A098147 -id:A098147 - cp's OEIS frontend

A098146 First odd semiprime > 10^n.

9, 15, 111, 1003, 10001, 100001, 1000001, 10000001, 100000001, 1000000013, 10000000003, 100000000007, 1000000000007, 10000000000015, 100000000000013, 1000000000000003, 10000000000000003, 100000000000000015

Offset: 0

Hugo Pfoertner, Aug 28 2004

nonn

			a(0)=9 because 9=3*3 is the first odd semiprime following 10^0=1.
a(13) = 10000000000015 = 5*2000000000003.

Dario Alpern, Factorization using the Elliptic Curve Method.

Cf. A046315 (odd semiprimes), A098147(n)=a(n)-10^n continuation of this sequence, A003717 (smallest n-digit prime).

osp[n_]:=Module[{k=1},While[PrimeOmega[n+k]!=2,k=k+2];n+k]; Join[{9}, Table[osp[10^i],{i,20}]] (* Harvey P. Dale, Jan 17 2012 *)

print1(9,","); for(n=1,10,forstep(i=10^n+1,10^(n+1)-1,2,f=factor(i); ms=matsize(f); if((ms[1]==1&&f[1,2]==2)||(ms[1]==2&&f[1,2]==1&&f[2,2]==1),print1(i,","); break))) /* Herman Jamke (hermanjamke(AT)fastmail.fm), Oct 21 2006 */

from sympy import factorint, nextprime
def is_semiprime(n): return sum(e for e in factorint(n).values()) == 2
def next_odd_semiprime(n):
    nxt = n + 1 + n%2
    while not is_semiprime(nxt): nxt += 2
    return nxt
def a(n): return next_odd_semiprime(10**n)
print([a(n) for n in range(20)]) # Michael S. Branicky, Sep 15 2021

A114137 Difference between first odd semiprime > 2^n and 2^n.

Original entry on oeis.org

8, 7, 5, 1, 5, 1, 1, 1, 3, 3, 3, 1, 1, 1, 3, 5, 9, 1, 5, 1, 1, 5, 7, 1, 3, 3, 3, 3, 1, 9, 25, 1, 1, 11, 7, 3, 7, 15, 19, 3, 1, 5, 3, 1, 31, 3, 7, 21, 3, 9, 7, 11, 3, 11, 3, 29, 9, 29, 25, 9, 45, 1, 3, 9, 1

Offset: 0

Jonathan Vos Post, Feb 03 2006

A098147 is difference between first odd semiprime > 10^n and 10^n. In this powers of 2 sequence, does 1 occur infinitely often? Does every odd number occur?

			a(0) = 8 (the only even value here) because 2^0 + 8 = 9 = 3^2 is an odd semiprime.
a(1) = 7 because 2^1 + 7 = 9 = 3^2 is an odd semiprime.
a(2) = 5 because 2^2 + 5 = 9 = 3^2 is an odd semiprime.
a(3) = 1 because 2^3 + 1 = 9 = 3^2 is an odd semiprime.
a(4) = 5 because 2^4 + 5 = 21 = 3 * 7 is an odd semiprime.
a(5) = 1 because 2^5 + 1 = 33 = 3 * 11 is an odd semiprime.
a(6) = 1 because 2^6 + 1 = 65 = 5 * 13 is an odd semiprime.
a(10) = 3 because 2^10 + 3 = 1027 = 13 * 79 is an odd semiprime.
a(30) = 25 because 2^30 + 25 = 1073741849 = 29 * 37025581 is an odd semiprime.

Cf. A001358, A098147.

a[n_] := Block[{z}, If[n == 0, z = 3, z = 2^n + 1]; While[ PrimeOmega[z] != 2, z += 2]; z - 2^n]; a /@ Range[0, 64] (* Giovanni Resta, Jun 14 2016 *)

a(n) = minimum integer k such that 2^n + k is an element of A046315. a(n) = minimum integer k such that A000079(n) + k is an element of A046315.

a(46) corrected by Giovanni Resta, Jun 14 2016

A114141 Difference between first odd semiprime > 3^n and 3^n.

Original entry on oeis.org

8, 6, 6, 6, 4, 4, 2, 2, 22, 6, 8, 2, 2, 20, 10, 4, 6, 8, 4, 4, 2, 4, 20, 10, 8, 6, 10, 2, 2, 8, 14, 12, 20, 10, 14, 20, 16, 6, 14, 4, 2, 8, 8, 12, 2, 24, 20, 10, 10, 4, 48, 40, 8, 34, 4, 4, 38, 56, 4, 28, 2, 14, 14, 22, 6, 8, 10, 4, 16, 4, 20, 2, 26, 56, 32

Offset: 0

Jonathan Vos Post, Feb 03 2006

A098147 is difference between first odd semiprime > 10^n and 10^n. How can we prove that there exists an a(n) for all n? In this powers of 3 sequence, does 2 occur infinitely often? Does every even integer k > 0 occur?

			a(0) = 8 because 3^0 + 8 = 9 = 3^2 is an odd semiprime.
a(1) = 6 because 3^1 + 6 = 9 = 3^2 is an odd semiprime.
a(2) = 6 because 3^2 + 6 = 15 = 3 * 5 is an odd semiprime.
a(3) = 6 because 3^3 + 6 = 33 = 3 * 11 is an odd semiprime.
a(4) = 4 because 3^4 + 4 = 85 = 5 * 17 is an odd semiprime.
a(5) = 4 because 3^5 + 4 = 247 = 13 * 19 is an odd semiprime.

Cf. A000244, A001358, A098147.

a[n_] := Block[{z}, z = 3^n + 2; While[ PrimeOmega[z] != 2, z += 2]; z - 3^n]; a /@ Range[0, 60] (* Giovanni Resta, Jun 14 2016 *)

a(n) = minimum integer k such that 3^n + k is an element of A046315. a(n) = minimum integer k such that A000244(n) + k is an element of A046315.

a(26) corrected and more terms from Giovanni Resta, Jun 14 2016

A114187 Difference between first semiprime >= n! and n!. Least k such that n!+k is semiprime.

Original entry on oeis.org

3, 3, 2, 0, 1, 1, 1, 1, 1, 5, 1, 3, 17, 1, 1, 7, 2, 3, 23, 1, 1, 11, 29, 3, 1, 1, 1, 37, 1, 41, 2, 19, 11, 11, 1, 7, 3, 41, 1, 13, 127, 47, 59, 2, 37, 5, 37, 59, 1, 2, 73, 59, 79, 73, 1, 1, 61, 118, 37, 1, 61

Offset: 0

Jonathan Vos Post, Feb 04 2006

a(n) = 1 when n!+1 is a factorial prime.
A098147 is difference between first odd semiprime > 10^n and 10^n.
In this sequence, does 1 occur infinitely often (next with n = 71, 75)? If not 0 (for n=3) or 1, a(n) = k must be a prime other than 5.
Does every odd prime but 5 occur? Some of these take longer to factor, when both prime factors are large, such as n = 37, 38, 42, 47, 50, 54.
Essentially the same as A085747. - Georg Fischer, Oct 07 2018

			a(0) = a(1) = 3 because 0! + 3 = 1! + 3 = 4 = 2^2 is semiprime (the only even example).
a(2) = 2 because 2! + 2 = 2 + 2 = 4 = 2^2 is semiprime.
a(3) = 0 because 3! + 0 = 6 = 2*3 is semiprime (6+3=9=3^2 so this term would be 3 if we required nonzero values).
a(4) = 1 because 4! + 1 = 24 + 1 = 25 = 5^2 is semiprime.
a(5) = 1 because 5! + 1 = 120 + 1 = 121 = 11^2 is semiprime.
a(6) = 1 because 6! + 1 = 720 + 1 = 721 = 7 * 103 is semiprime.
a(7) = 1 because 7! + 1 = 5040 + 1 = 5041 = 71^2 is semiprime.
a(8) = 1 because 8! + 1 = 40320 + 1 = 40321 = 61 * 661 is semiprime.
a(9) = 5 because 9! + 5 = 362880 + 1 = 362885 = 5 * 72577 is semiprime.
a(10) = 1 because 10! + 1 = 3628800 + 1 = 3628801 = 11 * 329891 is semiprime.

Cf. A000142, A001358, A098147.

a(n) = minimum integer k such that n! + k is an element of A001358. a(n) = minimum integer k such that A000142(n) + k is an element of A001358.

Data corrected by Giovanni Resta, Jun 14 2016

A114313 Difference between first odd semiprime >= 5^n and 5^n.

Original entry on oeis.org

8, 4, 0, 4, 4, 2, 2, 2, 4, 4, 2, 2, 4, 14, 4, 2, 18, 6, 2, 12, 16, 2, 4, 2, 42, 6, 4, 2, 22, 26, 12, 18, 18, 38, 12, 14, 2, 6, 36, 2, 16, 24, 6, 14, 12, 6, 28, 24, 24, 8, 16, 32, 16, 28, 12, 8, 16, 6, 16, 98

Offset: 0

Jonathan Vos Post, Feb 05 2006

A098147 is difference between first odd semiprime > 10^n and 10^n.

			a(0) = 8 because 5^0 + 8 = 9 = 3^2 is an odd semiprime; note that because 5^0 + 3 = 4 = 2^2 is an even semiprime, but we only care about odd semiprimes here.
a(1) = 4 because 5^1 + 4 = 9 = 3^2 is an odd semiprime.
a(2) = 0 because 5^2 + 0 = 25 = 5^2 is an odd semiprime; there are no more zero values.
a(3) = 4 because 5^3 + 4 = 129 = 3 * 43.
a(4) = 4 because 5^4 + 4 = 629 = 17 * 37.
a(5) = 2 because 5^5 + 2 = 3127 = 53 * 59.
a(6) = 2 because 5^6 + 2 = 15627 = 3 * 5209.
a(7) = 2 because 5^7 + 2 = 78127 = 7 * 11161.
a(8) = 4 because 5^8 + 4 = 390629 = 577 * 677 (brilliant).
a(9) = 4 because 5^9 + 4 = 1953129 = 3 * 651043.

Cf. A000351, A001358, A098147.

dfpsn[n_]:=Module[{n5=5^n,s},s=If[OddQ[n5],n5,n5+1];While[PrimeOmega[s] != 2,s=s+2];s-n5]; Array[dfpsn,60,0] (* Harvey P. Dale, Sep 04 2013 *)

a(n) = minimum integer k such that 5^n + k is an element of A046315. a(n) = minimum integer k such that A000351(n) + k is an element of A046315.

Corrected and extended by Harvey P. Dale, Sep 04 2013

A114314 Difference between the first odd semiprime > 7^n and 7^n.

Original entry on oeis.org

8, 2, 2, 12, 6, 10, 2, 6, 12, 2, 4, 12, 2, 10, 2, 4, 10, 6, 4, 4, 12, 24, 10, 10, 10, 4, 2, 30, 10, 4, 4, 30, 12, 2, 10, 4, 10, 10, 54, 10, 12, 22, 10, 24, 2, 4, 88, 16, 16, 22, 30, 6, 60, 12, 2, 8, 22, 30, 10, 10, 88, 6, 34, 6, 22, 22, 94, 40, 6, 30, 4, 100, 40, 16, 40, 4, 36, 20, 4, 16, 18

Offset: 0

Jonathan Vos Post, Feb 05 2006

A098147 is the difference between the first odd semiprime > 10^n and 10^n.

			a(0) = 8 because 7^0 + 8 = 9 = 3^2 is an odd semiprime.
a(1) = 2 because 7^1 + 2 = 9 = 3^2 is an odd semiprime.
a(2) = 2 because 7^2 + 2 = 51 = 3*17 is an odd semiprime.
a(3) = 12 because 7^3 + 12 = 355 = 5 * 71.
a(4) = 6 because 7^4 + 6 = 2407 = 29 * 83 (brilliant).
a(5) = 10 because 7^5 + 10 = 16817 = 67 * 251.

Cf. A000351, A001358, A098147.

a(n) = minimum integer k such that 7^n + k is an element of A046315. a(n) = minimum integer k such that A000420(n) + k is an element of A046315.

Corrected and extended by Don Reble, Oct 16 2017

A114321 Difference between first odd semiprime > 11^n and 11^n.

Original entry on oeis.org

8, 4, 2, 2, 6, 14, 4, 6, 6, 6, 4, 6, 32, 14, 10, 2, 12, 2, 2, 12, 36, 6, 28, 6, 12, 32, 8, 12, 28, 24, 48

Offset: 0

Jonathan Vos Post, Feb 05 2006

A098147 is difference between first odd semiprime > 10^n and 10^n.

			a(0) = 8 because 11^0 + 8 = 9 = 3^2 is an odd semiprime.
a(1) = 4 because 11^1 + 4 = 15 = 3 * 5 is an odd semiprime.
a(2) = 2 because 11^2 + 2 = 123 = 3 * 41.
a(3) = 2 because 11^3 + 2 = 1333 = 31 * 43 (brilliant).
a(4) = 6 because 11^4 + 6 = 14647 = 97 * 151.
a(5) = 14 because 11^5 + 14 = 161065 = 5 * 32213.
a(6) = 4 because 11^6 + 4 = 1771565 = 5 * 354313.
a(7) = 6 because 11^7 + 6 = 19487177 = 41 * 475297.
a(8) = 6 because 11^8 + 6 = 214358887 = 317 * 676211.
a(9) = 6 because 11^9 + 6 = 2357947697 = 7 * 336849671.
a(10) = 4 because 11^10 + 4 = 25937424605 = 5 * 5187484921.
a(30) = 48 because 11^30 + 48 = 17449402268886407318558803753849 = 7 * 2492771752698058188365543393407.

Cf. A001020, A001358, A098147.

a(n) = minimum integer k such that 11^n + k is an element of A046315. a(n) = minimum integer k such that A001020(n) + k is an element of A046315.

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A098146 First odd semiprime > 10^n.

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A114137 Difference between first odd semiprime > 2^n and 2^n.

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A114141 Difference between first odd semiprime > 3^n and 3^n.

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A114187 Difference between first semiprime >= n! and n!. Least k such that n!+k is semiprime.

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A114313 Difference between first odd semiprime >= 5^n and 5^n.

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A114314 Difference between the first odd semiprime > 7^n and 7^n.

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A114321 Difference between first odd semiprime > 11^n and 11^n.

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