Jens Kruse Andersen - cp's OEIS frontend

A217372 Initial prime in the first Ormiston n-tuple.

2, 1913, 11117123, 6607882123, 20847942560791

Offset: 1

Jens Kruse Andersen, Oct 20 2012

An Ormiston n-tuple is n consecutive primes containing the same decimal digits in different order. a(5) found by Giovanni Resta. a(6) may be 166389896360719.

			(1913, 1931) is the first case of two consecutive primes with the same digits. The first 3-, 4- and 5-tuples are: (11117123, 11117213, 11117321), (6607882123, 6607882213, 6607882231, 6607882321), (20847942560791, 20847942560917, 20847942560971, 20847942561079, 20847942561097).

Jens Kruse Andersen, Ormiston Tuples
A. Edwards, Ormiston Pairs, Australian Mathematics Teacher, Vol. 58, No. 2 (2002), pp 12-13.
E. W. Weisstein, Rearrangement Prime Pair at MathWorld.Wolfram.com.

Cf. A069567 (Ormiston pairs), A075093 (triples), A161160 (quadruples), A217797 (5-tuples)

A101042 a(n) is the smallest positive d such that the n-th prime is the smallest prime p for which p+d is also prime.

Original entry on oeis.org

1, 2, 6, 22, 116, 88, 470, 112, 284, 242, 202, 772, 1326, 718, 1334, 1328, 2558, 1762, 1642, 2402, 3274, 1732, 7094, 9512, 7984, 5246, 12688, 10532, 9952, 16766, 7702, 60458, 9974, 25708, 5888, 13528, 10342, 25678, 62156, 69518, 76838, 37666

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

Except for n=1, A020483(a(n)/2) is the first appearance of the n-th prime. It is conjectured that a(n) always exists. a(386) is the first number which must be above 10^12.

			a(3)=6 because: The 3rd prime is 5. 2+6, 3+6 is composite, 5+6 is prime. 6 is the smallest such number.

J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]

Cf. A020483, A101043, A101044, A101045, A101046.

A101046 d such that the smallest prime p for which p+d is also prime is larger than for any smaller d.

Original entry on oeis.org

1, 2, 6, 22, 88, 112, 202, 718, 1328, 1642, 1732, 5246, 5888, 10342, 25678, 37666, 59894, 76004, 103102, 108412, 180814, 359662, 651362, 872698, 2373478, 6088792, 7642528, 9244552, 13038352, 13591192, 24318988, 34857778, 55076404, 147838742

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

The numbers in A101042 which are smaller than all following numbers.

			Consider d=6. The smallest prime p for which p+6 is also prime, is p=5. All numbers below d=6 have a p<5 (or no p at all), so 6 is in the sequence.

J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]

Cf. A020483, A101042, A101043, A101044, A101045.

A101043 A101042 sorted. There exists a prime p for which a(n) is the smallest positive d such that p is the smallest prime where p+d is also prime.

Original entry on oeis.org

1, 2, 6, 22, 88, 112, 116, 202, 242, 284, 470, 718, 772, 1326, 1328, 1334, 1642, 1732, 1762, 2402, 2558, 3274, 5246, 5888, 7094, 7702, 7984, 9512, 9952, 9974, 10342, 10532, 12688, 13528, 16766, 25678, 25708, 37666, 59894, 60458, 61756, 62156

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

Except for n=1, A020483(a(n)/2) is the first appearance of a prime in A020483.

			d=6 is in the sequence because there exists the prime p=5 satisfying the required conditions: 2+6, 3+6 is composite and 5+6 is prime. 6 is the smallest such number.

J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]

Cf. A020483, A101042, A101044, A101045, A101046.

A101044 Primes corresponding to A101043 (which is A101042 sorted).

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 11, 31, 29, 23, 17, 43, 37, 41, 53, 47, 67, 79, 61, 71, 59, 73, 101, 149, 83, 127, 97, 89, 109, 137, 157, 107, 103, 151, 113, 163, 139, 181, 197, 131, 193, 167, 191, 173, 227, 199, 179, 223, 211, 307, 241, 349, 229, 239, 233, 257, 379, 277, 271

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

The order in which the primes (except 2) first appear in A020483. It is conjectured that all primes are in this sequence.

J. K. Andersen, Prime gaps (not necessarily consecutive).
Mike Oakes, Ed Pegg Jr, Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]

Cf. A020483, A101042, A101043, A101045, A101046.

A101045 Record size primes in A101044.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 31, 43, 53, 67, 79, 101, 149, 157, 163, 181, 197, 227, 307, 349, 379, 409, 431, 619, 631, 661, 691, 751, 757, 811, 829, 1093, 1117, 1217, 1279, 1423, 1453, 1481, 1531, 1549, 1579, 1759, 1877, 2239, 2273, 2287, 2383, 2447, 2659, 2671, 2707

Offset: 1

Jens Kruse Andersen, Nov 28 2004

nonn

This sequence (except 2) is also the record size primes in the longer A020483.

Conjecture: lim_{n->infinity} a(n)/n^2 = 1. - Ya-Ping Lu, Sep 24 2020

J. K. Andersen, Prime gaps (not necessarily consecutive).
Taras Goy and Mark Shattuck, Determinant Formulas of Some Hessenberg Matrices with Jacobsthal Entries, Applications and Appl. Math.: An Int'l J. (2021) Vol. 16, Issue 1, Art. 10.
Mike Oakes, Ed Pegg Jr, and Jens Kruse Andersen, Prime gaps (not necessarily consecutive), digest of 5 messages in primenumbers Yahoo group, Nov 26 - Nov 27, 2004. [Cached copy]

Cf. A020483, A101042, A101043, A101044, A101046.

from sympy import isprime, nextprime
m = p_max = 0
while m >= 0:
    p = 2
    while isprime(p + 2*m) == 0:
        p = nextprime(p)
    if p > p_max:
        print(p)
        p_max = p
    m += 1 # Ya-Ping Lu, Sep 24 2020

A075131 Picture-perfect numbers of form 57*p for p in A075130. The decimal reversal is equal to the sum of the reversed proper divisors.

Original entry on oeis.org

7980062073, 79862699373, 798006269373, 7986207926373, 79862079269373, 798062073062073, 798620736269373, 7980000620792073, 7980006269999373, 7980620730626373, 7980626373062073, 7980626379269373, 7986207306269373

Offset: 1

Jens Kruse Andersen, Sep 04 2002

All known picture-perfect numbers are of this form, except the first 4 in A069942. - Jens Kruse Andersen, May 06 2008

			a(4)=57*140108910989=7986207926373.

Discussion forum, Systematic Undertaking to Find Picture-Perfect Numbers
Joseph Pe, Picture-Perfect Numbers and Other Digit-Reversal Diversions

a(n) = 57*A075130(n)

Cf. A075130, A069942.

Edited by Jens Kruse Andersen, May 06 2008

A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.

Original entry on oeis.org

2, 3, 61, 193, 15121, 838561, 807905281, 19896463921, 3059220303001, 3931520917431241

Offset: 1

Rick L. Shepherd, Jun 30 2002 and Jens Kruse Andersen, Jul 28 2002

By definition, each term of this sequence is prime.
a(11) <= 1452591346605212407096281241 (Frederick Schneider), see primepuzzles link. - sent by amd64(AT)vipmail.hu, Dec 21 2007
Prime factors are counted with multiplicity. - Harvey P. Dale, Mar 09 2021

			a(3)=61 because 61 (prime), 62 (=2*31), 63 (=3*3*7) have exactly 1, 2, 3 prime factors respectively, and this is the smallest solution;
a(6)=807905281: 807905281 is prime; 807905281+1=2*403952641;
807905281+2=3*15733*17117; 807905281+3=2*2*1871*107951;
807905281+4=5*11*43*211*1619; 807905281+5=2*3*3*3*37*404357;
807905281+6=7*7*7*7*29*41*283; 807905281 is the smallest number m such that m+k is product of k+1 primes for k=0,1,2,3,4,5,6.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 61, p. 22, Ellipses, Paris 2008.

alt.math.recreational thread, Consecutive numbers with counting prime factors.
Carlos Rivera, Puzzle 425. Consecutive numbers, increasing quantity of prime factors, The Prime Puzzles & Problems Connection.

Cf. A001222, A093552, A093550, A086560, A124592.

a(1) = A000040(1), a(2) = A005383(1), a(3) = A112998(1), a(4) = A113000(1), a(5) = A113008(1), a(6) = A113150(1).

(* This program is not suitable to compute a large number of terms. *) nmax = 6; kmax = 10^6; a[1] = 2; a[n_] := a[n] = For[k = a[n-1]+n-1, k <= kmax, k++, If[AllTrue[Range[0, n-1], PrimeOmega[k+#] == #+1&], Return[k] ] ]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Sep 06 2017 *)

a(7) found by Mark W. Lewis
a(8) and a(9) found by Jens Kruse Andersen
a(10) found by Jens Kruse Andersen; probably a(11) > 10^20. - Aug 24 2002
Entry revised by N. J. A. Sloane, Jan 26 2007
Cross-references and editing by Charles R Greathouse IV, Apr 20 2010

A075130 Primes with decimal representation 140{{0}10{9}89}.

Original entry on oeis.org

140001089, 1401099989, 14000109989, 140108910989, 1401089109989, 14001089001089, 14010890109989, 140000010891089, 140000109999989, 140010890010989, 140010989001089, 140010989109989, 140108900109989

Offset: 1

Jens Kruse Andersen, Sep 04 2002

57 times one of these primes is a picture-perfect number (see A075131).

			a(4)=140108910989 because this is the 4th prime of the required form.

Sean A. Irvine, Table of n, a(n) for n = 1..1000
Discussion forum, Systematic Undertaking to Find Picture-Perfect Numbers
Joseph Pe, Picture-Perfect Numbers and Other Digit-Reversal Diversions
Rulthan P. Sumicad, On the Picture-Perfect Number, J. Math. Stat. Studies (2023).

A075131(n)=57*a(n)

Cf. A075131, A069942.

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User: Jens Kruse Andersen

A217372 Initial prime in the first Ormiston n-tuple.

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A101042 a(n) is the smallest positive d such that the n-th prime is the smallest prime p for which p+d is also prime.

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A101046 d such that the smallest prime p for which p+d is also prime is larger than for any smaller d.

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A101043 A101042 sorted. There exists a prime p for which a(n) is the smallest positive d such that p is the smallest prime where p+d is also prime.

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A101044 Primes corresponding to A101043 (which is A101042 sorted).

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A101045 Record size primes in A101044.

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A075131 Picture-perfect numbers of form 57*p for p in A075130. The decimal reversal is equal to the sum of the reversed proper divisors.

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A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.

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A075130 Primes with decimal representation 140{{0}10{9}89}.

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