A007662 -id:A007662 - cp's OEIS frontend

A291350 Numbers k such that k!4 + 2^9 is prime, where k!4 = k!!!! is the quadruple factorial number (A007662).

9, 11, 13, 17, 19, 21, 29, 31, 33, 35, 67, 103, 111, 129, 179, 355, 713, 799, 921, 1013, 1389, 1543, 2097, 2287, 3657, 4115, 7031, 10689, 11715, 16401, 16893, 19497, 29737, 35615

Offset: 1

Robert Price, Aug 22 2017

Corresponding primes are: 557, 743, 1097, 10457, 66347, 209357, 151413137, ...
a(35) > 10^5.
Terms > 35 correspond to probable primes.

			11!4 + 2^9 = 11*7*3*1 + 512 = 743 is prime, so 11 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!4+512.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A007662, A037082, A084438, A123910, A242994.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 4] + 2^9] &]

A291351 Numbers k such that k!4 + 2^10 is prime, where k!4 = k!!!! is the quadruple factorial number (A007662).

Original entry on oeis.org

9, 13, 23, 27, 33, 47, 61, 113, 145, 161, 191, 281, 291, 417, 869, 919, 1213, 1297, 1663, 2103, 2297, 2325, 3241, 3895, 4337, 6645, 7911, 8737, 13369, 13555, 19245, 34025, 47779, 48589, 54521, 91355

Offset: 1

Robert Price, Aug 22 2017

Corresponding primes are: 1069, 1609, 1515229, 40884559, 4996617649, ...
a(37) > 10^5.
Terms > 33 correspond to probable primes.

			13!4 + 2^10 = 13*9*5*1 + 1024 = 1609 is prime, so 13 is in the sequence.

Henri & Renaud Lifchitz, PRP Records. Search for n!4+1024.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A007662, A037082, A084438, A123910, A242994.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 50000], PrimeQ[MultiFactorial[#, 4] + 2^10] &]
Select[Range[10^3],PrimeQ[Times@@Range[#,1,-4]+2^10]&] (* The program generates the first 16 terms of the sequence. *) (* Harvey P. Dale, Feb 08 2025 *)

a(36)-a(37) from Robert Price, Sep 25 2019

A291687 Primes of the form k!4+4, where k!4 is the quadruple factorial number (A007662).

Original entry on oeis.org

5, 7, 3469, 9949, 65839, 1514209, 5221129, 40883539, 151412629, 44358635479, 16713607661375629, 36453104912477522894629, 1027438963906784290227656915629, 7419136758370889359733910587728129, 4551830726072842264843919206776501006328129

Offset: 1

Robert Price, Aug 29 2017

nonn

Robert Price, Table of n, a(n) for n = 1..18
Henri& Renaud Lifchitz, PRP Records.Search for n!4+4.
Joe McLean, Interesting Sources of Probable Primes
OpenPFGW Project, Primality Tester

Cf. A291122.

MultiFactorial[n_, k_] := If[n<1, 1, n*MultiFactorial[n-k, k]];
Select[Table[MultiFactorial[i, 4] + 4, {i, 0, 100}], PrimeQ[#]&]

A045747 Number of prime factors of n!!!! (A007662), with multiplicity.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 5, 3, 5, 3, 8, 4, 7, 5, 12, 5, 10, 6, 15, 7, 12, 7, 19, 9, 14, 10, 22, 10, 17, 11, 27, 12, 19, 13, 31, 13, 21, 15, 35, 14, 24, 16, 38, 17, 26, 17, 43, 19, 29, 19, 46, 20, 33, 21, 50, 22, 35, 22, 54, 23, 37, 25, 60, 25, 40, 26, 63, 27, 43, 27, 68, 28, 45, 30, 71

Offset: 1

Jeff Burch

Cf. A001222 (Omega), A007662 (n!!!!).

NFactorialM[n_Integer, m_Integer] := Block[{k = n, p = Max[1, n]}, While[k > m, k -= m; p *= k]; p]; Table[PrimeOmega[NFactorialM[n, 4]], {n, 100}] (* Wesley Ivan Hurt, May 26 2024 after Robert G. Wilson v *)

a(n) = A001222(A007662(n)). - Wesley Ivan Hurt, May 26 2024

A108895 Partial sums of quadruple factorial numbers n!!!! (A007662).

Original entry on oeis.org

1, 2, 4, 7, 11, 16, 28, 49, 81, 126, 246, 477, 861, 1446, 3126, 6591, 12735, 22680, 52920, 118755, 241635, 450480, 1115760, 2629965, 5579085, 10800210, 28097490, 68981025, 151556385, 302969010, 821887410, 2089276995, 4731688515

Offset: 0

Jonathan Vos Post, Feb 08 2006

Quadruple factorial numbers n!!!! = n*(n-4)!!!!, 0!!!! = 1!!!! = 1, 2!!!! = 2, 3!!!! = 3. The cumulative sum a(n) is prime for n = 1, 3, 4 and never again, as all values from a(8) = 81 are multiples of 3. The cumulative sum a(n) is semiprime for n = 2, 7 and never again, as all values from a(16) are divisible by both 3 and 5.

			a(31) = 1 + 1 + 2 + 3 + 4 + 5 + 12 + 21 + 32 + 45 + 120 + 231 + 384 + 585 + 1680 + 3465 + 6144 + 9945 + 30240 + 65835 + 122880 + 208845 + 665280 + 1514205 + 2949120 + 5221125 + 17297280 + 40883535 + 82575360 + 151412625 + 518918400 + 1267389585 = 2089276995 = 3 * 5 * 13 * 337 * 31793.

J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.

Cf. A000040, A001358, A007662, A114347.

NFactorialM[n_Integer, m_Integer] := Block[{k = n, p = Max[1, n]}, While[k > m, k -= m; p *= k]; p]; Table[ Sum[ NFactorialM[i, 4], {i, 0, n}], {n, 0, 33}] (* Robert G. Wilson v, Feb 21 2006 *)

a(n) = Sum_{i=0..n} i!!!!.

a(n) = Sum_{i=0..n} A007662(i).

A328454 Numbers k such that k![4] - 4 is prime, where k![4] = A007662(k) = quadruple factorial.

Original entry on oeis.org

7, 9, 11, 15, 17, 19, 39, 45, 57, 59, 63, 69, 81, 85, 127, 141, 149, 153, 163, 165, 201, 235, 259, 377, 457, 649, 815, 897, 1057, 1433, 1453, 1519, 1759, 3047, 3561, 4151, 7025, 11917, 11971, 15295, 18919, 19449, 20765, 70385, 71293

Offset: 1

Robert Price, Nov 06 2019

nonn

a(46) > 10^5.

The first 6 primes associated with this sequence are: 17, 41, 227, 3461, 9941, 65831.

C. K. Caldwell, The Prime Glossary, multifactorial prime
C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75
Ken Davis, Status of Search for Multifactorial Primes.
Joe McLean, Interesting Sources of Probable Primes

Cf. A007662, A283553.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[1000], (x = MultiFactorial[#, 4] - 4; x > 0 && PrimeQ[x]) &]

A329112 Numbers k such that k![4] - 8 is prime, where k![4] = A007662(k) = quadruple factorial.

Original entry on oeis.org

7, 9, 11, 13, 15, 19, 21, 23, 29, 35, 37, 55, 57, 77, 85, 139, 243, 251, 433, 667, 671, 895, 2127, 2263, 2293, 2645, 2733, 2845, 3675, 4381, 6453, 6825, 36557, 78531

Offset: 1

Robert Price, Nov 06 2019

nonn

a(35) > 10^5.

The first 6 primes associated with this sequence are: 13, 37, 223, 577, 3457, 65827.

C. K. Caldwell, The Prime Glossary, multifactorial prime
C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75
Ken Davis, Status of Search for Multifactorial Primes.
Joe McLean, Interesting Sources of Probable Primes

Cf. A007662, A283553.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[1000], (x = MultiFactorial[#, 4] - 8; x > 0 && PrimeQ[x]) &]

A329166 Numbers k such that k![4] - 16 is prime, where k![4] = A007662(k) = quadruple factorial.

Original entry on oeis.org

7, 9, 13, 15, 17, 29, 31, 35, 39, 105, 109, 147, 173, 239, 287, 293, 505, 711, 837, 947, 1015, 1025, 1977, 2917, 4035, 4935, 5935, 7693, 10911, 11367, 12029, 14155, 15221, 17921, 17961, 20521, 23053, 32821, 45147, 45351, 68057, 78315

Offset: 1

Robert Price, Nov 06 2019

nonn

a(43) > 10^5.

The first 5 primes associated with this sequence are: 5, 29, 569, 3449, 9929.

C. K. Caldwell, The Prime Glossary, multifactorial prime
C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75.
Ken Davis, Status of Search for Multifactorial Primes.
Joe McLean, Interesting Sources of Probable Primes

Cf. A007662, A283553.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[1000], (x = MultiFactorial[#, 4] - 16; x > 0 && PrimeQ[x]) &]

A329167 Numbers k such that k![4] - 32 is prime, where k![4] = A007662(k) = quadruple factorial.

Original entry on oeis.org

9, 11, 15, 25, 29, 47, 55, 67, 119, 171, 331, 475, 549, 819, 1151, 1543, 2303, 2749, 3303, 3649, 4065, 4261, 4497, 4873, 9105, 12749, 18677, 20121, 22459, 32489, 35765, 46971, 75843, 79585, 79731

Offset: 1

Robert Price, Nov 06 2019

nonn

a(36) > 10^5.

The first 4 primes associated with this sequence are: 13, 199, 3433, 5221093.

C. K. Caldwell, The Prime Glossary, multifactorial prime
C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75.
Ken Davis, Status of Search for Multifactorial Primes.
Joe McLean, Interesting Sources of Probable Primes

Cf. A007662, A283553.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[1000], (x = MultiFactorial[#, 4] - 32; x > 0 && PrimeQ[x]) &]

A329175 Numbers k such that k![4] - 64 is prime, where k![4] = A007662(k) = quadruple factorial.

Original entry on oeis.org

11, 13, 41, 45, 59, 85, 141, 283, 357, 419, 713, 1149, 1353, 1537, 1669, 2353, 2389, 2543, 5147, 5279, 12801, 30035, 39421, 46969, 61077

Offset: 1

Robert Price, Nov 07 2019

nonn

a(26) > 10^5.

The first 3 primes associated with this sequence are: 167, 521, 7579867420061.

C. K. Caldwell, The Prime Glossary, multifactorial prime
C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75.
Ken Davis, Status of Search for Multifactorial Primes.
Joe McLean, Interesting Sources of Probable Primes

Cf. A007662, A283553.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[1000], (x = MultiFactorial[#, 4] - 64; x > 0 && PrimeQ[x]) &]

cp's OEIS Frontend

A291350 Numbers k such that k!4 + 2^9 is prime, where k!4 = k!!!! is the quadruple factorial number (A007662).

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A291351 Numbers k such that k!4 + 2^10 is prime, where k!4 = k!!!! is the quadruple factorial number (A007662).

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A291687 Primes of the form k!4+4, where k!4 is the quadruple factorial number (A007662).

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A045747 Number of prime factors of n!!!! (A007662), with multiplicity.

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A108895 Partial sums of quadruple factorial numbers n!!!! (A007662).

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A328454 Numbers k such that k![4] - 4 is prime, where k![4] = A007662(k) = quadruple factorial.

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A329112 Numbers k such that k![4] - 8 is prime, where k![4] = A007662(k) = quadruple factorial.

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A329166 Numbers k such that k![4] - 16 is prime, where k![4] = A007662(k) = quadruple factorial.

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A329167 Numbers k such that k![4] - 32 is prime, where k![4] = A007662(k) = quadruple factorial.

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