A180330 -id:A180330 - cp's OEIS frontend

A180331 The larger amicable number corresponding to A180330(n).

2924, 10856, 66992, 2090656, 7684672, 1762592896, 750555392, 7076729344, 25941935104, 161905117184, 3949454626816, 56692308189184, 162225706778624, 5469803815141376, 21548132777132032, 48336776233877504, 2961919695995011072, 5591729105229316096

Offset: 2

T. D. Noe, Sep 07 2010

nonn

These numbers have the form 2^n * p * q, where p and q are distinct odd primes.

Sergei Chernykh, Amicable numbers list.

Cf. A002046, A063990, A180330.

a(18)-a(19) from Chernykh's database added by Amiram Eldar, Jul 26 2025

A180650 Irregular triangle in which row n consists of the smaller of pairs of amicable numbers of the form 2^n pq, with p and q odd primes.

Original entry on oeis.org

2620, 5020, 10744, 66928, 171856, 176272, 437456, 503056, 1392368, 2802416, 2082464, 15002464, 17908064, 7677248, 56055872, 1237888448, 1750776704, 16873473664, 749380864, 902335744, 4377991936, 74006870272, 243815958784

Offset: 2

T. D. Noe, Sep 25 2010

The length of row n is A180335(n). The first number in row n is A180330(n).

T. D. Noe, Rows n=2..24, flattened

Cf. A180651

A180329 Odd semiprimes pq such that there is another odd semiprime rs with (p+1)(q+1)=(r+1)(s+1) and p, q, r, and s distinct primes.

Original entry on oeis.org

33, 35, 51, 55, 69, 77, 87, 95, 115, 119, 123, 141, 143, 155, 159, 161, 177, 187, 203, 205, 209, 213, 221, 235, 249, 253, 267, 287, 295, 299, 303, 319, 321, 323, 329, 335, 339, 341, 355, 371, 391, 393, 395, 407, 413, 415, 437, 445, 447, 451, 473, 485, 493

Offset: 1

T. D. Noe, Sep 07 2010

nonn

These numbers are related to amicable pairs of the form (G * pq, G * rs), where G is coprime to pq and rs. The interesting case of G=2^n is shown in A180330.

			For pq = 33 = 3*11, the corresponding rs is 35 because (3+1)(11+1) = 48 = (5+1)(7+1).

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A180328 (all semiprimes with this property)

nn=1000; sp=Select[Range[3,4*nn/3,2], Last/@FactorInteger[ # ]=={1, 1}&]; prods=Table[Times@@(1+First/@FactorInteger[n]), {n,sp}]; dups=Select[Tally[prods], #[[2]]>1&]; goodProds=Sort[Transpose[dups][[1]]]; pos=Select[Range[Length[sp]], sp[[ # ]]<=nn && MemberQ[goodProds, prods[[ # ]]]&]; sp[[pos]]

A180335 Number of amicable pairs of the form (2^n * pq, 2^n * rs), where p, q, r, and s are distinct odd primes.

Original entry on oeis.org

0, 2, 1, 7, 3, 3, 2, 7, 1, 8, 4, 8, 7, 5, 6, 15, 4, 9, 9, 14, 8, 15, 8, 15

Offset: 1

T. D. Noe, Sep 25 2010

There are only a finite number of such pairs for each n. A180330(n) is the smallest value of 2^n * pq which is part of an amicable pair. The terms in this sequence equal the count of such amicable pairs in Pedersen's table up to n=22, after which the table stops being a complete list of all amicable pairs of the form 2^n * pq. Currently, Pedersen's table lists 4 such amicable pairs for n=23 when there are actually 8 pairs.

			Eight pairs for n=23, with new ones indicated by *:
(112072375885373577887744, 112072384861110016147456) *
(1268674663237438821892096, 1268674685089131637243904)
(207428765646084356836425728, 207428778650058836339064832) *
(3201546606177940316018966528, 3201546948253280471426793472)
(3246126368411500102994100224, 3246126753752370852634034176)
(30886958374694867104144818176, 30886959839775761339382759424) *
(58230121837690366602703273984, 58230128772645945195748130816)
(1564092755993481057609717907456, 1564092823984266648474085228544) *

J. M. Pedersen, Regular type (2,2) amicable pairs

Cf. A180650, A180651 (the amicable pairs for each n)

cp's OEIS Frontend

A180331 The larger amicable number corresponding to A180330(n).

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A180650 Irregular triangle in which row n consists of the smaller of pairs of amicable numbers of the form 2^n pq, with p and q odd primes.

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A180329 Odd semiprimes pq such that there is another odd semiprime rs with (p+1)(q+1)=(r+1)(s+1) and p, q, r, and s distinct primes.

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A180335 Number of amicable pairs of the form (2^n * pq, 2^n * rs), where p, q, r, and s are distinct odd primes.

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