A008864 -id:A008864 - cp's OEIS frontend

A033634 OddPowerSigma(n) = sum of odd power divisors of n.

1, 3, 4, 3, 6, 12, 8, 11, 4, 18, 12, 12, 14, 24, 24, 11, 18, 12, 20, 18, 32, 36, 24, 44, 6, 42, 31, 24, 30, 72, 32, 43, 48, 54, 48, 12, 38, 60, 56, 66, 42, 96, 44, 36, 24, 72, 48, 44, 8, 18, 72, 42, 54, 93, 72, 88, 80, 90, 60, 72, 62, 96, 32, 43, 84, 144, 68, 54, 96, 144

Offset: 1

Yasutoshi Kohmoto

Odd power divisors of n are all the terms of A268335 (numbers whose prime power factorization contains only odd exponents) that divide n. - Antti Karttunen, Nov 23 2017

The Mobius transform is 1, 2, 3, 0, 5, 6, 7, 8, 0, 10, 11, 0, 13, 14, 15, 0, 17, 0, 19, 0, 21, 22, 23, 24, 0, 26, ..., where the places of zeros seem to be listed in A072587. - R. J. Mathar, Nov 27 2017

			The divisors of 7 are 1^1 and 7^1, which have only odd exponents (=1), so a(8) = 1+7 = 8. The divisors of 8 are 1^1, 2^1, 2^2 and 2^3; 2^2 has an even prime power and does not count, so a(8) = 1+2+8=11. The divisors of 12 are 1^1, 2^1, 3^1, 2^2, 2^1*3^1 and 2^2*3; 2^2 and 2^2*3 don't count because they have prime factors with even powers, so a(12) = 1+2+3+6 = 12.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A000203, A008864, A072587, A181150, A268335, A295316.
Cf. also A126849.
Cf. A065463, A072691.

A033634 := proc(n)
    a := 1 ;
    for d in ifactors(n)[2] do
        if type(op(2,d),'odd') then
            s := op(2,d) ;
        else
            s := op(2,d)-1 ;
         end if;
        p := op(1,d) ;
        a := a*(1+(p^(s+2)-p)/(p^2-1)) ;
    end do:
    a;
end proc: # R. J. Mathar, Nov 20 2010

f[e_] := If[OddQ[e], e+2, e+1]; fun[p_,e_] := 1 + (p^f[e] - p)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, May 14 2019 *)

A295316(n) = factorback(apply(e -> (e%2), factorint(n)[, 2]));
A033634(n) = sumdiv(n,d,A295316(d)*d); \\ Antti Karttunen, Nov 23 2017

Let n = Product p(i)^r(i) then a(n) = Product (1+[p(i)^(s(i)+2)-p(i)]/[p(i)^2-1]), where si=ri when ri is odd, si=ri-1 when ri is even. Special cases:
a(p) = 1+p for primes p, subsequence A008864.
a(p^2) = 1+p for primes p.
a(p^3) = 1+p+p^3 for primes p, subsequence A181150.
a(n) = Sum_{d|n} A295316(d)*d. - Antti Karttunen, Nov 23 2017
a(n) <= A000203(n). - R. J. Mathar, Nov 27 2017
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 - 1/(p*(p+1))) = A072691 * A065463 = 0.5793804... . - Amiram Eldar, Oct 27 2022

A045924 Numbers n such that prime(n) == -1 (mod n).

Original entry on oeis.org

1, 2, 3, 4, 10, 70, 72, 182, 440, 1053, 6458, 6461, 6471, 40087, 40089, 251737, 251742, 637320, 637334, 637336, 1617173, 4124466, 10553445, 10553455, 10553569, 10553570, 10553574, 10553576, 10553819, 10553829, 27067100, 27067262, 69709705, 69709719, 69709734, 69709873

Offset: 1

Len Smiley

nonn

Same as n such that n divides A008864(n). - David James Sycamore, Jul 23 2018

Also numbers n such that prime(n) == n-1 (mod n). - Muniru A Asiru, Jul 24 2018

			10 is a member because the 10th prime, 29, is congruent to -1 mod 10.

Giovanni Resta, Table of n, a(n) for n = 1..108
E. Labos, Graph of prime(n) mod n

Cf. A004648 and esp. A023143.

Cf. A052013, A048891, A092044, A092045, A092046, A092047, A092048, A092049, A092050, A092051, A092052, A008864.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 1; Do[ If[Mod[p = NextPrim[p], n] == n - 1, Print[n]], {n, 1, 10^9}] (* Robert G. Wilson v, Feb 18 2004 *)

isok(n) = Mod(prime(n), n) == -1; \\ Michel Marcus, Jul 24 2018

More terms from Patrick De Geest, Nov 15 1999

Terms a(33) and beyond from Giovanni Resta, Feb 23 2020

A373674 Last element of each maximal run of powers of primes (including 1).

Original entry on oeis.org

5, 9, 11, 13, 17, 19, 23, 25, 27, 29, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239

Offset: 1

Gus Wiseman, Jun 16 2024

nonn

A run of a sequence (in this case A000961) is an interval of positions at which consecutive terms differ by one.
The first element of the same run is A373673.
Consists of all powers of primes k such that k+1 is not a power of primes.

			The maximal runs of powers of primes begin:
   1   2   3   4   5
   7   8   9
  11
  13
  16  17
  19
  23
  25
  27
  29
  31  32
  37
  41
  43
  47
  49

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

For prime antiruns we have A001359, min A006512, length A027833.
For composite runs we have A006093, min A008864, length A176246.
For prime runs we have A067774, min A025584, length A251092 or A175632.
For squarefree runs we have A373415, min A072284, length A120992.
For nonsquarefree runs we have min A053806, length A053797.
For runs of prime-powers:
- length A174965
- min A373673
- max A373674 (this sequence)
- sum A373675
For runs of non-prime-powers:
- length A110969 (firsts A373669, sorted A373670)
- min A373676
- max A373677
- sum A373678
For antiruns of prime-powers:
- length A373671
- min A120430
- max A006549
- sum A373576
For antiruns of non-prime-powers:
- length A373672
- min A373575
- max A255346
- sum A373679
A000961 lists all powers of primes (A246655 if not including 1).
A025528 counts prime-powers up to n.
A057820 gives first differences of consecutive prime-powers, gaps A093555.
A361102 lists all non-prime-powers (A024619 if not including 1).
Cf. A000040, A005381, A007674, A014963, A068780, A068781, A073051, A373400.

pripow[n_]:=n==1||PrimePowerQ[n];
Max/@Split[Select[Range[nn],pripow],#1+1==#2&]//Most

A377286 Numbers k such that there are no prime-powers between prime(k)+1 and prime(k+1)-1.

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82

Offset: 1

Gus Wiseman, Oct 25 2024

nonn

			Primes 18 and 19 are 61 and 67, and the interval (62, 63, 64, 65, 66) contains the prime-power 64, so 18 is not in the sequence.

The interval from A008864(n) to A006093(n+1) has A046933(n) elements.
For powers of 2 instead of primes see A013597, A014210, A014234, A244508, A304521.
The nearest prime-power before prime(n)-1 is A065514, difference A377289.
These are the positions of 0 in A080101, or 1 in A366833.
The nearest prime-power after prime(n)+1 is A345531, difference A377281.
For at least one prime-power we have A377057.
For one instead of no prime-powers we have A377287.
For two instead of no prime-powers we have A377288.
A000015 gives the least prime-power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A031218 gives the greatest prime-power <= n.
A246655 lists the prime-powers not including 1, complement A361102.
Cf. A001597, A002808, A024619, A053707, A064113, A065890, A075526, A095195, A276781, A376597, A377051, A377282.

Select[Range[100], Length[Select[Range[Prime[#]+1,Prime[#+1]-1],PrimePowerQ]]==0&]

from itertools import count, islice
from sympy import factorint, nextprime
def A377286_gen(): # generator of terms
    p, q, k = 2, 3, 1
    for k in count(1):
        if all(len(factorint(i))>1 for i in range(p+1,q)):
            yield k
        p, q = q, nextprime(q)
A377286_list = list(islice(A377286_gen(),66)) # Chai Wah Wu, Oct 27 2024

A060800 a(n) = p^2 + p + 1 where p runs through the primes.

Original entry on oeis.org

7, 13, 31, 57, 133, 183, 307, 381, 553, 871, 993, 1407, 1723, 1893, 2257, 2863, 3541, 3783, 4557, 5113, 5403, 6321, 6973, 8011, 9507, 10303, 10713, 11557, 11991, 12883, 16257, 17293, 18907, 19461, 22351, 22953, 24807, 26733, 28057, 30103, 32221

Offset: 1

Jason Earls, Apr 27 2001

Terms are divisible by 3 iff p is of the form 6*m+1 (A002476). - Michel Marcus, Jan 15 2017

			a(3) = 31 because 5^2 + 5 + 1 = 31.

Harry J. Smith, Table of n, a(n) for n = 1..1000
R. J. Mathar, No common terms in the sequences sigma(p^i) and sigma(p^(i+1)) as p runs through the primes.

Cf. A001248, A131991, A131992, A131993, A253905.

Cf. A008864, A000203. - Zak Seidov, Feb 13 2016

[p^2+p+1: p in PrimesUpTo(200)]; // Vincenzo Librandi, Mar 20 2014

A060800:= n -> map (p -> p^(2)+p+1, ithprime(n)):
seq (A060800(n), n=1..41); # Jani Melik, Jan 25 2011

#^2 + # + 1&/@Prime[Range[200]] (* Vincenzo Librandi, Mar 20 2014 *)

{ n=0; forprime (p=2, prime(1000), write("b060800.txt", n++, " ", p^2 + p + 1); ) } \\ Harry J. Smith, Jul 13 2009

a(n) = A036690(n) + 1.
a(n) = 1 + A008864(n)*A000040(n) = (A030078(n) - 1)/A006093(n). - Reinhard Zumkeller, Aug 06 2007
a(n) = sigma(prime(n)^2) = A000203(A000040(n)^2). - Zak Seidov, Feb 13 2016
a(n) = A000203(A001248(n)). - Michel Marcus, Feb 15 2016
Product_{n>=1} (1 - 1/a(n)) = zeta(3)/zeta(2) (A253905). - Amiram Eldar, Nov 07 2022

More terms from Larry Reeves (larryr(AT)acm.org), May 03 2001

A008333 Sum of divisors of p+1, p prime.

Original entry on oeis.org

4, 7, 12, 15, 28, 24, 39, 42, 60, 72, 63, 60, 96, 84, 124, 120, 168, 96, 126, 195, 114, 186, 224, 234, 171, 216, 210, 280, 216, 240, 255, 336, 288, 336, 372, 300, 240, 294, 480, 360, 546, 336, 508, 294, 468, 465, 378, 504, 560, 432, 546, 744, 399, 728, 528, 720, 720, 558

Offset: 1

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 1..999 (offset adapted by Michel Marcus)

Cf. A000203, A008864, A051027.

for i from 1 to 500 do if isprime(i) then print(sigma(i+1)); fi; od;

lst={};Do[AppendTo[lst, DivisorSigma[1, Prime[n]+1]], {n, 6!}];lst (* Vladimir Joseph Stephan Orlovsky, Sep 20 2008 *)

a(n) = sigma(prime(n)+1); \\ Michel Marcus, May 11 2016

a(n) = A000203(A008864(n)). - Michel Marcus, May 11 2016

a(n) = A000203(A000203(prime(n))) = A051027(prime(n)). - Michel Marcus, May 11 2016

Offset 1 from Michel Marcus, Apr 19 2020

A187813 Numbers n whose base-b digit sum is not b for all bases b >= 2.

Original entry on oeis.org

0, 1, 2, 4, 8, 14, 30, 32, 38, 42, 44, 54, 60, 62, 74, 84, 90, 98, 102, 104, 108, 110, 114, 128, 138, 140, 150, 152, 158, 164, 168, 174, 180, 182, 194, 198, 200, 212, 224, 228, 230, 234, 240, 242, 252, 270, 278, 282, 284, 294, 308, 312, 314, 318, 332, 338, 348

Offset: 1

Tom Edgar, Aug 30 2013

Except for 1, every number is even.
No number ends in 6.
Numbers neither in A018900 nor in A226636 nor in A226969 nor in A227062 nor in A227080 nor ... . - R. J. Mathar, Sep 02 2013
From Hieronymus Fischer, Mar 27 2014, May 09 2014: (Start)
A079696 and this sequence have no terms in common.
Numbers which satisfy m == 1 (mod j) and m > j^2 for any j > 1 are not terms.
Example 1: m = 10^k, k>1, is not a term since 10^k == 1 (mod 9) and 10^k > 9^2.
Example 2: m = 1 + 3k, k > 3, is not a term, since m > 3(1+3) > 3^2.
This is the complement of the disjunction of A079696 with A239708.
Disregarding the first 3 terms, these are the numbers which are in A008864 but not in A239708. This leads to the following characterization: A number m > 2 is a term, i.e., satisfies digitalSum_b(m) <> b for all b > 1, if and only m is a prime number + 1 and m is not the sum of two distinct powers of 2.
a(6) is the only term such that a(n) = Prime(n) + 1. For n < 6, we have a(n) < Prime(n) + 1, and for n > 6, we have a(n) > Prime(n) + 1.
(End)

			8 has binary expansion (1,0,0,0) whose digit sum 1 is not 2,
ternary expansion (2,2) whose digit sum 4 is not 3,
quaternary expansion (2,0) whose digit sum 2 is not 4,
5-ary expansion (1,3) whose digit sum 4 is not 5,
6-ary expansion (1,2) whose digit sum 3 is not 6,
7-ary expansion (1,1) whose digit sum 2 is not 7,
8-ary expansion (1,0) whose digit sum 1 is not 8,
and b-ary expansion (8) when b>8 whose digit sum is 8 not b. Therefore, 8 is in the sequence.
3 has binary expansion (1,1) whose digit sum is 2, so 3 is not in the sequence.
From _Hieronymus Fischer_, Apr 10 2014: (Start)
a(10) = 42 (the 13th prime + 1)
a(100) = 618 (the 113th prime + 1)
a(1000) = 8172 (the 1026th prime + 1)
a(10^4) = 105254 (the 10042nd prime + 1)
a(10^5) = 1300464 (the 100056th prime + 1)
a(10^6) = 15486872 (the 1000063th prime + 1)
a(10^7) = 179425944 (the 10000071st prime + 1)
a(10^8) = 2038076324 (the 10^8 +84th prime + 1)
a(10^9) = 22801765334 (the 10^9 +92nd prime + 1)
a(10^10) = 252097803264 (the 10^10 +102nd prime + 1)
[calculation for large numbers processed with Smalltalk method A187813With: estimate; see Prog section]
(End)

Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Cf. A018900, A052224.
Cf. A007953, A079696, A008864, A239708, A000040.
Cf. A239703, A239705.

Q@n_:=AllTrue[Table[{b,Plus@@IntegerDigits[n,b]},{b,2,n}],#[[1]]!=#[[2]]&];
Select[Range[0, 1000], Q] (* Hans Rudolf Widmer, Oct 08 2022 *)

from itertools import count, islice
from sympy import isprime
def A187813_gen(startvalue=0): # generator of terms >= startvalue
    yield from filter(lambda n:n<3 or (isprime(n-1) and n.bit_count()!=2), count(max(startvalue,0)))
A187813_list = list(islice(A187813_gen(startvalue=20),30)) # Chai Wah Wu, Mar 24 2025

n=1000 #change n for more terms
S=[]
for i in [0..n]:
    test=False
    for b in [2..i]:
        if sum(Integer(i).digits(base=b))==b:
            test=True
            break
    if not test:
        S.append(i)
S
# From Hieronymus Fischer, Apr 10 2014: (Start)

A187813NextTerm
  "Calculates the next term of A187813 greater than the receiver, i.e., calculates a(n+1) from a(n).
  Usage: a(n) A187813NextTerm
  Answer: a(n+1)
  Version 1: Using numOfBasesWithDigitalSumEQBase from A239703 ==> fast calculation, since only the divisors of  have to tested to be candidates for bases b with base-b digital sum equal to b"
  | an |
  an := self + 1.
  [an numOfBasesWithDigitalSumEQBase > 0]
  whileTrue: [an := an+1].
  ^an
-----------
A187813NextTerm
  "Calculates the next term of A187813 greater than the receiver, i.e., calculates a(n+1) from a(n).
  Usage: a(n) A187813NextTerm
  Answer: a(n+1)
  Version 2: Using the equivalence with A008864 and A239708 ==> even much more faster calculation"
  | p q |
  self < 0 ifTrue: [^0].
  self = 0 ifTrue: [^1].
  self = 1 ifTrue: [^2].
  p := (self - 1) nextPrime.
  q := p+1-(2 raisedToInteger: (p+1 integerFloorLog: 2)).
  [q > 0 and: [(2 raisedToInteger: (q integerFloorLog: 2)) - q = 0]] whileTrue: [p := p nextPrime.
                   q := p + 1 - (2 raisedToInteger: (p + 1 integerFloorLog: 2))].
  ^p + 1
-----------
A187813
  "Calculates the n-th term of A187813, iteratively.
  Usage: n A187813
  Answer: a(n)"
  | an n |
  n := self.
  n < 3 ifTrue: [^#(0 1) at: n].
  an := 2.
  4 to: n do: [:i |an := an A187813NextTerm].
  ^an
-----------
A187813rec
  "Calculates the n-th term of A187813, using the recursive method <A187813With: param>
  Usage: n A187813
  Answer: a(n)"
  self < 3 ifTrue: [^#(0 1) at: self].
  ^self A187813With: self prime
-----------
A187813With: estimate
"Method to calculate the n-th term of A187813 based on the value estimate, recursively. The n-th prime is a adequate estimate. Valid for n > 2.
  Usage: n A187813With: estimate
  Answer: a(n)"
  | x m |
  (x:=((m:= estimate A239708inv)+self-3) prime + 1) = estimate
      ifFalse: [^self A187813With: x].
  (m + 1) A239708 = x
      ifTrue: [^self A187813With: x + 4].
  ^x
[End]

From Hieronymus Fischer, Mar 27 2014: (Start)
A239703(a(n)) = 0.
a(n+1) = min (p > a(n) | A239703(p) = 0)
[for a Smalltalk implementation see Prog section, method A187813NextTerm version 1].
a(n+1) = 1 + min (p > a(n) | p is prime AND ((q := p+1 - 2^floor(log_2(p+1)) = 0) OR (2^floor(log_2(q)) <> q)))
[for a Smalltalk implementation see Prog section, method A187813NextTerm version 2].
a(n) > Prime(n), for n > 5.
a(n - m) < Prime(n), for n > 1, where m := i*(i-1)/2 + j - 1, i := floor(log_2(Prime(n))), j := floor(log_2(Prime(n) - 2^i)).
a(n - m) < Prime(n), for n > 32, where m := i*(i-1)/2 + j - 16 with i and j above.
a(n) = Prime(n + m - 3) + 1, where m = max ( k | A239708(k) < a(n)), n > 3.
Remark: This identity can be used to calculate a(n) recursively. For a Smalltalk implementation see Prog section, methods A187813rec and A187813With: estimate.
With same conditions: a(n) = A008864(n + m - 3).
a(n - m + 3) = Prime(n) + 1, where m = max ( k | A239708(k) < Prime(n)), n > 3, provided Prime(n) + 1 is not a term of A239708.
(End)

A344695 a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.

Original entry on oeis.org

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 3, 74, 114, 4, 20, 96, 168, 80, 6, 1, 126, 84, 32, 108

Offset: 1

Antti Karttunen and Peter Munn, May 26 2021

nonn

This is not multiplicative. The first point where a(m*n) = a(m)*a(n) does not hold for coprime m and n is 108 = 4*27, where a(108) = 8, although a(4) = 1 and a(27) = 4. See A344702.
A more specific property holds: for prime p that does not divide n, a(p*n) = a(p) * a(n). In particular, on squarefree numbers (A005117) this sequence coincides with sigma and psi, which are multiplicative.
If prime p divides the squarefree part of n then p+1 divides a(n). (For example, 20 has square part 4 and squarefree part 5, so 5+1 divides a(20) = 6.) So a(n) = 1 only if n is square. The first square n with a(n) > 1 is a(196) = 21. See A344703.
Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes). 5 does not appear as a term until a(366025) = 5, where 366025 = 5^2 * 11^4. At this point, the missing numbers less than 22 are 2, 10 and 17. 17 appears at the latest by a(17^2 * 103^16) = 17.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to sigma(n)

Cf. A000203, A001615, A005117, A244963, A344696, A344697, A344702, A344703 (numbers k for which a(k^2) > 1).
Cf. also A048250, A291750, A291751, A340070, A323363.
Subsets of range: A008864, A065091 (conjectured).

Table[GCD[DivisorSigma[1,n],DivisorSum[n,MoebiusMu[n/#]^2*#&]],{n,100}] (* Giorgos Kalogeropoulos, Jun 03 2021 *)

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A344695(n) = gcd(sigma(n), A001615(n));
(Python 3.8+)
from math import prod, gcd
from sympy import primefactors, divisor_sigma
def A001615(n):
    plist = primefactors(n)
    return n*prod(p+1 for p in plist)//prod(plist)
def A344695(n): return gcd(A001615(n),divisor_sigma(n)) # Chai Wah Wu, Jun 03 2021

a(n) = gcd(A000203(n), A001615(n)).
For prime p, a(p^e) = (p+1)^(e mod 2).
For prime p with gcd(p, n) = 1, a(p*n) = a(p) * a(n).
a(A007913(n)) | a(n).
a(n) = gcd(A000203(n), A244963(n)) = gcd(A001615(n), A244963(n)).
a(n) = A000203(n) / A344696(n).
a(n) = A001615(n) / A344697(n).

A341605 Square array A(n,k) = A017665(A246278(n,k)), read by falling antidiagonals; numerator of the abundancy index as applied onto prime shift array A246278.

Original entry on oeis.org

3, 7, 4, 2, 13, 6, 15, 8, 31, 8, 9, 40, 48, 57, 12, 7, 32, 156, 96, 133, 14, 12, 26, 72, 400, 168, 183, 18, 31, 16, 248, 16, 1464, 252, 307, 20, 13, 121, 84, 684, 216, 2380, 360, 381, 24, 21, 124, 781, 144, 1862, 280, 5220, 480, 553, 30, 18, 104, 342, 2801, 240, 3294, 432, 7240, 720, 871, 32

Offset: 1

Antti Karttunen, Feb 16 2021

Ratio A341605(row, col)/A341606(row, col) shows the abundancy index when applied to the natural numbers > 1 as ordered in the prime shift array A246278:
  n =       1            2        3              4            5              6
2n  =       2            4        6              8           10             12
----+--------------------------------------------------------------------------
  1 |     3/2,         7/4,     2/1,          15/8,         9/5,           7/3,
  2 |     4/3,        13/9,     8/5,         40/27,       32/21,         26/15,
  3 |     6/5,       31/25,   48/35,       156/125,       72/55,       248/175,
  4 |     8/7,       57/49,   96/77,       400/343,       16/13,       684/539,
  5 |   12/11,     133/121, 168/143,     1464/1331,     216/187,     1862/1573,
  6 |   14/13,     183/169, 252/221,     2380/2197,     280/247,     3294/2873,
  7 |   18/17,     307/289, 360/323,     5220/4913,     432/391,     6140/5491,
we see that when going down in each column, the magnitude of the ratio decreases monotonically, which follows because the abundancy index of prime(i+1)^e is less than that of prime(i)^e (see A336389). The first ratio that is < 2 (corresponding to the first deficient number obtained when 2*n is successively prime shifted) is found at row number 1+A336835(2*n) = 1+A378985(n) for column n.
Each ratio r at row n and column k is a product of the topmost ratio (on row 1), and the product of all ratios on rows 1..(row-1) given in arrays A341626/A341627:
  n =       1            2        3              4            5              6
2n  =       2            4        6              8           10             12
----+--------------------------------------------------------------------------
  1 |     8/9,       52/63,     4/5,         64/81,     160/189,         26/35,
  2 |    9/10,     279/325,     6/7,     1053/1250,     189/220,       372/455,
  3 |   20/21,   1425/1519,   10/11,   12500/13377,     110/117,     4275/4774,
  4 |   21/22,     343/363,   49/52,   62769/66550,     351/374,     2401/2574,
  5 |   77/78, 22143/22477,   33/34, 791945/804102,   6545/6669, 199287/205751,
  6 | 117/119, 51883/52887, 130/133, 573417/584647, 13338/13685, 518830/531981,
In other words, if r(row,col) = A341605(row,col)/A341606(row,col) and d(row,col) = A341626(row,col)/A341627(row,col), then r(row+1,col) = r(row,col)*d(row,col), that is, each column in the latter arrays of ratios gives the first quotients of ratios in the corresponding columns in the former array, and they are all < 1.
See also comments and examples in A341606.
By lemma given in A341529, the ratio A341626/A341627 stays in open interval (0.5 .. 1). - Antti Karttunen, Jan 02 2025

			The top left corner of the array:
  k=   1    2    3      4    5      6    7       8      9     10    11      12
2k =   2    4    6      8   10     12   14      16     18     20    22      24
----+--------------------------------------------------------------------------
n=1 |  3,   7,   2,    15,   9,     7,  12,     31,    13,    21,   18,      5,
  2 |  4,  13,   8,    40,  32,    26,  16,    121,   124,   104,   56,     16,
  3 |  6,  31,  48,   156,  72,   248,  84,    781,   342,   372,  108,   1248,
  4 |  8,  57,  96,   400,  16,   684, 144,   2801,   152,   114,  160,   4800,
  5 | 12, 133, 168,  1464, 216,  1862, 240,  16105,  2196,  2394,  288,  20496,
  6 | 14, 183, 252,  2380, 280,  3294, 336,  30941,  4298,  3660,  420,   2520,
  7 | 18, 307, 360,  5220, 432,  6140, 540,  88741,  6858,  7368,  576, 104400,
  8 | 20, 381, 480,  7240, 600,  9144, 640, 137561, 11060, 11430,   40, 173760,
  9 | 24, 553, 720, 12720, 768, 16590, 912, 292561, 20904, 17696, 1008, 381600,
etc.

Cf. A017665, A246278, A341626, A341627, A378985, A379500.
Cf. A008864 (column 1), A378995 (row 1).
Cf. A341606 (denominators), A341626 (numerators of the columnwise first quotients of A341605/A341606), A341627 (and their denominators), A355925, A355927.
Cf. also A336389, A336834, A336835, A337473, A337474.

up_to = 105;
A246278sq(row,col) = if(1==row,2*col, my(f = factor(2*col)); for(i=1, #f~, f[i,1] = prime(primepi(f[i,1])+(row-1))); factorback(f));
A017665(n) = numerator(sigma(n)/n);
A341605sq(row,col) = A017665(A246278sq(row,col));
A341605list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A341605sq(col,(a-(col-1))))); (v); };
v341605 = A341605list(up_to);
A341605(n) = v341605[n];

A(n, k) = A017665(A246278(n, k)).
A(n, k) = A355927(n, k) / A355925(n, k). - Antti Karttunen, Jul 22 2022
A(n, k) = A379500(n, k) / A341606(n, k). - Antti Karttunen, Jan 04 2025

A377466 Numbers k such that there is more than one perfect power x in the range prime(k) < x < prime(k+1).

Original entry on oeis.org

4, 9, 11, 30, 327, 445, 3512, 7789, 9361, 26519413

Offset: 1

Gus Wiseman, Nov 02 2024

Perfect powers (A001597) are numbers with a proper integer root, the complement of A007916.
Is this sequence finite?
The Redmond-Sun conjecture (see A308658) implies that this sequence is finite. - Pontus von Brömssen, Nov 05 2024

			Primes 9 and 10 are 23 and 29, and the interval (24,25,26,27,28) contains two perfect powers (25,27), so 9 is in the sequence.

For powers of 2 see A013597, A014210, A014234, A188951, A244508, A377467.
For no prime-powers we have A377286, ones in A080101.
For a unique prime-power we have A377287.
For squarefree numbers see A377430, A061398, A377431, A068360, A224363.
These are the positions of terms > 1 in A377432.
For a unique perfect power we have A377434.
For no perfect powers we have A377436.
A000015 gives the least prime power >= n.
A000040 lists the primes, differences A001223.
A000961 lists the powers of primes, differences A057820.
A001597 lists the perfect powers, differences A053289, seconds A376559.
A007916 lists the non-perfect-powers, differences A375706, seconds A376562.
A046933 counts the interval from A008864(n) to A006093(n+1).
A081676 gives the greatest perfect power <= n.
A131605 lists perfect powers that are not prime-powers.
A246655 lists the prime-powers not including 1, complement A361102.
A366833 counts prime-powers between primes, see A053607, A304521.
A377468 gives the least perfect power > n.
Cf. A000720, A023055, A031218, A045542, A052410, A053706, A069623, A116086, A116455, A216765, A308658, A336416, A345531, A375740, A376560, A376561, A377057.

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Select[Range[100],Count[Range[Prime[#]+1, Prime[#+1]-1],_?perpowQ]>1&]

from itertools import islice
from sympy import prime
from gmpy2 import is_power, next_prime
def A377466_gen(startvalue=1): # generator of terms >= startvalue
    k = max(startvalue,1)
    p = prime(k)
    while (q:=next_prime(p)):
        c = 0
        for i in range(p+1,q):
            if is_power(i):
                c += 1
                if c>1:
                    yield k
                    break
        k += 1
        p = q
A377466_list = list(islice(A377466_gen(),9)) # Chai Wah Wu, Nov 04 2024

a(n) = A000720(A116086(n)) = A000720(A116455(n)) for n <= 10. This would hold for all n if there do not exist more than two perfect powers between any two consecutive primes, which is implied by the Redmond-Sun conjecture. - Pontus von Brömssen, Nov 05 2024

a(10) from Pontus von Brömssen, Nov 04 2024

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