*r range from 1713592 to 173482552.
The a(4) = 18 PWN of the form 2^4*p*q*r range from 15126992 to 6587973136.
The a(5) = 41 PWN of the form 2^5*p*q*r range from 569494624 to 297512429728.

Cf. A258883, A002975.

A299401(n,k=3,m=2^n,P=3,cnt=0,s)={if(k>1,forprime(p=P,,(s=sigma(m*p,-1))<2||next;p>P&&s*(1+1/p)^(k-1)<2&&break;/*printf("%d",[k,p]);*/cnt+=A299401(n,k-1,m*p,p)),s=sigma(m);my(p=1\(2*m/s-1)+1,d);while(PA005835(m*p,d=divisors(m*p),s+(s-m)*p,#d-1)&&cnt++));cnt}

A298157 Number of primitive abundant numbers (A071395) with n prime factors, counted with multiplicity.

Original entry on oeis.org

0, 0, 2, 25, 906, 265602, 13232731828

Offset: 1

Gianluca Amato, Feb 15 2018

This uses the first definition of primitive abundant numbers, A071395: having only deficient proper divisors. The second definition (A091191: having no abundant proper divisors) would yield infinite a(3), since all numbers 6*p, p > 3, are in that sequence.
See A287728 for the number of ODD primitive abundant numbers with n prime factors, counted with multiplicity and A295369 for the number of squarefree primitive abundant numbers with n distinct prime factors.
It appears that a(n) is just slightly larger than A295369(n).

			For n=3, the only two primitive abundant numbers (PAN) are 2*2*5 = 20 and 2*5*7 = 70. The latter is also a primitive weird number, see A002975.
For n=4, the 25 PAN range from 2^3*11 = 88 to 2*5*11*53 = 5830.

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

Cf. A071395 (primitive abundant numbers), A091191 (alternative definition), A287728 (counts of odd PAN), A295369 (counts of squarefree PAN).

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

A362916 Least weird number (A006037) with exactly n divisors that are weird numbers, or -1 if no such number exists.

Original entry on oeis.org

70, 10430, 1554070, 5681270, 34501908070, 1436692670

Offset: 1

Amiram Eldar, May 26 2023

a(8) = 5702250610, and there are no more terms below 10^11.

			Table of the weird divisors for the first 6 terms:
  n  a(n)         weird divisors
  -  -----------  ----------------------------------------------
  1  70           70
  2  10430        70, 10430
  3  1554070      70, 10430, 1554070
  4  5681270      70, 19390, 20510, 5681270
  5  34501908070  70, 10430, 1554070, 231556430, 34501908070
  6  1436692670   70, 14770, 32270, 3116470, 6808970, 1436692670

Cf. A006037, A002975.

cp's OEIS Frontend

A298973 Squarefree primitive abundant numbers (first definition: having only deficient proper divisors).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

A320875 Least d > 0 such that both Q = M + 2d and R = M + (M^2-1)/(Q-M) are prime, where M = 2^n - 1 = A000225(n), or 0 if there is no such d.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

A265418 a(1)=2; for n>1, a(n) is the least prime q greater than p = a(n-1) such that p/q reaches a new minimum.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

A265726 Primitive weird numbers whose abundance is a record.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A295001 a(n) = nextprime(1/(2/sigma[-1](P(n)) - 1)) where P(n) = Product_{0 <= k < n} a(k), sigma[-1](x) = sigma(x)/x, a(0) = 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A357050 Number of ways A005101(n)+1 can be written as sum of a subset of the proper divisors of A005101(n), the n-th abundant number.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

A299401 Number of primitive weird numbers (PWN) of the form 2^n*p*q*r, where p,q,r are odd primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A298157 Number of primitive abundant numbers (A071395) with n prime factors, counted with multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

SageMath

A299401 Number of primitive weird numbers (PWN) of the form 2^npq*r, where p,q,r are odd primes.