A057588 -id:A057588 - cp's OEIS frontend

A276092 a(n) = Product_{i=1..n} prime(i)^(prime(i)-1), a(0)=1.

1, 2, 18, 11250, 1323551250, 34329510752434301250, 799811863723341113907011901401250, 38919798565076223182552300534870824616780123359001250, 4052615498709835178737678586220586796222761283625319842830388618784835051250, 3679152532021669595137666762315244807517735994898621013565758767014111825486079213219685771368099483111250

Offset: 0

Antti Karttunen, Aug 22 2016

nonn

Cumulative product of A036878 (after a(0)). - Rick L. Shepherd, Aug 23 2016

			For n=0 we have an empty product, thus a(0) = 1.
For n=1, a(1) = 2^1.
For n=2, a(2) = 2^1 * 3^2 = 18.
For n=3, a(3) = 2^1 * 3^2 * 5^4 = 11250.

Cf. A000040, A002110, A057588, A036878, A276086.

Subsequence of A048103.

Table[Product[Prime[i]^(Prime[i] - 1), {i, n}], {n, 0, 9}] (* Michael De Vlieger, Aug 31 2016 *)

A276092(n) = prod(i=1, n, prime(i)^(prime(i) - 1)) \\ Rick L. Shepherd, Aug 23 2016

(define (A276092 n) (let outloop ((i n) (t 1)) (if (zero? i) t (let ((p (A000040 i))) (let inloop ((j (- p 1)) (t t)) (if (zero? j) (outloop (- i 1) t) (inloop (- j 1) (* t p))))))))
;; Or as a recurrence:
(definec (A276092 n) (if (zero? n) 1 (* (A276092 (- n 1)) (expt (A000040 n) (- (A000040 n) 1)))))

a(n) = Product_{i=1..n} prime(i)^(prime(i)-1).
a(0) = 1; and for n >= 1, a(n) = a(n-1) * A000040(n)^(A000040(n)-1).
a(n) = A276086(A057588(n)).

A328616 Number of digits in primorial base expansion of n that are maximal possible in their positions.

Original entry on oeis.org

0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 2, 3, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1, 1, 2, 0, 1, 0, 1

Offset: 0

Antti Karttunen, Oct 22 2019

			In primorial base (see A049345), the maximum digit value that can occur in the k-th position from the right (with k=1 standing for the rightmost, i.e., the least significant digit position) is A000040(k)-1, and it is for the terms of A057588 (primorial numbers minus one) where all significant digits are maximal allowed for their positions, e.g. 209 is written as "6421" because 209 = 6*30 + 4*6 + 2*2 + 1*1, thus a(209) = 4.
87 is written as "2411" because 87 = 2*A002110(3) + 4*A002110(2) + 1*A002110(1) + 1*A002110(0) = 2*30 + 4*6 + 1*2 + 1*1. Only the digit positions 1 and 3 are occupied with maximum digits allowed in those positions (that are 1 and 4, being one less than the corresponding primes, 2 and 5), thus a(87) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base.

Cf. A002110, A049345, A276086, A328114, A328614, A328615.
Cf. A057588 (positions of records, and the first occurrence of each n > 0).
Cf. also A260736.

a[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; Count[Prime[Range[1, Length[s]]] - s, 1]]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Mar 13 2024 *)

A328616(n) = { my(s=0, p=2); while(n, s += ((p-1)==(n%p)); n = n\p; p = nextprime(1+p)); (s); };

For all n >= 1, a(A057588(n)) = n.

A376411 a(n) is the number of terms less than A276086(n) in the range of A276086, where A276086 is the primorial base exp-function.

Original entry on oeis.org

0, 1, 2, 4, 6, 13, 3, 7, 11, 21, 32, 64, 18, 36, 54, 108, 162, 325, 90, 180, 271, 541, 812, 1624, 450, 902, 1354, 2707, 4061, 8122, 5, 10, 15, 30, 45, 91, 25, 50, 75, 151, 227, 454, 126, 253, 378, 758, 1137, 2274, 632, 1264, 1895, 3790, 5685, 11370, 3158, 6317, 9475, 18952, 28428, 56856, 35, 70, 106, 212, 318, 637

Offset: 0

Antti Karttunen, Nov 13 2024

nonn

Number of terms of A048103 that are less than A276086(n).
Permutation of nonnegative integers.
Troughs are at primorials, A002110, and the local maxima occur just before, at A057588.

Cf. A048103, A057588, A276086, A343048, A359550, A377982.
Cf. A376413 (inverse permutation, but note the different offsets and ranges).
Cf. also A064273 (analogous permutation for base-2).

up_to = (2*210)-1; \\ Must be one of the terms of A343048.
A276085(n) = { my(f = factor(n), pr=1, i=1, s=0); for(k=1, #f~, while(i <= primepi(f[k, 1])-1, pr *= prime(i); i++); s += f[k, 2]*pr); (s); };
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
A376411list(up_to) = { my(size=up_to, v=vector(size), m=A276086(size), s=1, j); for(i=2,m,if(!(m%i), j=A276085(i); v[j] = s; print1("i=",i," v[",j,"]=",s", ");); s += A359550(i)); (v); };
v376411 = A376411list(up_to);
A376411(n) = if(!n,n,v376411[n]);

\\ For incremental computing, less efficient than above:
A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
A359550(n) = { my(pp); forprime(p=2, , pp = p^p; if(!(n%pp), return(0)); if(pp > n, return(1))); };
memoA376411 = Map(); \\ We use k=A276086(n) as our key. kvs will be a list of key-value-pairs sorted into descending order by the key. We search the largest key in it < k, and continue summing from that:
A376411(n) = if(n<=2,n,my(v, k=A276086(n)); if(mapisdefined(memoA376411,k,&v), v, my(kvs = vecsort(Mat(memoA376411)~,(x,y) -> sign(y[1]-x[1])), ss=si=0); for(i=1, #kvs, if(kvs[1,i]A359550(i)); mapput(memoA376411,k,v); (v)));

a(n) = A377982(A276086(n))-1 = Sum_{i=1 .. A276086(n)-1} A359550(i).
For all n >= 1, a(A376413(n)) = n-1, and for all n >= 0, A376413(1+a(n)) = n.
a(i)/a(j) ~ A276086(i)/A276086(j), and particularly, a(2*n+1) ~ 2*a(2*n).

A286615 Square array A(n,k) = A276945(n,k)-1, read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

0, 1, 2, 5, 7, 3, 29, 35, 11, 4, 209, 239, 59, 13, 6, 2309, 2519, 419, 65, 31, 8, 30029, 32339, 4619, 449, 215, 37, 9, 510509, 540539, 60059, 4829, 2339, 245, 41, 10, 9699689, 10210199, 1021019, 62369, 30239, 2549, 269, 43, 12, 223092869, 232792559, 19399379, 1051049, 512819, 32549, 2729, 275, 61, 14

Offset: 1

Antti Karttunen, Jun 30 2017

A permutation of nonnegative integers.

			The top left 8 X 15 corner of the array:
   0,  1,   5,   29,   209,    2309,    30029,    510509
   2,  7,  35,  239,  2519,   32339,   540539,  10210199
   3, 11,  59,  419,  4619,   60059,  1021019,  19399379
   4, 13,  65,  449,  4829,   62369,  1051049,  19909889
   6, 31, 215, 2339, 30239,  512819,  9729719, 223603379
   8, 37, 245, 2549, 32549,  542849, 10240229, 233303069
   9, 41, 269, 2729, 34649,  570569, 10720709, 242492249
  10, 43, 275, 2759, 34859,  572879, 10750739, 243002759
  12, 61, 425, 4649, 60269, 1023329, 19429409, 446696249
  14, 67, 455, 4859, 62579, 1053359, 19939919, 456395939
  15, 71, 479, 5039, 64679, 1081079, 20420399, 465585119
  16, 73, 485, 5069, 64889, 1083389, 20450429, 466095629
  17, 89, 629, 6929, 90089, 1531529, 29099069, 669278609
  18, 91, 635, 6959, 90299, 1533839, 29129099, 669789119
  19, 95, 659, 7139, 92399, 1561559, 29609579, 678978299

Transpose: A286616.
One less than A276945.
Row 1: A057588.

(define (A286615 n) (A286615bi (A002260 n) (A004736 n)))
(define (A286615bi row col) (+ -1 (A276945bi row col))) ;; For A276945bi see under A276945.

A(n,k) = A276945(n,k)-1.

A321683 Numbers with distinct digits in primorial base.

Original entry on oeis.org

0, 1, 2, 4, 5, 10, 13, 14, 19, 20, 22, 23, 25, 26, 28, 29, 52, 58, 79, 80, 85, 86, 95, 100, 103, 104, 115, 116, 118, 119, 125, 130, 133, 134, 139, 140, 142, 143, 155, 160, 163, 164, 169, 170, 172, 173, 175, 176, 178, 179, 185, 190, 193, 194, 199, 200, 202, 203

Offset: 1

Rémy Sigrist, Nov 17 2018

This sequence is a variant of A010784 (numbers with distinct digits in decimal). The final term of that sequence is 9876543210. This sequence, by contrast, has infinitely many terms (for example, all the terms of A057588 belong to this sequence).

			13 in primorial base is 201, which has no repeated digits, hence 13 is in the sequence.
14 in primorial base is 210, which has no repeated digits, hence 14 is also in the sequence.
15 in primorial base is 211, so 15 is not in the sequence on account of the digit 1 appearing twice in its primorial base representation.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base.

See A321682 for the factorial base variant.

Cf. A049345, A057588, A276150, A321682.

q[n_] := Module[{k = n, p = 2, s = {}, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, AppendTo[s, r]; p = NextPrime[p]]; UnsameQ @@ s]; Select[Range[0, 210], q] (* Amiram Eldar, Mar 13 2024 *)

is(n) = my (s=0); forprime (p=2, oo, if (n==0, return (1)); my (d=n%p); if (bittest(s,d), return (0), s+=2^d; n\=p))

A366809 The sum of the divisors of prime(n)#-1 where p# is the product of all the primes from 2 to p inclusive.

Original entry on oeis.org

1, 6, 30, 240, 2310, 30030, 518940, 9943560, 230876448, 6551588160, 200561595684, 7471933410000, 304250263527210, 13082853940673340, 618109122639794688, 32589631537463089128, 1922760350251477679196, 117386696543681561301312, 7906535060701218163040640

Offset: 1

Sean A. Irvine, Oct 23 2023

nonn

			a(4)=240 because the divisors of 7#-1 = 209 are {1, 11, 19, 209}.

Cf. A057588, A000230, A054989, A366809, A064145.

seq(numtheory[sigma](mul(ithprime(k), k=1..n)-1), n=1..30);

a(n) = sigma(prime(n)#-1) = A000230(A057588(n)).

A125191 Primes of the form k# + (k+1)# +- 1, where k# = A002110(k) = primorial(k).

Original entry on oeis.org

2, 7, 37, 239, 241, 2521, 32341, 540539, 540541, 232792559, 232792561, 207030183359, 311671001662019, 41287621429375723111588738861, 5801527386969669153864265802424086050777441586253956297278498679

Offset: 1

Tomas Xordan, Jan 12 2007

nonn

Prime numbers of the form (prime(k+1) + 1)*k# +- 1.

			Let k = 1; then 1#+2# = 2+6 = 8, 8-1 = 7 is prime (hence a term of the sequence) but 8+1 = 9 is nonprime.
Let k = 3; then 3#+4# = 30+210 = 240, 240-1 = 239 is prime and 240+1 = 241 is also prime, so both are terms.

Cf. A002110 (primorial numbers), A006862 (Euclid numbers), A057588 (Kummer numbers).

A002110 := 1 : A000040 := 2 : for n from 1 to 38 do if isprime(A002110*(1+A000040)-1) then printf("%d,",A002110*(1+A000040)-1) ; fi ; if isprime(A002110*(1+A000040)+1) then printf("%d,",A002110*(1+A000040)+1) ; fi ; A002110 := A002110*A000040 : A000040 := nextprime(A000040) : od : # R. J. Mathar, Jan 26 2007

plim=45;k= FoldList[Times, 1, Prime[Range[plim]]];m=Table[k[[l]]+k[[l+1]],{l,plim}];Sort[Select[Join[m+1,m-1],PrimeQ]] (* James C. McMahon, Dec 15 2024 *)
Join[{2},Select[Sort[Flatten[#+{1,-1}&/@(Total/@Partition[FoldList[Times,Prime[Range[40]]],2,1])]],PrimeQ]] (* Harvey P. Dale, Jul 15 2025 *)

{m=37;for(n=0,m,p=primorial(n)+primorial(n+1);if(isprime(a=p-1),print1(a,","));if(isprime(a=p+1),print1(a,",")))} \\ Klaus Brockhaus, Jan 25 2007

genit(maxx)={arr=List();for(n=0, maxx, p=factorback(primes(n))+factorback(primes(n+1));if(ispseudoprime(p-1),listput(arr,p-1));if(ispseudoprime(p+1),listput(arr,p+1)));arr} \\ Bill McEachen, Jun 21 2021 (from David A. Corneth's code at A002110)

Edited, corrected and extended by Klaus Brockhaus and R. J. Mathar, Jan 25 2007

A286616 Transpose of square array A286615.

Original entry on oeis.org

0, 2, 1, 3, 7, 5, 4, 11, 35, 29, 6, 13, 59, 239, 209, 8, 31, 65, 419, 2519, 2309, 9, 37, 215, 449, 4619, 32339, 30029, 10, 41, 245, 2339, 4829, 60059, 540539, 510509, 12, 43, 269, 2549, 30239, 62369, 1021019, 10210199, 9699689, 14, 61, 275, 2729, 32549, 512819, 1051049, 19399379, 232792559, 223092869

Offset: 1

Antti Karttunen, Jun 30 2017

A permutation of nonnegative integers.

			The top left 7 X 7 corner of the array:
      0,      2,       3,       4,       6,        8,        9
      1,      7,      11,      13,      31,       37,       41
      5,     35,      59,      65,     215,      245,      269
     29,    239,     419,     449,    2339,     2549,     2729
    209,   2519,    4619,    4829,   30239,    32549,    34649
   2309,  32339,   60059,   62369,  512819,   542849,   570569
  30029, 540539, 1021019, 1051049, 9729719, 10240229, 10720709

Transpose: A286615.
One less than A276943.
Column 1: A057588.

(define (A286616 n) (A286615bi (A004736 n) (A002260 n))) ;; For the rest, see under A286615, A276945, etc.

A(n,k) = A286615(k,n) = A276943(n,k)-1.

A286942 Irregular triangle read by rows: numbers 1 <= k <= (A002110(n) - 1) where gcd(k, A002110(n - 1)) = 1.

Original entry on oeis.org

1, 2, 1, 3, 5, 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 1, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, 151, 157, 161, 163, 167, 169, 173

Offset: 1

Jamie Morken and Michael De Vlieger, May 16 2017

From Michael De Vlieger, May 18 2017: (Start)
Row n of a(n) is the list of numbers 1 <= k <= A002110(n) that are coprime to A002110(n-1).
A286941(n) and A279864(n) are subsets of a(n) such that the terms of the rows of each sequence combined and sorted comprise all the terms of a(n).
Row lengths = A005867(n) + A005867(n-1): {2, 3, 10, 56, 528, 6240, 97920, ...}.
1 is coprime to all n thus delimits the rows of a(n).
The smallest prime q in row n of a(n) is gpf(primorial(n)) = A006530(A002110(n)) = prime(n) by definition of primorial.
The smallest composite x in row n of a(n) is q^2 = A001248(n).
The Kummer number A057588(n) = A002110(n) - 1 is the largest term in row n of a(n). (End)

			The triangle starts:
1, 2;
1, 3, 5;
1, 5, 7, 11, 13, 17, 19, 23, 25, 29
Example1:
To find row n of the irregular triangle A286942, take a running sum for each value in the irregular triangle row n-1 of A286941 with A002110(n-1) b-1 times, where b is the largest prime factor in A002110(n).
For example to find row 3 of A286942: Take a running sum for both 1 and 5 in row n-1 of A286941 with A002110(3-1)=6, 5-1=4 times, where b is the largest prime factor 5 in A002110(3).
Result:
1 5
7 11
13 17
19 23
25 29
Equal to row 3 of A286942: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29.
Example2:
To find row n of the irregular triangle A279864, multiply each value in row n-1 of A286941 with the largest prime factor b in A002110(n).
Example for n=3: b=5.
1*5=5
5*5=25
Example3:
To find row n of the irregular triangle A286941, remove the values that are in row n of the irregular triangle A279864 from the values that are in row n of the irregular triangle A286942.
For n=3.
A286942 row n = 1, 5, 7, 11, 13, 17, 19, 23, 25, 29.
A279864 row n = 5, 25.
Removing values 5, 25 from the values in A286942 row n gives row n of A286941: 1, 7, 11, 13, 17, 19, 23, 29.

Michael De Vlieger, Table of n, a(n) for n = 1..6839 (rows 1 <= n <= 6).
Eric Weisstein's World of Mathematics, Relatively Prime - _Michael De Vlieger_, May 18 2017
Seqfan, Formula for the sequence.

Cf. A002110, A005867, A279864, A286941, A286942.

Table[Select[Range@ #2, Function[k, CoprimeQ[k, #1]]] & @@ Map[Times @@ # &, {Most@ #, #}] &@ Prime@ Range@ n, {n, 4}] // Flatten (* Michael De Vlieger, May 18 2017 *)

a(n) = union(A286941(n), A279864(n)) where n consists of all terms in row n of each sequence. - Michael De Vlieger, May 18 2017

More terms from Michael De Vlieger, May 18 2017

A341812 Primes p such that p#-1 is divisible by the next prime after p.

Original entry on oeis.org

3, 7, 176078267

Offset: 1

Jeppe Stig Nielsen, Feb 20 2021

Here p# means A034386(p).

The next prime after p is then in A341804.

			The prime 7 is a member of this sequence because 2*3*5*7-1=209 is divisible by 11, the next prime immediately following 7.
11 is NOT a member of this sequence because 2*3*5*7*11-1=2309 is not divisible by 13 (leaves a remainder of 8), the next prime after 11.

Carlos Rivera, Puzzle 117. Certain p#+1 values, contains message from Wilfred Whiteside mentioning this sequence.

Cf. A057588, A058233, A341804, A341805.

isok(p) = if (isprime(p), my(a=Mod(1,nextprime(1+p))); forprime(q=2,p,a*=q); (a == 1)); \\ Michel Marcus, Mar 03 2021

cp's OEIS Frontend

A276092 a(n) = Product_{i=1..n} prime(i)^(prime(i)-1), a(0)=1.

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A328616 Number of digits in primorial base expansion of n that are maximal possible in their positions.

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A376411 a(n) is the number of terms less than A276086(n) in the range of A276086, where A276086 is the primorial base exp-function.

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A286615 Square array A(n,k) = A276945(n,k)-1, read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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A321683 Numbers with distinct digits in primorial base.

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A366809 The sum of the divisors of prime(n)#-1 where p# is the product of all the primes from 2 to p inclusive.

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A125191 Primes of the form k# + (k+1)# +- 1, where k# = A002110(k) = primorial(k).

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A286616 Transpose of square array A286615.