Giovanni Resta - cp's OEIS frontend

A386965 Number of permutations p of [2n] such that there is at least one index i in [2n-1] with p(i+1) = n + p(i).

1, 10, 294, 16296, 1458120, 191751120, 34807535280, 8337722440320, 2547572372311680, 966944845408147200, 446304490431888211200, 246166572372916851532800, 159902551429370021259187200, 120818209587660157360960972800, 105060730670227917425027835648000

Offset: 1

Giovanni Resta, Aug 11 2025

Problem 6 at IMO '89 essentially asks to show that a(n) > (2*n)!/4.

			The 10 permutations corresponding to a(2) are 1243, 1324, 1342, 2134, 2413, 2431, 3124, 3241, 4132, 4213.

30th International Mathematical Olympiad (1989), Problem 6.
Index to sequences related to Olympiads.

Cf. A002674, A324361, A159960.

a[n_] := Sum[(-1)^(k+1) Binomial[n, k] (2 n - k)!, {k, n}]; Array[a, 15]

a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (2*n - k)!.
a(n) = (2*n)! * (1 - 1F1(-n; -2*n; -1)).
a(n) = n! * A324361(n).

A367600 Numbers that are not the comma-successor of any number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 32, 37, 38, 39, 40, 41, 42, 43, 49, 50, 51, 52, 53, 54, 60, 62, 63, 64, 65, 70, 74, 75, 76, 80, 86, 87, 90, 98, 200, 300, 400, 500, 600, 700, 800, 900, 2000, 3000, 4000, 5000, 6000, 7000

Offset: 1

Giovanni Resta, Nov 23 2023

These are the positive integers that do not appear in A367338.

All terms > 98 are of the form c*10^i for i >= 2 and 2 <= c <= 9; see proof in links. - Michael S. Branicky, Nov 28 2023

Michael S. Branicky, Table of n, a(n) for n = 1..110 (all terms < 10^9)
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Fibonacci Quarterly 62:3 (2024), 215-232.
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides
Michael S. Branicky, Comma-Predecessor Theorem

Cf. A121805, A337338, A367614.

from itertools import count, islice
def A367338(n):
    nn = n + 10*(n%10)
    return next((nn+y for y in range(1, 10) if str(nn+y)[0] == str(y)), -1)
def agen():
    A367338_set = set()
    for n in count(1):
        A367338_set.add(A367338(n))
        if n not in A367338_set:
            yield n
        # A367338_set.discard(n-100) # uncomment if memory is an issue
print(list(islice(agen(), 86))) # Michael S. Branicky, Nov 28 2023

A361336 Smallest decimal number containing n palindromic substrings (Version 2). See Comments for precise definition.

Original entry on oeis.org

0, 10, 11, 100, 1002, 111, 1000, 10002, 10001, 1111, 10000, 100002, 100001, 1000012, 11111, 100000, 1000002, 1000001, 10000012, 10000010, 111111, 1000000, 10000002, 10000001, 100000012, 100000010, 110111111, 1111111, 10000000, 100000002, 100000001, 1000000012, 1000000010

Offset: 1

N. J. A. Sloane, Apr 01 2023, based on postings to the Sequence Fans Mailing list by Eric Angelini, Mar 28 2023 (definition), and Giovanni Resta, Mar 28 2023 (terms)

Suppose m has decimal expansion d_1 d_2 ... d_k. A palindromic substring here is any substring d_i, d_{i+1}, ..., d_j with 1 <= i <= j <= n which is palindromic. In this version d_i can be 0 even if j>i. For example, if m = 10^3 + 1 = 1001 there are six substrings: 1, 0, 0, 1, 00, and 1001. See A361335 for Version 1.

Giovanni Resta, Table of n, a(n) for n = 1..99

Cf. A361335.

A361335 Smallest decimal number containing n palindromic substrings (Version 1). See Comments for precise definition.

Original entry on oeis.org

0, 10, 11, 101, 1001, 111, 1110, 10101, 10111, 1111, 11110, 102222, 101111, 1001111, 11111, 111110, 1022222, 1011111, 10011111, 11101111, 111111, 1111110, 10222222, 10111111, 100111111, 111111212, 110111111, 1111111, 11111110, 102222222, 101111111, 1001111111, 1111111212, 1101111111, 10101111111, 11111111, 111111110

Offset: 1

N. J. A. Sloane, Apr 01 2023, based on postings to the Sequence Fans Mailing list by Eric Angelini, Mar 28 2023 (definition), and Giovanni Resta, Mar 28 2023 (terms)

Suppose m has decimal expansion d_1 d_2 ... d_k. A palindromic substring here is any substring d_i, d_{i+1}, ..., d_j with 1 <= i <= j <= n which is palindromic, except that if d_i = 0 then i = j. For example, if m = 10^3 + 1 = 1001 there are five substrings: 1, 0, 0, 1, 1001 (but not 00). See A361336 for Version 2.

Giovanni Resta, Table of n, a(n) for n = 1..99

Cf. A361336.

A335817 a(k) is the index of the first occurrence of 2*k-1 in A208884, or 0 if it does not occur.

Original entry on oeis.org

1, 2, 13, 4, 6, 75, 22975, 11, 15, 10, 1417, 37, 17, 12, 573, 19, 16, 49, 28, 227, 34, 18, 26, 31, 1371, 20, 41, 36, 8181, 339, 38, 30, 40, 207, 43, 70, 1113, 91, 235, 32, 63, 203, 50, 60, 103, 48, 189, 42, 421, 57, 74, 98, 65, 259, 11997, 155, 44, 54, 199, 67

Offset: 1

Giovanni Resta, Jun 25 2020

nonn

It is conjectured that A208884 contains all the odd numbers. If so, a(n) > 0 for every n.

			The first terms of A208884 are 1, 3, 3, 7, 3, so a(2*1-1) = 1, a(2*2-1) = 2, and a(2*4-1) = 4.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A208884.

b[1]=1; b[n_] := b[n] = #/2^IntegerExponent[#, 2] &@ (n + b[n-1]); A = Transpose@ {b /@ #, #} &@ Range[25000]; A = DeleteDuplicatesBy[ Sort@ A, First]; Last /@ TakeWhile[ A, A[[ (First[#] + 1)/2, 1]] == First@# &]

A335498 a(n) is the least odd number k such that Omega(k) = n and Omega(k+2) = n+1, where Omega(k) is the number of prime factors of k (A001222).

Original entry on oeis.org

1, 7, 25, 873, 1375, 41875, 903123, 1015623, 49671873, 200921875, 1157734375, 41898828123, 496308203125, 10506958984375, 7739037109375, 382999267578123, 17016876976778523, 46804302197265625, 80713609326109375

Offset: 0

Giovanni Resta, Jun 11 2020

			a(3) = 873 because Omega(873) = Omega(3^2*97) = 3, Omega(873+2) = Omega(5^3*7) = 4 and 873 is the smallest such integer.

Cf. A001222, A322300, A335496.

a[n_] := Block[{ov=0, v=1, k=3}, While[ov != n || v != n+1, ov = v; k += 2; v = PrimeOmega@ k]; k-2]; a /@ Range[0, 6]

generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p,ceil(A/m)), B\m, my(t=m*q); if(bigomega(t-2) == k, listput(list, t-2))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n+1, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jul 08 2023

a(12)-a(18) from Daniel Suteu, Jul 08 2023

A335496 a(n) is the least odd number k such that Omega(k) = n, Omega(k+2) = n+1, and Omega(k+4) = n+2, where Omega(k) is the number of prime factors of k (A001222).

Original entry on oeis.org

23, 871, 5423, 229955, 13771373, 558588875, 21549990623, 1325878234371, 17040894859373, 429205867309373

Offset: 1

Zak Seidov and Giovanni Resta, Jun 11 2020

a(n) mod 81 for n = 1..8: {23, 61, 77, 77, 77, 77, 77, 0}.

			23 is a term: 23 is a prime, 25=5*5 is a semiprime, 27=3*3*3 is a triprime.
871 is a term: 871 = 13*67 (semiprime), 873 = 3*3*97 (triprime), 875 = 5*5*5*7 (quadprime).

Cf. A335498.

generate(A, B, n, k) = A=max(A, 2^n); (f(m, p, n) = my(list=List()); if(n==1, forprime(q=max(p, ceil(A/m)), B\m, my(t=m*q); if(bigomega(t-4) == k && bigomega(t-2) == k+1, listput(list, t-4))), forprime(q=p, sqrtnint(B\m, n), list=concat(list, f(m*q, q, n-1)))); list); vecsort(Vec(f(1, 3, n)));
a(n) = my(x=2^n, y=2*x); while(1, my(v=generate(x, y, n+2, n)); if(#v >= 1, return(v[1])); x=y+1; y=2*x); \\ Daniel Suteu, Jul 08 2023

a(n) >= A335498(n). - Daniel Suteu, Jul 08 2023

a(9)-a(10) from Daniel Suteu, Jul 08 2023

A335205 Numbers of m digits which are equal to the absolute value of the sum of the m-th powers of their digits, with alternating signs.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 48, 407, 5920, 5921, 2918379, 18125436, 210897052, 11647261846, 18107015789, 27434621679, 31332052290, 4986706842391, 485927682264092, 1287253463537089, 1217506990394433558, 11008589751726485523, 107765279704274410345, 109462377410000145640, 109462377410000145641, 118620909850977982494, 319591187568367788829

Offset: 1

Giovanni Resta, May 26 2020

Numbers n equal to |Sum_{j=1..k} (-1)^j*d_j^k| where d_1 d_2 ... d_k is the decimal expansion of n. A variant of narcissistic numbers (A005188), they are finite as well.
The last term is smaller than 1.2*10^50. - Jinyuan Wang, May 28 2020
Note that a(14) = a(13) + 1, a(29) = a(28) + 1 - Chai Wah Wu, Jun 03 2020

			5921 is a term because |5^4 - 9^4 + 2^4 - 1^4| = 5921.

Cf. A005188, A335151.

s[n_] := Block[{d = IntegerDigits@ n}, Abs@ Total[d^Length[d] (-1)^Range@ Length@ d]]; Select[ Range[0, 3*10^6], s[#] == # &]

is(k)= my(v=digits(k)); abs(sum(i=1, #v, (-1)^i*v[i]^#v))==k; \\ Jinyuan Wang, May 28 2020

a(24)-a(25) from Chai Wah Wu, May 31 2020
a(26) from Chai Wah Wu, Jun 01 2020
a(27)-a(31) from Chai Wah Wu, Jun 03 2020

A334885 Let q = p | p' be the digit concatenation of a prime p with its prime successor. If the result is a prime repeat the construction setting p = q. a(n) is the smallest prime for which this can be repeated exactly n times.

Original entry on oeis.org

3, 2, 13681, 467, 127787377, 200603842261

Offset: 0

Giovanni Resta, May 14 2020

a(6) > 10^13.

			Let "|" denote concatenation.
3 | 5 = 35, which is not prime, so a(0) = 3.
2 | 3 = 23 (prime), 23 | 29 = 2329 (composite), so a(1) = 2.
13681 | 13687 (prime), 1368113687 | 1368113699 (prime), 13681136871368113699 | 13681136871368113711 (composite), so a(2) = 13681.

Carlos Rivera, Puzzle 29. P_i = P_(i-1) & nxtprm(P_(i-1)), P_i = prime for i => 1, The Prime Puzzles and Problems Connection.

Cf. A030459, A034593, A036339, A036340, A036341.

a[n_] := Block[{pp=1, p, q, c=-1}, While[ c!=n, c=0; p = pp = NextPrime@ pp; While[ PrimeQ[ q = FromDigits[ Join @@ IntegerDigits@{p, NextPrime@ p}]], c++; p = q]]; pp]; a /@ Range[0, 3]

A334726 a(k) is the earliest start of sequence of exactly k primes generated according to the rules stipulated in A005150.

Original entry on oeis.org

1, 2, 3, 7, 373, 1223, 233, 19972667609, 75022592087629

Offset: 0

Giovanni Resta, May 09 2020

			The sequence starting at 7 is 7 (prime), 17 (prime), 1117 (prime), and 3117 (composite), so a(3) = 7.

Carlos Rivera, Puzzle 36. Sequences of "descriptive primes", The Prime Puzzles and Problems Connection.
Carlos Rivera, Puzzle 999. In Memoriam to John Horton Conway, The Prime Puzzles and Problems Connection.

Cf. A005150, A079637, A037033, A038131, A038132.

from sympy import isprime, nextprime
from itertools import count, groupby, islice
def LS(n):
    return int(''.join(str(len(list(g)))+k for k, g in groupby(str(n))))
def f(n): return 0 if not isprime(n) else 1 + f(LS(n))
def agen(startn=0, startk=1):
    n, adict = startn, {i:-1 for i in range(startn)}
    for k in count(startk):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 7))) # Michael S. Branicky, Jul 27 2022

cp's OEIS Frontend

User: Giovanni Resta

A386965 Number of permutations p of [2*n] such that there is at least one index i in [2*n-1] with p(i+1) = n + p(i).

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A367600 Numbers that are not the comma-successor of any number.

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A361336 Smallest decimal number containing n palindromic substrings (Version 2). See Comments for precise definition.

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A361335 Smallest decimal number containing n palindromic substrings (Version 1). See Comments for precise definition.

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A335817 a(k) is the index of the first occurrence of 2*k-1 in A208884, or 0 if it does not occur.

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A335498 a(n) is the least odd number k such that Omega(k) = n and Omega(k+2) = n+1, where Omega(k) is the number of prime factors of k (A001222).

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A335496 a(n) is the least odd number k such that Omega(k) = n, Omega(k+2) = n+1, and Omega(k+4) = n+2, where Omega(k) is the number of prime factors of k (A001222).

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A335205 Numbers of m digits which are equal to the absolute value of the sum of the m-th powers of their digits, with alternating signs.

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A334885 Let q = p | p' be the digit concatenation of a prime p with its prime successor. If the result is a prime repeat the construction setting p = q. a(n) is the smallest prime for which this can be repeated exactly n times.

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A386965 Number of permutations p of [2n] such that there is at least one index i in [2n-1] with p(i+1) = n + p(i).